Formulas and Chapter Summaries Physics 123 Ross L. Spencer
Transcript of Formulas and Chapter Summaries Physics 123 Ross L. Spencer
Formulas and Chapter Summaries
Physics 123
Ross L Spencer Harold T Stokes
With additions from the textbook by Allred
1
Serway Chapter 14 Fluid Mechanics
Density and Pressure
The density of a material is defined to be the ratio between the mass of a small piece of it divided
by the volume of the small piece
m
V
Water has a density of 1000 kgm3 while air has a density of 13 kgm3 at sea level and 11
kgm3 in Provo Sea water has a density of 103 of water Ethanol (ethyl alcohol) is 0806
The pressure exerted on a small surface is defined to be the force applied to it divided by its
area
FP
A
At sea level atmospheric pressure is 1013 times 105 pascals where 1 Pa = 1 Nm2 Here in Provo
atmospheric pressure is about 86 times 104 Pa Sometimes pressure will be expressed in pounds-per-
square-inch (psi) or sometimes in torr 1 psi= 6895 times 103 Pa and 1 torr= 1333 Pa
Hydrostatics
If h measures distance below the surface of a liquid then the pressure as a function of the depth h
is given by
oP P gh
where Po is the pressure at the surface of the liquid usually atmospheric pressure In 123 liquids
are imcompressible ρ does not vary with depth and g = 980 m2s
Pascalrsquos Principle
A change in the pressure applied to an enclosed fluid is transmitted undiminished to every
portion of the fluid and to the walls of the containing vessel (Blaise Pascal 1652)
Archimedes Principle
A body wholly or partially immersed in a fluid will be buoyed up by a force equal to the weight
of the fluid that it displaces (Archimedes ca 220 BC) This principle is a handy computational
tool but it does not explain the real reason that objects in a fluid experience an upward force
The reason is that the pressure on the bottom of the object is greater than the pressure at the top
so there is a net upward pressure force The buoyant force is given by the formula
B Vg
2
where ρ is the density of the fluid V is the submerged volume of the object and g = 980 ms2 is
the acceleration of gravity If the object is floating on the liquid then the submerged volume V is
only that portion which is below the liquid surface
Equation of Continuity
For a fluid whose density is constant the rate of fluid flow or volume flux Φ measured in units
of m3s for a cross-section of fluid streamlines having area A and flow speed v is given by
Av
Bernoullirsquos Equation
The principle of conservation of energy for steady fluid flow states Along each streamline of the
flow the following quantity has a constant value
21
2P v gy
where y is measured upward against gravity Note that y in this equation and h in the hydrostatic
pressure equation have opposite signs This law works for gases and liquids The demos are
mostly for gases and are set up so that ρ does not vary
3
Serway Chapter 16
Longitudinal and Transverse Waves
A longitudinal wave is one in which the direction of vibration of the medium and the direction of
propagation of the wave are in the same direction Sound waves in the air in water and in solids
are longitudinal waves
A transverse wave is one in which the direction of vibration of the medium is perpendicular to
the direction of propagation of the wave Ocean waves waves on ropes and electromagnetic
waves are transverse waves
Wavelength and Wavenumber
The wavelength λ is the period in space of the wave ie it is the distance from one wave crest to
the next The wavenumber k (units reciprocal length) is related to the wavelength by the formula
2k
Frequency and Angular Frequency
The period T is the period in time of the wave ie it is the time between the arrival of one wave
crest and the next at some point in space The frequency f is the number of crests that arrive per
second which is just 1T and the angular frequency ω is 2π times the frequency Both have units
of per second (reciprocal time)
1 22f f
T T
Wave Speed
The wave speed of a wave is defined to be the the speed at which the wave crests or troughs
move through the medium This speed is related to the frequency and the wavelength by the
formulas
v fk
where k is the wavenumber of the wave v is also called the phase velocity
Traveling Waves
Traveling waves are oscillations in which the wave crests and troughs glide smoothly through the
medium Waves on oceans and lakes are traveling waves The mathematical form of a harmonic
(sinusoidal) traveling wave is
4
siny x t A kx t
where A is the amplitude of the wave As usual ω and k are related by the relation ω = kv where
v is the wave speed the speed at which the crests travel the minus sign means that the wave is
traveling to the right towards higher values of x φ is the phase and constrols where the maxima
and minima occur
Speed of Transverse Waves on a Rope
The velocity of transverse waves on a perfectly flexible rope is given by
Tv
where T is the tension in the rope and where μ is its linear mass density (mass per unit length)
Common Test Questions
A common question is to be given an equation for a wave and to be asked what is amplitude
velocity etc Another common type of question is to be given facts about a wave like its
direction velocity and magnitude at a given place and time and to be asked which of several
equations correctly describe the wave
5
Serway Chapter 17
Sound Speed in Solids Liquids and Gases
The speed of sound in a liquid or solid of bulk modulus B and volume mass density ρ
Bv
The speed of sound in air is
331 1 273v m s T C
The speed of sound in air at room temperature is about 343 ms Notice it doesnrsquot depend on
pressure or frequency
Power Intensity and Loudness
The intensity of a sound wave is defined to be the power per unit area ie wattsm2 in the wave
For spherical waves traveling away from a small source emitting waves with average power Рav
the intensity falls off with distance from the source r according to the inverse-square law
av24
Ir
Hence the intensity I2 at distance r2 is related to the intensity I1 at a different distance r1 by
21 2
22 1
I r
I r
Experiments on human hearing have shown that we hear intensity differences logarithmically so
the decibel loudness scale for sound intensity was invented The loudness β of a sound in
decibels is related to its intensity I in Wm2 by the formula
1010logo
I
I
Where Io is a sound intensity near the threshold of hearing defined to be Io = 10-12 Wm2 Note
that on this scale a sound is made 10 times more intense by adding 10 decibels Remember that
intensity is proportional to velocity squared Amplitude is the A is the previous chapterrsquos
equation for a traveling wave If you make the A 10 times as much the intensity will increase by
100 times (20 dB)
Doppler Effect
If sound waves are traveling through a medium and if either the receiver of the waves or the
source of waves is moving then the frequency received is related to the frequency emitted by
6
0
s
v vf f
v v
where fprime is the frequency detected by the observer f is the frequency emitted by the source v is
the speed of the waves vo is the speed of the observer and vs is the speed of the source This
formula assumes that the source and receiver are either moving directly toward each other or
directly away from each other To know which signs to use remember that when observer and
source approach each other the observed frequency is higher while if they move away from each
other it is lower Just examine the signs in the formula and make the answer come out right
For electromagnetic waves (light radio waves X-rays) traveling in vacuum Einsteinrsquos theory of
relativity (and careful experiments) show that the Doppler shift is given by
r
r
c vf f
c v
where vr is the relative speed between the source and the observer
For all kinds of waves (sound light etc) if the relative speed of the source and the observer is
small compared to the speed of the waves then there is a simple approximation to the Doppler
effect For example if the relative speed is 1 of the wave speed then the frequency shifts by
1 Remember that this is only an approximation
Shock Waves
When an object moves through a medium at a speed greater than the speed of waves a V-shaped
shock wave is produced The V-shaped wake behind a speeding boat is a good example of this
effect and the cone of sonic-boom behind a supersonic aircraft is another The angle that V-line
makes with the direction of travel of the source is given by
sins
v
v
where v is the wave speed and vs is the source speed
7
Serway Chapter 18
Principle of Linear Superposition
We say that a system obeys the principle of linear superposition if two or more different motions
of the system can simply be added together to find the net motion of the system Light waves
obey this principle as they propagate through the air as can be seen by shining two flashlights so
that their beams cross The beams propagate along without affecting each other (Light sabers are
a spectacular but unfortunately fictional example of systems that do not obey the principle of
superposition) Wave pulses on an ideal rope also obey this principle two different pulses pass
through each other without change Standing waves are an example of this effect being simply
the linear superposition of two traveling waves of the same frequency but moving in opposite
directions Light waves in matter do not always obey this principle For instance two powerful
laser beams could be made to cross in a piece of glass in such a way that their combined heating
effect in the crossing region could melt the glass and scatter the beams in complicated ways This
is an example of a nonlinear effect
Interference
When two or more waves are present in the same medium at the same time their net effect may
often be obtained simply by adding them at each point in the medium according to the principle
of linear superposition (Note this wonrsquot work if the medium is nonlinear) When this addition
makes the total amplitude be greater than the individual amplitudes of the various waves we say
that the interference is constructive When the addition produces cancellation and an amplitude
less than the amplitudes of the separate waves we have destructive interference
Standing Waves
A standing wave is the superposition of two identical traveling waves moving in opposite
directions Nodes are places where the two waves perfectly destructively interfere to produce
zero amplitude at all times Anti-nodes are places where the two waves perfectly constructively
interfere to produce an amplitude maximum The distance between nodes is λ2 Standing waves
on a string fixed at both ends have nodes at each end of the string Standing waves in an air
column enclosed in a tube have displacement anti-nodes at open ends of the tube and
displacement nodes at closed ends The frequency of the standing wave with the lowest possible
frequency is called the fundamental frequency Standing waves on strings or in air columns all
have frequencies which are integer multiples of the fundamental frequency and are called
8
harmonics (The fundamental is called the ldquofirst harmonicrdquo)
Beats
Beats are heard when two waves with slightly different frequencies f1 and f2 are combined The
waves constructively interfere for a number of cycles then destructively interfere for a number
of cycles We hear a periodic ldquowah-wahrdquo frequency equal to the difference of the two wave
frequencies
1 2bf f f
Musical Instruments
Musical instruments produce tones by exciting standing waves on strings (violins piano) and in
tubes (trumpet organ) The fundamental frequency of the standing wave is called the pitch of the
tone The pitch of concert A is 440 Hz by definition Two tones are an octave apart if one pitch
has twice the frequency of the other In written music there are 12 intervals in each octave with
the ratio between successive intervals equal to 2112 = 105946 The ratios for each tone in an
octave starting at A and ending at the next higher A are
A A B C C D D E F F G G A
1 10595 11225 11892 12599 13348 14142 14983 15874 16818 17818 18877 2
A musical tone is actually a superposition of the fundamental frequency and the higher
harmonics The tone quality of a musical instrument is determined by the amplitudes of the
various harmonics that it produces A violin and a trumpet can play the same pitch but they
donrsquot sound at all alike to our ears The difference between them is in the various amplitudes of
their harmonics
9
Serway Chapter 19
Temperature
Formally temperature is what is measured by a thermometer Roughly high temperature is what
we call hot and low temperature is what we call cold On the atomic level temperature refers to
the kinetic energy of the molecules A collection of molecules is called ldquohotrdquo if the molecules
have rapid random motion while a collection of molecules is called ldquocoldrdquo if the random motion
is slow When two bodies are placed in close contact with each other they exchange molecular
kinetic energy until they come to the same temperature This is the microscopic picture of the
Zeroth Law of Thermodynamics
Absolute Zero
Absolute zero is the lowest possible temperature that any object can have This is the temperature
at which all of the energy than can be removed an object has been removed (This removable
energy we call thermal energy) There is still motion at absolute zero Electrons continue to orbit
around atomic nuclei and even atoms continue to move about with a small amount of kinetic
energy but this small energy cannot be removed from the object For example at absolute zero
helium is a liquid whose atoms still move and slide past each other
Temperature Scales
Kelvin Scale Absolute zero is at T = 0 K water freezes at T = 27315 K room temperature is
around T = 295 K and water boils at T = 373 K Note we donrsquot use a deg symbol Kelvin is
prefered SI Unit
Celsius Scale Absolute zero is at T = -273degC water freezes at T = 0degC room temperature is
around T = 22degC and water boils at T = 100degC
Fahrenheit Scale Absolute zero is at T = -459degF water freezes at TF = 32degF room temperature
is around T = 72degF and water boils at TF = 212degF TF =18TC + 32 Notice that temperature
differences are the same for the Kelvin and Celsius scales
Thermal Expansion
When materials are heated they usually expand and when they are cooled they usually contract
(Water near freezing is a spectacular counterexample it works the other way around) The
coefficient of linear expansion is defined by the relation
1
i
L
L T
10
where Li is the initial length of a rod of the material and ΔL is the change in its length due to a
small temperature change ΔT The coefficient of volume expansion is defined similarly
1
i
V
V T
where Vi is the initial volume of a piece of material and where ΔV is the change in its volume due
to a small temperature change ΔT
Avogadrorsquos Number (N or NA)
One mole of any substance corresponds to 6022 times 1023 molecules The atomic mass given on the
periodic table (p A32-33 in the text) is the mass in grams of one mole of atoms For example
the atomic mass of oxygen is 15999 gmol The mass of a single atom is given by the same
number in units of atomic mass units (u) For example the mass of an oxygen atom is 15999 u
(on the average)
Ideal Gas Law (an example of an equation of state)
When the molecules of a gas are sufficiently inert and widely separated that interactions between
them are negligible we say that it is an ideal gas The pressure P volume V and temperature T
(in kelvins) of such a gas are State Variables and are related by the ideal gas law
Bor PV=NkPV nRT T
where n is the number of moles of the gas where R is the gas constant
8314 Jmol KR
where N is the number of molecules and where kB is Boltzmannrsquos constant
231380 10 J KBk
It works well for air at atmosphere pressure and even better for partial vaccuums The relative
ease of measuring pressure and the linear relationship between pressure and temperature (if V
and n are held fixed) makes an ideal gas an ideal thermometer Thermometers can be based on
properties of solids or liquids but the behavior of these materials with temperature is more
complicated
11
Serway Chapter 20
Heat
Heat is energy that flows between a system and its environment because of a tempera-
ture difference between them The units of heat are Joules as expected for an energy
Unfortunately there are several competing units of energy They are related by
1 cal 4186 J 1 Cal 4186 J 1 Btu 1054 J
Heat Capacity
There is often a simple linear relation between the heat that flows in or out of part of a
system and the temperature change that results from this energy transfer When this
linear relation holds it is convenient to define the heat capacity C and the specific
heat c as follows
For the entire object the heat Q it absorbs is given by Q = C(Tf - Ti)
For a particular material the specific heat is defined by c = Cm which is the heat
capacity per unit mass so that
Q = mc(Tf ndash Ti)
Note C has units of energy (J or Cal)(Kelvin kg)
Heats of Transformation or Latent Heat Q = plusmn mL
When a substance changes phase from solid to liquid or from liquid to gas it absorbs
heat without a change in temperature The latent heat or heat of transformation is
usually given per unit mass of the substance For example for water the heat of fusion
(melting) is L = 333 kJkg while the heat of vaporization (boiling) is L = 2260 kJkg
Note that heat for boiling is considerably bigger than melting for water You have to
be careful with signs heat is given off (negative) if you go down in temperature and
condense steam
Work
In general the small amount of work done on a system as a force Fon is exerted on it
through a vector displacement dx is given by
on xdW d F
12
But if the displacement is done very slowly (as we always assume in thermodynamics)
then the force exerted on the system and the force exerted by the system are in
balance so the force exerted by the system is ndash Fon In thermodynamics it is more
convenient to talk about the force exerted by the system so we change the above
formula for the work done on the system to
xdW d F
where F is the force exerted by the system This has confused students for more than a
century now but this is the way your book and many other books do it so you are
stuck You will need to memorize the minus sign in this definition of the work to be
able to use your textbook
There are many chances to get signs wrong in this and the next two chapters (Mosiah
2 )
When an external agent changes the volume of a gas at pressure P by a small amount
dV the (small amount) of work done on the system is given by
dW PdV
Notice that this minus sign is just what we need to make dW be positive if the external
agent compresses the gas for then dV is negative If on the other hand the external
agent gives way allowing the gas to expand against it then dV is positive and we say
that the work done on the gas is negative
The work done on the system (eg by the gas in a cylinder) in a thermodynamic
process is the area under the curve in a PV diagram It is positive for compressions
and negative for expansions If the volume of gas remains constant in a process then
no work is done by the gas
Cyclic processes are important For cyclic processes represented by PV diagrams the
magnitude of the net work during one cycle is simply the area enclosed by the cycle
on the diagram Be careful to keep track of signs when you are calculating that
enclosed area In cyclic process Q = ndashW for a cycle Put a PV diagram in your notes
1
Path A-B B-C C-D D-A A to A net
Q
W
ΔU
ΔS
Internal Energy
The energy stored in a substance is called its internal energy Eint This energy may be
stored as random kinetic energy or as potential energy in each molecule (stretched
chemical bonds electrons in excited states etc) For ideal gases all states with the
same temperature will have the same Eint
First Law of Thermodynamics
The change ΔEint in the internal energy of a system is given by
intE Q W
where Q is the heat absorbed by the system and where W is the work done on the
system Hence if a system absorbs heat (and if Wge0) the internal energy increases
Likewise if the system does work (W on the system is negative) and if Qge0 the
internal energy decreases Potential Pitfall Many times people talk about work done
by the system It is the minus of W on the system Donrsquot get tripped up
Processes
Adiabatic process No heat is exchanged Q = 0 requiring that the system be insulated
from the environment A process may be approximately adiabatic if it happens so
rapidly that heat does not have time to enter or leave the system Work + or ndash is done
and ΔEint = W
Adiabatic free expansion A gas is allowed to expand into a vacuum without pushing
on anythingndashit just rushes into the vacuum In this process Q = W = 0 so the internal
energy and temperature does not change (Note the difference between an adiabatic
process and a free expansion is that NO work is done in the adiabatic free expansion)
Isobaric process The pressure is held fixed ΔP = 0 For example usually the
pressure increases when a gas is heated but if it were allowed to expand during the
2
heating process in just the right way its pressure could remain fixed In isobaric
processes both Q and W are non-zero (Work is eacy = ndashP(Vf ndash Vi)
Isovolumetric process The volume is held fixed ΔV = 0 The work done by the gas is
then zero and so we have ΔEint = Q
Isothermal process The temperature is held fixed ΔT = 0 There is no change in Eint
so Q = ndashW for isothermal processes For an ideal gas PV = nRT so PV = constant in
an isothermal process The work done on the gas is then given by
lnf
i
Vi
Vf
VW PdV nRT
V
Heat Conduction
The quantity P is defined to be the rate at which heat flows through an object and is a
power having units of watts It is analogous to electric current which is the rate at
which charge flows through an object If the flow of heat through a slab of length L
and cross-sectional area A is steady in time then P is given by the equation
h cT TdQkA
dt L
P =
where Th and Tc are the (hot and cold) temperatures of the two ends of the slab The
heat flows of course because of this temperature difference The quantity k is called
the thermal conductivity and is a constant that is characteristic of the material It is
analogous to the electrical conductivity h cT TL is sometimes called the temperature
gradient and is written dTd dTdx
R-Values
It is common to have the heat-conducting properties of materials described by their R-
values especially for insulating materials like fiberglass batting The connection
between k and R is R = Lk where L is the material thickness In this country R-values
always have units of 2ft F hourBtu
Polyurethane foam 6 (1 in thick) Air space 1 (35 in thick) Fiber glass batting 11
(35 in thick) Wood about 1 (1 in thick) Glass 09 (0125 in thick)
The heat flow rate through a slab of area A is given by
3
h cT TA
R
P
in units of Btuhour Note that A must be in square feet and the temperatures must be
in degrees Fahrenheit
Convection
Convection is the transfer of thermal energy by flow of material For instance a home
furnace doesnrsquot heat a house by waiting for the heat from the burner to slowly conduct
throughout the house instead it quickly pumps warm air to all of the rooms
Generally convection is a much faster way to transfer heat than conduction
Radiation
Electromagnetic radiation can also transfer heat When you warm yourself near a
campfire which has burned itself down into a bed of glowing embers you are
receiving radiant heat from the infrared portion of the electromagnetic spectrum The
rate at which an object emits radiant heat is given by Stefanrsquos law
4AeTP
where P is the radiated power in watts σ is a constant
8 2 45696 10 W m K
A is the surface area of the object in m2 and T is the temperature in kelvins The
constant e is called the emissivity and it varies from substance to substance A perfect
absorber (think black velvet) has e = 1 while a perfect reflector (think mirror) has e =
0 Hence black objects radiate very well while shiny ones do not Also an object that
is hotter than its surroundings radiates more energy than it absorbs whereas an object
that is cooler than its surroundings absorbs more energy than it radiates
Terminology
Transfer variables vs state variables
Energy transfer by heat as well as work done depends on the initial final and
intermediate states of the system They are transfer variables But their sum (Q + W =
4
Eint) is a state variable
Figure 205
5
Serway Chapter 21
Kinetic Theory
The ideal gas law works for all atoms and molecules at low pressure It is rather
amazing that it does Kinetic theory explains why The properties of an ideal gas can
be understood by thinking of it as N rapidly moving particles of mass m As these
particles collide with the container walls momentum is imparted to the walls which
we call the force of gas pressure In this picture the pressure is related to the average
of the square of the particle velocity 2v by
22 1( )
3 2
NP mv
V
Using the ideal gas law we obtain the average translational kinetic energy per
molecule
21 3
2 2 Bmv k T
The rms speed is then given by
2rms
3 3Bk T RTv v
m M
where M is the molecular mass in kgmol
Degrees of Freedom
Roughly speaking a degree of freedom is a way in which a molecule can store energy
For instance since there are three different directions in space along which a molecule
can move there are three degrees of freedom for the translational kinetic energy
There are also three different axes of rotation about which a polyatomic molecule can
spin so we say there are three degrees of freedom for the rotational kinetic energy
There are even degrees of freedom associated with the various ways in which a
molecule can vibrate and with the different energy levels in which the electrons of
the molecule can exist
Internal Energy and Degrees of Freedom The internal energy of an ideal gas made
up of molecules with J degrees of freedom is given by
int 2 2 B
J JE nRT Nk T
6
Heat Capacities of an Ideal Gas The heat capacity of a gas is described by means of
molar heat capacities CV and CP These are the heat capacities per mole and the
subscript V on CV means that the volume is being held constant while for CP the
pressure is held constant For example to raise the temperature of n moles of a gas
whose pressure is held constant by 10 K we would have to supply an amount of heat Q
= nCP (10) K
Molar Specific Heat of an Ideal Gas at Constant Volume
VQ nC T
3monatomic
2VC R
5diatomic
2VC R
5polyatomic
2VC
Real gases deviate from these formulas because in addition to the translational and ro-
tational degrees of freedom they also have vibrational and electronic degrees of
freedom These are unimportant at low temperatures due to quantum mechanical
effects but become increasingly important at higher temperatures The rough rule is
No of degrees of freedom
2VC R
Molar Specific Heat of an Ideal Gas at Constant Pressure
PQ nC T
P VC C R
The internal energy of an ideal gas depends only on the temperature
int VE nC T
Adiabatic Processes in an Ideal Gas
7
An adiabatic process is one in which no heat is exchanged between the system and the
environment When an ideal gas expands or contracts adiabatically not only does its
pressure change as expected from the ideal gas law but its temperature changes as
well Under these conditions the final pressure Pf can be computed from the initial
pressure Pi and from the final and initial volumes Vf and Vi by
or constantf f i iP V PV PV
where γ = CPCV The quantity γ is called the adiabatic exponent Note that this
doesnrsquot mean that the ideal gas law no longer holds it does and in fact it can be
combined with the adiabatic law for pressure given above to obtain the adiabatic law
for temperatures
1 constantTV
Compressions in sound waves are adiabatic because they happen too rapidly for any
appreciable amount of heat to flow This is why the adiabatic exponent appears in the
formula for the speed of sound in an ideal gas
RTv
M
Note that v depends only on T and not on P Because it depends only on the
temperature the speed of sound is the same in Provo as at sea level in spite of the
lower pressure here due to the difference in elevation
Equipartition of Energy
Every kind of molecule has a certain number of degrees of freedom which are
independent ways in which it can store energy Each such degree of freedom has
associated with it ndash on average ndash an energy of 12 Bk T per molecule (or 1
2 RT per mole)
(Note since a molecule has so many possible degrees of freedom it would seem that
there should be a lot of 12 sBk T to spread around But because energy is quantized
some of these degrees of freedom are not ldquoactiverdquo until the temperature becomes high
enough that 12 Bk T is as big as the lowest quantum of energy
8
Serway Chapter 22
Second Law of Thermodynamics
There are several equivalent forms of this important law
Kelvin It is not possible to change heat completely into work with no other change
taking place Or in other words there are no perfect heat engines
Clausius It is not possible for heat to flow from one body to another body at a higher
temperature with no other change taking place Or in other words there are no
perfect refrigerators
Entropy In any thermodynamic process that proceeds from one equilibrium state to
another the entropy of the system + environment either remains unchanged or
increases The total entropy never decreases This law is a bit of an oddity among the
laws of physics because it is not absolute Things are forbidden by the second law not
because it is impossible for them to happen but because it is extremely unlikely for
them to happen (See below for more information about entropy)
Reversible and Irreversible Processes
A reversible process is one which occurs so slowly that it is in thermal equilibrium (or
very nearly so) at all times A hallmark of such processes is that a motion picture of
them looks perfectly normal whether run forward or backward Imagine for instance
the slow expansion of a gas at constant temperature in a cylinder whose volume is being
increased by a slowly moving piston Run the movie backwards and what do you see
You see the slow compression of a gas at constant temperature which looks perfectly
normal
An irreversible process is one which occurs in such a way that thermal equilibrium is
not maintained throughout the process The mark of this kind of process is that a motion
picture of it looks very odd when run backward Imagine the sudden expansion of a gas
into a previously evacuated chamber because a hole was punched in the wall between a
pressurized chamber and the evacuated one Run the movie backward and what do you
see You see the gas in the soon-to-be-evacuated chamber gather itself together and
stream through a tiny hole into a chamber in which there is already plenty of gas If you
have ever seen this happen get in touch with the support group for those who have
witnessed the spontaneous combustion of large mammals by calling 1-800-PYROCOW
9
Heat Engines
Because of the vexing difference in sign between work done on and system and work done
by a system we will invent a new work variable Weng Heat engines do work and so the net
W for these engines is negative But in engineering applications hidden minus signs are
regarded as evil so for heat engines we donrsquot talk about W instead we talk about its
magnitude engW W So for heat engines the first law is
int engE Q W
But for heat pumps and refrigerators work is done on the system so we use the usual work
W when we talk about these systems
A heat engine is a machine that absorbs heat converts part of it to work and exhausts the
rest The heat must be absorbed at high temperature and exhausted at low temperature If the
absorbed heat is Qh the exhausted heat is Qc and the work done by the engine is Weng then
eng h cW Q Q
and the efficiency of the engine is defined to be
eng
h
We
Q
A perfect engine would convert the heat hQ completely into work Weng giving an effi-
ciency of e = 1 Energy conservation alone allows a perfect engine but the second law
requires e lt 1
Refrigerators and Heat Pumps
A refrigerator is a machine that absorbs heat at low temperature and exhausts it at high
temperature the ldquobackwardsrdquo heat transfer being driven by the work done on the machine
by some source of power A heat pump is a machine that either works like a refrigerator
keeping a place cold by transferring heat from this cold place to a higher temperature
environment (cooling mode like an air conditioner) or it functions as a heater
transferring heat into a warm place from a cooler one (heating mode like a window unit
that heats a house by extracting thermal energy from the cold outdoors) The coefficient
of performance of a refrigerator or of a heat pump in cooling mode is defined to be
COP cooling modecQ
W
10
For a heat pump in heating mode the coefficient of performance is
COP heating modehQ
W
Note that we donrsquot have to use the engineering work here because in these systems
work is done on the system and W is naturally positive
A perfect refrigerator would take in heat Qc and exhaust the same amount of heat without
doing any work giving an infinite coefficient of performance Energy conservation alone
allows a perfect refrigerator but the second law requires COP lt infin A perfect heat pump
in heating mode would transfer Qh into the house without doing any work and so would
also have an infinite coefficient of performance The second law forbids this too A good
coefficient of performance for a real device would be around 5 or 6
Carnot Cycle
The most efficient of all possible engines is one that uses the Carnot cycle This cycle
employs an ideal gas has no friction and operates very slowly so that the gas can be in
thermal equilibrium at all parts of the cycle This means of course that it canrsquot
possibly be built and even if it could be built it would not run fast enough to be useful
Nevertheless this cycle is very important because it gives an upper bound on the
efficiency of real engines There cannot possibly be an engine that is more efficient
than one based on the Carnot cycle This cycle consists of the following four steps
1 The ideal gas absorbs heat Qh at constant temperature Th while the gas increases its
volume The reason that heat is absorbed is that expansion tends to cool the gas but
thermal contact with the environment at Th keeps the temperature high by heat
conduction into the ideal gas
2 The ideal gas further increases its volume by an adiabatic expansion This expansion
causes the gas to cool so at the end of this part of the cycle the gas is at temperature Tc
3 The gas exhausts heat Qc at constant temperature Tc while the gas decreases in
volume The reason that heat is exhausted is that compression tends to heat the gas
but thermal contact with the environment at Tc keeps the temperature low by heat
conduction out of the ideal gas
4 The gas is adiabatically compressed back to its original volume (the volume it started
with in step 1) This compression heats the gas from Tc up to Th
11
The efficiency of a Carnot engine is given by the very simple formula
1 cC
h
Te
T
where the temperatures must in be Kelvin No real engine can be more efficient than
this
The coefficient of performance of a Carnot refrigerator or heat pump in cooling mode
is given by
COP cooling mode cC
h c
T
T T
and no real refrigerator can have a coefficient of performance greater than this
The coefficient of performance of a Carnot heat pump in heating mode is
COP heating mode hC
h c
T
T T
So why donrsquot we just use these wonderful Carnot engines and have perfect efficiency
To make the reversible steps in the cycle really reversible they would have to occur
infinitely slowly So the price you pay for making a perfect engine is that it takes
forever to get it to do any work
Entropy
The entropy of a system is defined in terms of its molecular makeup and measures
roughly the disorder of the system If the system is packed into a very small volume
then it is quite ordered and the entropy will be low If it occupies a large volume the
entropy is high (To see what this has to do with disorder note that socks in a drawer
occupy a small volume while socks on the bed in the corner by the door and
hanging from the chandelier occupy a large volume) If the system is very cold then
the molecules hardly move and may even reach out to each other and form a crystal
This is a highly ordered state and therefore has low entropy If the system is very hot
with rapidly speeding molecules crashing into the container walls and bouncing off
each other things are disordered and the entropy is high
It is possible to calculate the entropy of a system in terms of its macroscopic thermody-
namic properties ie pressure volume temperature number of moles etc The key to this
calculation is the concept of a reversible process A reversible process is one that is
12
carried out without friction and so slowly that the process can be reversed at any stage by
making an infinitesimal change in the environment of the system The slow expansion of the
gas at Th in the Carnot cycle with tiny amounts of heat being transferred to the environment
is reversible If we turned around and began slowly to compress the gas would just slowly
exhaust heat to the environment in the exact reverse way that it absorbed it during expansion
Most processes however are irreversible For example if a gas-filled box were suddenly
increased in size so that the particles were free to wander into the void created by the sudden
expansion then the gas would eventually fill the new volume uniformly at the same
temperature as before the expansion (The temperature is unchanged in this imaginary
process because the kinetic energy of the molecules would be unaffected by such an
instantaneous expansion of the container walls) This imaginary but highly thought-
stimulating process is called a free expansion and it is impossible to reverse it During the
expansion we didnrsquot push on any of the molecules so reversing this process would mean
making them go back into their original volume without pushing on them they simply will
not cooperate to this extent Another way to see that just pushing them back where they came
from does not reverse the free expansion is to think about what would happen if we just
compressed either adiabatically or isothermally An adiabatic compression back to the
original volume would heat the gas above its original temperature and an isothermal
compression would require that heat be exhausted to the environment But the free expansion
involved neither temperature changes nor heat exchanges so neither of these two processes
is the reverse of the free expansion It is simply impossible to reverse this rapid expansion in
a way that takes us back to the initial state
It is possible to calculate the change of entropy for both reversible and irreversible processes
Letrsquos consider a reversible process first In a reversible process the entropy change is given
by the formula
dQS
T
where dQ is the amount of heat added to the system during a small step of the process
The total energy change during the process may then simply be calculated by integration
f f
f i i i
dQS S S dS
T
13
(Just as in the case of energy we are mainly interested in differences rather than in
absolute magnitudes)
This integration method does not work for irreversible processes For instance in the
rapid free expansion discussed above no heat is added to the system but its disorder
obviously goes up We need to find some other way to calculate the entropy The key
is the fact that the entropy of a system depends only on its current state and not at all
on how it arrived there So to calculate the entropy change in an irreversible process
first find out what the initial and final conditions of the process are Then invent a
reversible process that takes the system from the initial state to the final state Since
the entropy depends only on the state of system and not on the process the entropy
change for the reversible process is the same as that for the irreversible process
Entropy is thus like pressure or temperature if the pressure changes from Pl to to P2
then the pressure difference is P2 - Pl regardless of how the pressure change was
made This seems obvious for pressure but not for entropy only because you arent
comfortable with entropy yet
Entropy of an Ideal Gas
For n moles of an ideal gas the difference in entropy between a state with temperature
T and volume V and some standard state with temperature To and volume Vo is given
by
ln lno o Vo o
T VS T V S T V nC nR
T V
Entropy in a Phase Change
Calculating entropy change in a phase change like melting or boiling is easy It is
QT Where Q is in the latent heat for example the flows in to cause the melting and
T is the temperature of the phase change There is NO integral to do
14
Serway Chapter 35
Angle of Reflection
If plane waves are incident on a reflecting surface with the propagation direction of
the waves making angle θ1 with the normal direction to the surface then the reflected
angle θ1prime relative to the surface normal is simply
1 1
ie the incident angles and reflected angles are the same
Refraction
If plane waves traveling through medium 1 are incident on a plane interface between
medium 1 and medium 2 then the angle of incidence of the incoming wave θ1 and
the angle of refraction of the transmitted wave θ2 are related by Snellrsquos law
2 2
1 1
sin
sin
v
v
where v1 and v2 are the wave speeds in medium 1 and medium 2 The angle of
incidence and the angle of refraction are both measured between the wave propagation
direction and the normal to the interface In terms of indices of refraction in the case
of light waves Snellrsquos law takes the more familiar form
1 1 2 2sin sinn n
where ni = cvi
Total Internal Reflection
If a wave is incident from a medium of low wave speed into a medium of high wave
speed the law of refraction requires that the angle of refraction be greater than the
angle of incidence If the angle of refraction is required to be greater than 90deg then no
refracted wave can exist and total internal reflection occurs The critical incident
angle θc beyond which total internal reflection occurs is given by
2
1
sin c
n
n
15
Dispersion of Light
In addition to the speed of light varying from material to material it also varies with
wavelength within each material This means that the index of refraction is generally a
function of wavelength
c
nv
Since the wave speed is not constant such a medium is dispersive meaning in this
context that refraction actually disperses white light into its various colors because
Snellrsquos law gives a different angle for each wavelength In most materials the
variation with wavelength is quite small but this small effect is responsible for some
of the most spectacular color effects we ever see including rainbows a flashing
crystal chandelier and the colored fire of a diamond solitaire by candlelight
16
Serway Chapter 36
Real and Virtual Images
When light rays are focused at a certain plane producing an image if a sheet of white
paper is placed there we call the image a real image The images produced by film
projectors and overhead projectors are examples of real images
When light rays appear to come from a certain location but no image is produced
when a screen is placed there we say that there is a virtual image at that location For
instance when you look in a mirror it appears that someone is behind the mirror but
a screen placed back there in the dark would show nothing Your image in the mirror
is a virtual image
Ray Tracing
There are lots of rules about how to find the images in optical systems but the best
way to keep things straight is to learn how to draw the principal rays for curved
mirrors and lenses The rays for convex and concave mirrors are shown in Fig 3615
The rays for converging and diverging thin lenses are shown in Fig 3627 You should
memorize the principal rays and know how to use them to locate images
Curved Mirrors
The focal length of a curved mirror with radius of curvature R is given by
2
Rf
The relation between object distance p image distance q and focal length f is
1 1 1
p q f
If the mirror is a diverging mirror f should be negative and if q should turn out to be
negative the image is virtual
Lateral Magnification
The lateral magnification in an optical system is defined by the ratio of the image size
to the object size
17
Image height
Object height
qM
p
for curved mirrors
As usual there are sign conventions here too but this formula just gives the
magnitude It is better to keep keep track of upright images versus inverted images by
means of ray diagrams rather than by memorizing sign conventions
Thin Lenses
The focal length of a thin lens is related to the radii of curvature of the two faces R1
and R2 of the lens by
1 2
1 1 11n
f R R
Note that this formula differs from Eq (3611) in the text by not having a minus sign
between the two R-terms We like this form better because for a simple converging
lens like a magnifying glass we just use positive values of R for both surfaces If one
of the faces is concave producing divergence use a negative value for R And if a
surface is flat use R = infin If the face is flat the radius is infinite
The relation between the image object and focal distances for a thin lens is the same
as that for a curved mirror
1 1 1
q p f
Use a negative focal length if the lens is diverging
The lateral magnification for a thin lens is the same as for a curved mirror
Image height
Object height
qM
p
Camera
The lens system in a camera projects a real image of an object onto the film (or CCD
array in a digital camera) The position of the image is adjusted to be on the film by
moving the lens into or out of the camera
18
Eye
The eye is like a camera in that a real image is formed on the retina Unlike a camera
the image position is adjusted by changing the focal length of the lens This is done by
the ciliary muscle which squeezes the lens changing its shape
Near Point The near point is the closest distance from the eye for which the lens can
focus an image on the retina It is usually 18-25 cm for young persons
Far Point The far point is the greatest distance from the eye for which the lens can
focus an image on the retina For a person with normal vision the far point is at
infinity
Nearsightedness A person is nearsighted if their far point is at some finite distance
less than infinity This condition can be corrected with a lens that takes an object at
infinity and produces a virtual image at the personrsquos far point
Farsightedness and Presbyopia A person is farsighted if their near point is too far
away for comfortable near work like reading or knitting This can be corrected by a
lens which takes an object at a normal near point distance of 18-25 cm and produces a
virtual image at the personrsquos natural near point Presbyopia involves a similar
problem which nearly all people experience as they age The ciliary muscle becomes
too weak and the lens becomes too stiff to allow the eye to provide for both near and
far vision The solution for this problem is either reading glasses or bifocal lenses
Reading classes are just weak magnifying glasses mounted on eyeglass frames
Bifocal lenses are split into upper and lower halves The lower half is a lens which
gives the proper correction for near work and the upper half is a different lens for
proper focusing at infinity
Angular Size
When an object is brought closer to the eye it appears to be larger because the image
on the retina is larger The size of this image is directly proportional to the objectrsquos
angular size which is the angle subtended by the object measured from the center of
the lens of the eye In optical instruments which are to be used with the eye the
angular size of the final image is whatrsquos important because it determines how large
the image will appear to the viewer
19
Simple Magnifier
A simple magnifier is a single converging lens or magnifying glass It takes an object
closer to the eye than a normal near point and produces a virtual image at or beyond
this near point The angular magnification is defined to be the ratio of the angular size
when viewed through the lens to the angular size of the object when viewed at the
normal near point (without aid of the lens)
Microscope
This instrument has two lenses (1) The objective is near the object being viewed and
produces a greatly magnified real image (2) The eyepiece is a simple magnifier which
the viewer uses to closely examine the image from (1)
Telescope
This instrument also has two lenses (1) The objective at the front of the telescope
takes light from a distant object and produces a real inverted image (which is rather
small) near its focal point (2) This small real image is then examined by the eyepiece
functioning as a simple magnifier to produce a virtual image with a larger angular
size
20
Serway Chapter 37
Two-Slit Interference
If light is incident on two closely spaced narrow slits a pattern of light and dark
stripes is produced beyond the slits The bright stripes or fringes are caused by
constructive interference of the two waves coming from the slits Constructive
interference occurs whenever two waves arrive at a location in phase with each other
This occurs when the distance x1 from slit 1 to a point P on the screen and the
distance x2 from slit 2 to point P differ by in integral number of wavelengths
1 2 where 0 1 2x x m m
where λ is the wavelength of the light When x1 and x2 are much larger than the slit
spacing d this condition reduces to
sind m
where θ is the angle between the direction of the incident light and the direction of the
light arriving at the screen
Thin Films
When light is partially reflected and partially transmitted by a thin film of transparent
material it is possible to have interference between the wave reflected from the front
of the film and light reflected from the back of the film (The colored reflections from
the thin film of oil on the water in a rain-soaked parking lot are an example of this
effect) It is difficult to write down formulas that will work in all cases so we will
just review the important principles here
1 If the two reflected waves are in phase with each other the film has enhanced
reflection (constructive interference) but if the two reflected waves are out of phase
with each other reflection is diminished (destructive interference) Phase shifts occur
due to reflection and due to the extra path length through the film of the wave
reflected from the back of the film
2 The phase change due to reflection is determined by the difference in index of
refraction between the two media involved in the reflection If the wave is incident
21
from a medium with a low index of refraction into a medium with a high index of
refraction a phase change of 180deg occurs and the reflected wave is inverted If
incident from high to low no phase shift occurs and the reflected wave is non-
inverted
3 The extra path length through the film of the wave reflected from the back of the
film is equal to 2t where t is the thickness of the film (The incident light is assumed
to be normal to the surface of the film) The number of wavelengths contained in the
extra path length is equal to 2tλn where λn = λn is the wavelength of the light in the
film
4 Rules for reflection from thin films
If one ray is inverted and the other is not then we have
12 constructive
2 nt m
2 destructivent m
If either both rays are inverted or both are non-inverted we have
2 constructivent m
12 destructive
2 nt m
22
Serway Chapter 38
Diffraction Grating
A diffraction grating is simply a fancy version of two-slit interference with the two
slits replaced by thousands of slits Just as in the two-slit case the bright fringes
occur at angles given by
sin md
but in this case d the distance between neighboring slits is made to be very small
The effect of having many slits instead of two is to make each bright fringe highly
localized with wide dark regions between neighboring maxima
Single Slit Diffraction
When light passes through an opening in an opaque screen an interference pattern is
produced beyond the opening To understand why we may replace the single opening
by many small coherent sources of light These many sources interfere with each
other producing a pattern known as a diffraction pattern (Note that many authors do
not distinguish between interference and diffraction treating them as interchangeable
terms) If the opening is a slit of width a then the diffraction pattern far from the slit
will have a bright central maximum with a succession of minima and weaker maxima
on either side The angle between the incident direction and the minima is given by
sin ma
where m = plusmn1 plusmn2
If the opening is circular with diameter D the angle between the incident direction
and the first minimum is given by
sin 122D
Optical Resolution and Rayleighrsquos Criterion
Two point sources can just be resolved (distinguished from each other) if the peak of
the diffraction image of the first source overlies the first minimum of the diffraction
image of the second source For circular holes of the kind usually encountered in
23
optical devices this condition is approximately satisfied when the angular separation
between the two sources as viewed from the optical instrument is greater than or equal
to the critical angle
min 122D
where λ is the wavelength of the light and where D is the diameter of the aperture in
the instrument
Polarization
We say that an electromagnetic wave is polarized if its electric field vector doesnrsquot
change direction in a random fashion The simplest kind of polarization is linear
polarization in which the electric field vector oscillates back and forth along the
same axis in space Polarized light can be produced from normal unpolarized light by
selective absorption (as in Polaroid sunglasses) by passing light through crystals that
have different indices of refraction for different polarizations (double refraction) by
scattering (the blue sky is polarized) and by reflection (glare) Polarization by
reflection occurs when light reflects from a shiny insulating (non-metallic) surface
The amount of polarization is greatest for reflection at Brewsterrsquos angle
2
1
tan P
n
n
where θP is the incidence angle of light from medium 1 onto medium 2 and where n1
and n2 are the indices of refraction for the media (In our everyday experience n1 = 1
since the light comes in through the air and n2 is the index of refraction of the shiny
insulating material producing the glare eg water glass plastic paint etc)
Malusrsquos Law
The intensity of transmitted polarized light through a perfect polarizer is related to the
incident intensity of polarized light by Malusrsquos law
2cosoI I
where Io is the intensity of the incident light and where θ is the angle between the
electric field vector in the incident wave and the transmission axis the polarizer
24
Serway Chapter 39
Principles of Relativity
All of the weirdness of relativity flows from two simple principles
(1) The laws of physics must be the same in all inertial (non-accelerating) reference
frames
(2) The speed of light in vacuum has the same value in all inertial reference frames
Note that (1) seems reasonable but (2) is very odd It says that if two space ships are
approaching each other at nearly the speed of light and a laser pulse is shot from ship
1 toward ship 2 then when the people on ship 2 measure the speed of the pulse as it
goes by it is moving at 3 times 108 ms the same as if the ships were stationary
Simultaneity
If observer 1 sees two events in her own frame as simultaneous at two different
locations a moving observer 2 will see these two events happening at different times
Relativistic gamma
The factor γ (gamma) appears regularly in the formulas of relativity
2
2
1
1 vc
where v is the relative speed between two inertial frames
Time Dilation
If observer 1 sees two events at the same location in space separated by time Δtp in his
own frame then observer 2 moving at speed v relative to observer 1 will see these
two events separated by a longer time Δt
pt t
Moving clocks run slow
Length Contraction
If observer 1 measures the length of an object along the x-direction at some instant of
time in her frame to be Lp then observer 2 moving in the x-direction at speed v will
measure the length L of the object to be shorter
pLL
25
Moving meter sticks are short
Velocity Addition
If a particle is moving at velocity vac relative to frame c and if frame c is moving at
speed vcb with respect to frame b (with both velocities directed along the same line)
what would be the velocity of the particle in frame b In classical physics the answer is
ab ac cbv v v
but if any of these velocities are an appreciable fraction of the speed of light the
answer changes
21ac cb
abac cb
v Vv
v v c
In using this equation make sure that vac has the same sign in the numerator and in the
denominator and that vbc also has the same sign in both places Note that this equation
is different (and in our opinion easier to understand) than the equations in the text
Momentum and Energy
The momentum and total energy of a particle moving at velocity v (its speed is v v )
are given by
22
2 2
22
1 1 vvc c
m mcm E mc
vp v
Rest Energy
When a particle is at rest (v = 0 and γ = 1) then E = mc2 is called the rest energy It
includes the potential energy of the particle so that if we change the potential energy
of a particle we change its rest mass
Kinetic Energy The kinetic energy is the difference between the total energy and the
rest energy
2 2K mc mc
For v ltlt c this reduces to K = mv22
General Relativity
Einsteinrsquos principle of equivalence in the theory of general relativity is In the vicinity
of a point a gravitational field is equivalent to an accelerated frame of reference in
26
the absence of gravitational effects For example there is no difference whatever in
the physics experienced by an astronaut on earth and one who is in interstellar space
far away from any stars or planets but who has just turned on her spaceshiprsquos engines
and is accelerating at 98 ms2
27
Serway Chapter 40
Photons
We now know that even though light behaves like a wave it is better described as
consisting of small packets of energy called photons The energy of a photon is related
to its frequency by
E hf
where h is Planckrsquos constant
346626 10 J sh
Photons also carry momentum given by
hf hp
c
Since the amount of energy in each photon is fixed the difference between dim light
and intense light (of the same wavelength) is that dim light consists of fewer photons
than intense light
Photoelectric Effect
In the photoelectric effect light shining onto a metal surface gives the electrons in the
metal enough energy to escape and be detected It requires a certain amount of energy
(called the work function typically = 3 ndash 5 eV) just to kick the electron out of the
metal so the light must deliver at least this much energy to an electron to produce the
effect It is observed that red light no matter how intense never produces electrons
But ultraviolet light even if quite dim will eject electrons from the metal Since red
light consists of 2 eV photons and ultraviolet light has photons with energies around
4-6 eV the photon idea explains the behavior of the photoelectric effect The
maximum energy that an ejected electron can have is
maxK hf
where f is the photon frequency The minimum frequency that light can have and cause
electrons to be ejected is called the cutoff energy and is found by setting Kmax = 0
cf h
28
Compton Effect
When high-frequency light interacts with free electrons the scattered light does not
have the same wavelength as the incident light contrary to what classical
electromagnetism would predict But the photon picture in which photons have
momentum and energy allows us to treat scattering as a collision between the photon
and the electron In this collision the electron and the scattered photon both have a
different momentum and energy than they did before And since p = hλ for a photon
if the momentum is different the wavelength will also be different
after before 1 cose
h
m c
where the angle θ is the angle between the incoming photon direction and the
direction of the scattered photon
Particles are Waves
Since photons behave like particles sometimes it is not surprising that elementary
particles can behave like waves sometimes The wavelength of a particle with
momentum p is given by
h
p
Wave-particle Duality
Both photons and elementary particles have a dual nature sometimes they behave like
particles and sometimes they behave like waves It is hard for us to comprehend the
nature of such an object by making mental pictures but experiments definitively show
that this is the case Since we have no direct experience with photons and elementary
particles (because their energies are so much smaller than the energies of the everyday
objects we encounter) it is perhaps not surprising that we have a hard time forming a
mental picture of how they behave
Electron Interference
Since an electron is both a particle and a wave just like a photon it should be able to
produce an interference pattern This is observed If an electron beam is shot at two
closely-spaced slits and if the electrons that pass through the slits are detected
downstream it is found that there are some locations where electrons are never detected
29
and others where lots of electrons are detected The pattern is exactly the same as the
one observed for light waves provided that we use the electron wavelength λ = hp in
place of the wavelength of light This pattern is observed even though each electron is
detected as a single dot on the screen Only after many such dots are collected does the
pattern emerge And if we try to understand how this effect could possibly work by
looking closely at each slit to see which one the electron came through the pattern
disappears the act of measurement destroys the interference This means that each
single electron somehow comes through both slits (as a wave would)
Uncertainty Principle
In classical physics we always imagine that the positions and momenta of moving
particles have definite values It might be hard to measure them but surely at each
instant of time a particle should be precisely located at some point in space and have a
similarly precise momentum This turns out not to be true Instead both position and
momentum are required to be uncertain with their uncertainties Δx and Δp satisfying
the Heisenberg uncertainty relation
2x p
So if the particle were known to be precisely at some particular location (so that Δx =
0) we couldnrsquot know anything about its momentum (Δp = infin) and if its momentum
were exactly known we couldnrsquot know its position
There is a similar relation involving the particlersquos energy E and the time interval Δt
over which this energy is measured
2E t
This means that energy is not actually conserved if we are considering very small time
intervals and this brief non-conservation of energy has been observed
30
Serway Chapter 41
Wave Function ψ and Probability
Quantum mechanics does not predict exactly what an electron or a photon will do Instead
it specifies the wave function or probability amplitude ψ of an electron or a photon
This wave function is a complex-valued function of space and time whose squared
magnitude is the probability density P for finding a particle at a particular place in
space at a certain time
2P
where is the complex conjugate of ψ
For example the wave function of an electron with perfectly specified momentum p
would have a wavelength given by
h
p
and its probability amplitude would be proportional to
2i x ipxe e
The corresponding probability density would then be
21ipx ip ipxP e e e
which means that the electron is equally probable to be anywhere along the x axis
This is in accord with the uncertainty principle since we specified the momentum
precisely we canrsquot have any idea about the position of the electron
As another example you have probably seen ldquofuzzy ballrdquo drawings of electron
orbitals in a chemistry book These fuzzy balls are meant to indicate the distribution
of the probability density 2
P in the orbital
Particle in a Box
A simple example in which we can calculate the wave function is the case of a particle of
mass m constrained to be inside a 1-dimensional box between x = 0 and x = L In this
case the general wave function is a linear superposition of wave functions ψn of the form
sin niE tn
n xx t A e
L
31
where A is a positive constant where n = 1 2 3 and where the energy associated
with each of the quantum states ψn is given by
22
28n
hE n
mL
This wave function is zero at x = 0 and x = L which means that the particle will never
be found at the walls of the box The wave function has maximum values in the
interior and at these places the particle is most likely to be found and it also has
places where it is zero and at these places the particle will also never be found as
expressed by the formula
2sinn x
PL
The particle in the box is interfering with itself producing a probability interference
pattern across the box just like the interference patterns we studied with light and
sound
32
Serway Chapter 42
Atoms
Many people picture an atom as a miniature solar system where electrons orbit around
a massive nucleus at the center This picture is misleading because in atoms the wave
nature of electrons dominates The electrons form 3-dimensional standing waves
(called orbitals) centered at the nucleus And if you ask what is it thatrsquos waving the
answer is ψ the probability amplitude (see Chapter 41)
Standing waves on a string can occur only for certain frequencies (the fundamental
and higher harmonics) Similarly atomic orbitals occur only for certain energies For
the hydrogen atom the energies of the orbitals have the a particularly simple form
2
1136 eVnE
n
where n = 1 2 3 is called the principal quantum number For other atoms the
determination of the orbital energies requires numerical calculation by computers
Atomic Spectra
If an electron is somehow given extra extra energy (we say that it is excited) so that it
occupies a higher orbital it will eventually ldquofallrdquo back down to a lower orbital Each
time an electron falls to a lower orbital it loses the difference in energy between the
two orbitals in the form of a photon Since the orbital energies are discrete so are the
energy differences and so are the wavelengths of the emitted light The entire set of
these discrete wavelengths is called the atomic spectrum and it is unique to each type
of atom For hydrogen the spectrum can be simply written as
H 2 2
1 1 1
f i
Rn n
where RH = 10973732 times 107 m-1 For other atoms the spectrum cannot be expressed as
a formula
Orbital Angular Momentum Electrons in atoms also have quantized values of
angular momentum The orbital quantum number ℓ specifies the value of this
quantized angular momentum through the formula
1L
33
If we want to know the value of the angular momentum along some direction in space
say the z direction the answer is not L but rather
zL m
where mℓ is another quantum number which runs from
1 1m
This quantum number is important when an atom sits in a magnetic field
Spin Angular Momentum It has been found experimentally that electrons and other
charged particles also carry internal angular momentum which we call spin
Electrons have an intrinsic spin angular momentum s along a specified axis that is
extremely quantized it can only take on 2 possible values
1
2zs s
We thus say that electrons have spin s = 12 or that electrons are rdquospin one-half parti-
cles Since s is an angular momentum it obeys the same rule as that for orbital angular
momentum namely that if its value along some axis is s then its total magnitude is
given by
31
2S s s
I know this seems weird but quantum mechanics is weird The only excuse for this
bizarre way of looking at the world is that it predicts what happens in experiments
Exclusion Principle The answer to the question of how many electrons (or any other
spin one-half particle) can be in one particular quantum state was discovered by
Wolfgang Pauli and is called the exclusion principle
ldquoNo two electrons can ever be in the same quantum state therefore no two electrons
in the same atom can have the same set of quantum numbersrdquo
This is the reason that we have atoms with different properties instead of every atom
simple having all of its electrons in the ground state All of the variety we see around
us in the world is the result of chemical differences and these differences would not
exist unless electrons obeyed this important principle The entire structure of the
periodic table (see pages 1377-1379 in Serway) is an expression of this principle
34
Serway Chapter 44
Nuclear Properties
The nucleus is composed of protons with charge +e and mass mp = 1007226 u and of
neutrons with zero charge and mass mn = 1008665 u where u is one atomic mass unit
-271 u=1660540 10 kg
The atomic number Z counts the number of protons in a nucleus while the neutron
number N counts the number of neutrons The mass number A is the sum of the two
A N Z
Protons and nuetrons have about the same mass and some times called baryons (heavy
ones) or nucleons (partcles in the nucleus) A is sometimes called the baryon
number The chemical elements are distinguished by Z ie hydrogen has Z = 1 iron
has Z = 26 uranium has Z = 92 etc But for a given Z there might be several nuclei
with different numbers of neutrons These nuclear siblings are called isotopes
These numbers are used to label nuclei according to the pattern
5626 ie FeA
Z X
denotes the isotope of iron with N = 56 ndash 26 = 30 neutrons
The nucleus is roughly spherical with a radius given approximately by
1 3 150 0where 12 10 mr r A r
Nuclear Stability
Because the positively charged protons electrically repel each other with an enormous
force at distances as small as 10-15 m there must be some really strong force that that
overcomes electrical repulsion to hold protons and neutrons together This force is
called with some lack of imagination the strong force It is a very short-range force
(it only acts over a distance of about 2 x 10-15 m) and attracts protons to protons
neutrons to neutrons and neutrons to protons But in spite of this strong nuclear force
the coulomb repulsion of the protons is still present so anything that might keep the
protons from being right next to each other would help keep the nucleus from
35
exploding This role is played by the neutrons and for nuclei with Z le 20 the stable
nuclei roughly have N = Z
For Z gt 20 the coulomb repulsion force is more powerful and more neutrons are
needed to dilute it up to about N = 15Z around Z = 80 For Z gt 83 no amount of
neutrons can help and these nuclei are unstable (radioactive) There is an ldquoisland of
(relative) stabilityrdquo around Z=90-92 (thorium and uranium respectively) These
elements have one or two isotopes with half-lifes of billions of years so there are
substantial amounts of such elements on Earth This fact makes it possible to have
practical fission devices
Radioactive Decay
There are three types of radioactive decay
Alpha decay the nucleus kicks out a helium nucleus (N = 2 Z = 2 A = 4) So if the
original nucleus is called X and the new nucleus is called Y then the decay would look
like this
4 42 2X Y+ HeA A
Z Z
Beta decay the nucleus either kicks out an electron (endash) or its positively-charged
antimatter twin the positron (e+) plus either an electron neutrino v or an electron
anti-neutrino v
1X Y+e electron decayA AZ Z v
1X Y+e positron decayA AZ Z v
A neutrino is a particle with no charge hardly any mass (much less than the electron
mass) and interacts so weakly with matter that most neutrinos upon encountering the
planet earth just pass right through it as if it werenrsquot there
Gamma decay the nucleons in the nucleus X are in an excited energy state X
(perhaps as a result of having undergone alpha or beta decay) and they drop down to a
lower energy state shedding the energy as a high frequency photon
X XA AZ Z
36
This process is exactly analogous to the way that the electrons in atoms emit photons
Decay Rate and Half Life
There is no way to predict exactly when an unstable or excited nucleus will decay but
there is an average rate at which this decay occurs called the decay constant λ The
meaning of this constant is that if there are a large number N of nuclei in a sample
then the number of decays per second that will be observed (called the decay rate R)
is R = λN In mathematical language
dMR N
dt
This simple differential equation has for its solution
0tN t N e
where N0 is the number of nuclei in the sample at time t = 0
The half-life is the time it takes for half of the nuclei in the sample to decay and is
related to the decay constant by
1 2
ln 2 0693T
Disintegration Energy
When a nucleus decays it is making a transition to an overall state of lower energy
which means according to Einsteinrsquos famous formula E = mc2 that the sum of the
masses after the decay must be less than the mass before with the lost mass appearing
as kinetic energy among the decay products For example in alpha decay this kinetic
energy called the disintegration energy Q is given by
2X YQ M M M c
37
Serway Chapter 45
Nuclear energy
There are two ways to extract energy from the nucleus fission and fusion For nuclei
with Z greater than 26 breaking the nucleus apart into pieces leads to a lower
2mc energy than the original nucleus so energy can be extracted by fission For
nuclei with Z less than 26 a lower energy is achieved by combining nuclei so energy
can be extracted by fusion
Fission
Since neutrons have no charge they are not repelled from nuclei as protons are for
this reason their behavior is the key to understanding how fission works
Fast neutrons mostly bounce off other nuclei with each collision slowing the neutron
down This elastic energy loss is most effective if the other nuclei have low mass (like
hydrogen) and these materials are called moderators because of their ability to slow
down fast neutrons
The reason that slowing neutrons is important is that slow neutrons are much more
likely to be absorbed by a nucleus which then leads to nuclear reactions of various
kinds For a few very large nuclei like uranium-235 and some plutonium isotopes
absorption of a slow neutron causes the nucleus to split into two large fragments plus
2 or 3 fast neutrons (energy is released in the process too) Thus one slow neutron
can produce energy plus 2 or 3 more neutrons which if moderated from fast to slow
can split 2 or 3 more nuclei leading to a runaway chain reaction If the reaction is
allowed to proceed unchecked it produces a large explosion If a neutron absorbing
material is added to the mix (like the cadmium in reactor control rods) it is possible to
keep the reaction under control and to extract the released energy as heat to drive
steam turbines and produce electricity
Fusion
Fusion involves mashing two nuclei together and since they are both charged and repel
each other this reaction is much harder to make go The nuclei must have enough energy to
overcome the coulomb repulsion which is why this reaction requires a high temperature
(hundreds of millions of degrees K) like that in the sun (which is burning hydrogen to
helium via fusion) or like that in the center of the fission explosion that is used to detonate
38
a hydrogen bomb
This reaction is of interest for power production in spite of this difficult temperature
requirement because of the abundance of fusion fuel on the planet There are about
012 g of deuterium ( 21H ) in every gallon of water on earth and it only costs about 4
cents to extract it The fusion energy available from this minuscule amount of
deuterium would run a 1000-Megawatt power plant for 10 seconds Water is so
abundant on earth that if fusion were to work we would have an essentially
inexhaustible source of energy
So why donrsquot we have fusion power plants Well the fuel is cheap but the match is
incredibly expensive The only way we know to control this difficult high-temperature
reaction is with large and expensive pieces of equipment involving either large
magnetic fields and complex high-power electromagnetic antennas or with gigantic
(football-fieldsized) laser facilities involving more than a hundred of the highest-
energy lasers ever built Power plants based on these current methods for controlling
fusion are unattractive to the fiscally-minded people who run the electric power
industry Hopefully better designs will be discovered as experiments continue
39
c actinium 68 Er erbium 101 Md mendelevium 104 Rf rutherfordium l aluminum 63 Eu europium 80 Hg mercury 62 Sm samarium m americium 100 Fm fermium 42 Mo molybdenum 21 Sc scandium b antimony 9 F fluorine 60 Nd neodymium 106 Sg seaborgium r argon 87 Fr francium 10 Ne neon 34 Se selenium s arsenic 64 Gd gadolinium 93 Np neptunium 14 Si silicon t astatine 31 Ga gallium 28 Ni nickel 47 Ag silver a barium 32 Ge germanium 41 Nb niobium 11 Na sodium k berkelium 79 Au gold 7 N nitrogen 38 Sr strontium e beryllium 72 Hf hafnium 102 No nobelium 16 S sulfur i bismuth 108 Hs hassium 76 Os osmium 73 Ta tantalum h bohrium 2 He helium 8 O oxygen 43 Tc technetium
boron 67 Ho holmium 46 Pd palladium 52 Te tellurium r bromine 1 H hydrogen 15 P phosphorus 65 Tb terbium d cadmium 49 In indium 78 Pt platinum 81 Tl thallium a calcium 53 I iodine 94 Pu plutonium 90 Th thorium f californium 77 Ir iridium 84 Po polonium 69 Tm thulium
carbon 26 Fe iron 19 K potassium 50 Sn tin e cerium 36 Kr krypton 59 Pr praseodymium 22 Ti titanium s cesium 57 La lanthanum 61 Pm promethium 74 W tungsten l chlorine 103 Lr lawrencium 91 Pa protactinium 92 U uranium r chromium 82 Pb lead 88 Ra radium 23 V vanadium o cobalt 3 Li lithium 86 Rn radon 54 Xe xenon u copper 71 Lu lutetium 75 Re rhenium 70 Yb ytterbium m curium 12 Mg magnesium 45 Rh rhodium 39 Y yttrium b dubnium 25 Mn manganese 37 Rb rubidium 30 Zn zinc y dysprosium 109 Mt meitnerium 44 Ru ruthenium 40 Zr zirconium s einsteinium
40
UNITS The SI units are given in parentheses Other commonly used units are given in terms of the SI units Symbols conform with the recommendations of the American National Standards Institute (ANSI) and the American Institute of Physics (AIP) Commonly used multiples of SI units
prefix symbol factor tera T 1012 giga G 109
mega M 106 kilo k 103
centi c 10-2 milli m 10-3
micro μ 10-6 nano n 10-9 pico p 10-12
femto f 10-15
length meter (m) angstrom 1 Ǻ = 10-10 m inch 1 in = 254 x 10-2 m foot 1 ft = 03048 m mile 1 mi = 1609 m
mass kilogram (kg) atomic mass unit 1 u = 1661 x 10-27 kgslug 1 slug = 1459 kg
force newton (N = kg bull ms2) dyne 1 dyn = 10-5 N pound 1 lb = 4448 N
pressure Pascal (Pa = kgm bull s2) atmosphere 1 atm = 1013 x 105 Pa poundssquare inch 1 psi = 6895 Pa cm of mercury 1 cm Hg = 1333 Pa bar 1 bar = 1000 x 105 Pa torr 1 torr = 1333 Pa
time second (s) minute 1 min = 60 s hour 1 h =3600 s
frequency hertz (Hz = s-1) radianssecond 1 rads = 12π Hz
energy joule (J = kg bull m2s2) erg 1 erg = 10-7 J electron volt 1 eV = 1602 x 10-19 J calorie 1 cal = 4187 J kilowatt-hour 1 kWh = 36 x 106 J British thermal unit 1 Btu = 1055 J
power watt (W = kg bull m2s3) horsepower 1 hp = 7457 W
charge coulomb (C = A bull s)
electric potential volt (V = kg bull m2 s3 bull A)
current ampere (A)
resistance ohm (Ω = kg bull m2s3 bull A2)
capacitance farad (F = s4 bull A2kg bull m2)
magnetic field tesla (T = kgs2 bull A) gauss 1 G = 10-4 T
magnetic flux weber (Wb = kg m2s2 bull A) maxwell 1 Mx = 10-8 Wb
magnetic inductance henry (H = kg bull m2s2 bullA2)
temperature kelvin (K) degrees Celsius 0degC = 27315 K
angle radian (rad) degree 1 deg = π180 rad revolution 1 rev = 2π rad
41
Some Physical Constantsa Quantity Symbol Valueb Atomic mass unit μ 1660 538 73 (13) x 10-27 kg
931494 013 (37) MeVc2 Avogadros number NA 6022 141 99 (47) x 1023 particlesmol Avogadrorsquos number 6022 x 1023 mol Bohr magneton
2B e
em 9274 008 99 (37) x 10-24 JT
Bohr magneton μB 927 x 10-24 JT Bohr radius
0
2
2e em e k
a 5291 772 083 (19) x 10-11 m
Boltzmanns constant B A
RNk 1380 650 3 (24) x 10-23 JK
Boltzmannrsquos constant kB 1380 x 10-23 JK Compton wavelength
C eh
m c 2426 310 215 (18) x 10-12 m
Deuteron mass md 3343 583 09 (26) x 10-27 kg 2013 553 212 71 (35) u
electron charge e 1602 x 10-19 C Electron mass me 9109 381 88 (72) x 10-31 kg
5485 799 110 (12) x 10-4 u 0510 998 902 (21) MeVc2
electron mass 911 x 10-31 kg Electron volt eV 1602 176 462 (63) x 10-19 J Elementary charge e 1602 176 462 (63) x 10-19 C Gas constant R 8314 472 (15) JKmol Gravitational constant G 6673 (10) x 10-11 Nm2kg2 Neutron mass mn 1674 927 16 (13) x 10-27 kg
1008 664 915 78 (55) u 939565 330 (38) MeVc2
neutron mass 1675 x 10-27 kg Nuclear magneton
2n p
em 5050 783 17 (20) x 10-27 JT
permeability constant μ0 1257 x 10-6 Hm permittivity constant ε0 8854 x 10-12 Fm Plancks constant h
2h
6626 068 76 (52) x 10-34 Js 1054 571 596 (82) x 10-34 Js
Planckrsquos constant h ħ
6626 x 10-34 Js 1055 x 10-34 Js
Proton mass mp 1672621 58 (13) x 10-27 kg 1007 276 466 88 (13) u 938271 998 (38) MeVc2
proton mass 1673 x 10-27 kg Rydberg constant RH 1097 373 156 854 9 (83) x 107 m-1 Speed of light in vacuum c 2997 92458 x 108 ms (exact) speed of light c 300 x 108 ms a
These constants are the values recommended in 1998 by CODATA based on a least-squares adjustment of data from different measurements For a more complete list see P J Mohr and B N Taylor Rev Mod Phys 72351 2000 b The numbers in parentheses for the values above represent the uncertainties of the last two digits
NOTE The ones in red are the ones from Appendix 13 The ones in yellow are the ones that are from the Serway book table but that were already on the Appendix 13 table
42
INDEX
Absolute zero 9
Adiabatic 17
Adiabatic exponent 18
Adiabatic process 17
Alpha decay 46
Amplitude 4
Angle of reflection 25
Angular frequency 3
Angular magnification 30
Angular momentum atomic 43
Angular size 29
Archimedes Principle 1
Atomic spectra 43
Atoms 43
Avogadrorsquos number 10
Beats 8
Bernoullirsquos Equation 2
Beta decay 46
Bifocals 29
Brewsterrsquos angle 34
British Thermal Unit Btu 11
Buoyancy 2
calorie 11
Calorie 11
Camera 28
Carnot cycle 21
Carnot efficiency 22
Celsius scale 9
Ciliary muscle 29
Coefficient of performance 21
Compton effect 39
Constructive interference 7
Continuity equation of 2
Contraction length 35
Convection 14
Curved mirrors 27
Decay constant 46
Decay rate radioactive 46
Decibel scale 5
Degrees of freedom 16
Density 1
Destructive interference 7
Diffraction single slit 33
Diffraction grating 33
Dilation time 35
Disintegration energy 47
Dispersion of light 26
Doppler effect 6
Double slit interference 31
Efficiency 20
Electron interference 39
Emissivity 15
Energy nuclear decay 47
Energy relativistic 36
43
Engineering work 20
Entropy 22
Entropy ideal gas 24
Equipartition of energy 18
Equivalence principle 36
Exclusion principle 44
Expansion thermal 9
Expansion coefficient linear 9
Eye 29
Fahrenheit scale 9
Far point 29
Farsightedness 29
First Law of Thermodynamics 12
Fission 48
Flux volume 2
Free expansion 23
Fringes 31
Fusion 48
Fusion heat of 11
Gamma relativistic 35
Gamma decay 46
General relativity 36
Half-life 46
Heat 11
Heat capacity 11
Heat conduction 13
Heat engine 20
Heat of fusion 11
Heat of vaporization 11
Heat pump 20
Heats of transformation 11
Hydrostatics 1
Ideal Gas Law 10
Images real and virtual 27
Intensity sound 5
Interference 7
Interference two-slit 31
Internal energy 12
Internal energy degrees of freedom 16
Irreversible process 19 23
Isotopes 45
Joule 11
Kelvin scale 9
Kinetic energy relativity 36
Kinetic theory 16
Latent heat 11
Length contraction 35
Linear expansion coefficient 9
Linear polarization 34
Linear superposition 7
Longitudinal wave 3
Loudness 5
Magnification lateral 27
Magnifying glass 30
Malusrsquos law 34
Microscope 30
Momentum relativistic 36
Muscle ciliary 29
44
Musical instruments 8
Musical scale 8
Near point 29
Nearsightedness 29
Nonlinear 7
Nuclear energy 48
Nuclear properties 45
Nuclear stability 45
Octave 8
Optical resolution 33
Orbital quantum number ℓ 43
Orbitals 43
Particle in a Box 41
Particles are waves 39
Pascalrsquos Principle 1
Period 3
Photoelectric effect 38
Photons 38
Pitch 8
Polarization 34
Power sound 5
Presbyopia 29
Pressure 1
Principal quantum number n 43
Principle of equivalence 36
Principle of linear superposition 7
Probability amplitude ψ 41
Processes thermodynamic 13
R-value 14
Radiation thermal 14
Radioactive decay 46
Radioactive decay rate 46
Ray tracing 27
Rayleighrsquos criterion 33
Reading glasses 29
Real image 27
Refraction 25
Refrigerator 20
Relativistic gamma 35
Relativity principles 35
Resolved for light sources 33
Rest energy 36
Reversible process 19 22
Rope wave speed 4
Second Law of Thermodynamics 19
Shock waves 6
Simple magnifier 30
Simultaneity 35
Single slit diffraction 33
Snellrsquos law 25
Sound speed 5
Specific heat 11
Spin Angular momentum 44
Standing waves 7
Stefanrsquos law 14
Strong force 45
Telescope 30
Temperature 9
45
Temperature Scales 9
Thermal conductivity 14
Thermal energy 12
Thermal expansion 9
Thin film interference 31
Thin lenses 28
Time dilation 35
Tone musical 8
Total internal reflection 25
Transverse wave 3
Traveling Waves 3
Two-slit interference 31
Uncertainty principle 40
Vaporization heat of 11
Velocity addition relativity 36
Virtual image 27
Volume flux 2
Wave function 41
Wave Function ψ and Probability 41
Wave speed 3
Wave-particle duality 39
Wavelength 3
Wavenumber 3
Wien Displacement Law 16
Work 11
1
Serway Chapter 14 Fluid Mechanics
Density and Pressure
The density of a material is defined to be the ratio between the mass of a small piece of it divided
by the volume of the small piece
m
V
Water has a density of 1000 kgm3 while air has a density of 13 kgm3 at sea level and 11
kgm3 in Provo Sea water has a density of 103 of water Ethanol (ethyl alcohol) is 0806
The pressure exerted on a small surface is defined to be the force applied to it divided by its
area
FP
A
At sea level atmospheric pressure is 1013 times 105 pascals where 1 Pa = 1 Nm2 Here in Provo
atmospheric pressure is about 86 times 104 Pa Sometimes pressure will be expressed in pounds-per-
square-inch (psi) or sometimes in torr 1 psi= 6895 times 103 Pa and 1 torr= 1333 Pa
Hydrostatics
If h measures distance below the surface of a liquid then the pressure as a function of the depth h
is given by
oP P gh
where Po is the pressure at the surface of the liquid usually atmospheric pressure In 123 liquids
are imcompressible ρ does not vary with depth and g = 980 m2s
Pascalrsquos Principle
A change in the pressure applied to an enclosed fluid is transmitted undiminished to every
portion of the fluid and to the walls of the containing vessel (Blaise Pascal 1652)
Archimedes Principle
A body wholly or partially immersed in a fluid will be buoyed up by a force equal to the weight
of the fluid that it displaces (Archimedes ca 220 BC) This principle is a handy computational
tool but it does not explain the real reason that objects in a fluid experience an upward force
The reason is that the pressure on the bottom of the object is greater than the pressure at the top
so there is a net upward pressure force The buoyant force is given by the formula
B Vg
2
where ρ is the density of the fluid V is the submerged volume of the object and g = 980 ms2 is
the acceleration of gravity If the object is floating on the liquid then the submerged volume V is
only that portion which is below the liquid surface
Equation of Continuity
For a fluid whose density is constant the rate of fluid flow or volume flux Φ measured in units
of m3s for a cross-section of fluid streamlines having area A and flow speed v is given by
Av
Bernoullirsquos Equation
The principle of conservation of energy for steady fluid flow states Along each streamline of the
flow the following quantity has a constant value
21
2P v gy
where y is measured upward against gravity Note that y in this equation and h in the hydrostatic
pressure equation have opposite signs This law works for gases and liquids The demos are
mostly for gases and are set up so that ρ does not vary
3
Serway Chapter 16
Longitudinal and Transverse Waves
A longitudinal wave is one in which the direction of vibration of the medium and the direction of
propagation of the wave are in the same direction Sound waves in the air in water and in solids
are longitudinal waves
A transverse wave is one in which the direction of vibration of the medium is perpendicular to
the direction of propagation of the wave Ocean waves waves on ropes and electromagnetic
waves are transverse waves
Wavelength and Wavenumber
The wavelength λ is the period in space of the wave ie it is the distance from one wave crest to
the next The wavenumber k (units reciprocal length) is related to the wavelength by the formula
2k
Frequency and Angular Frequency
The period T is the period in time of the wave ie it is the time between the arrival of one wave
crest and the next at some point in space The frequency f is the number of crests that arrive per
second which is just 1T and the angular frequency ω is 2π times the frequency Both have units
of per second (reciprocal time)
1 22f f
T T
Wave Speed
The wave speed of a wave is defined to be the the speed at which the wave crests or troughs
move through the medium This speed is related to the frequency and the wavelength by the
formulas
v fk
where k is the wavenumber of the wave v is also called the phase velocity
Traveling Waves
Traveling waves are oscillations in which the wave crests and troughs glide smoothly through the
medium Waves on oceans and lakes are traveling waves The mathematical form of a harmonic
(sinusoidal) traveling wave is
4
siny x t A kx t
where A is the amplitude of the wave As usual ω and k are related by the relation ω = kv where
v is the wave speed the speed at which the crests travel the minus sign means that the wave is
traveling to the right towards higher values of x φ is the phase and constrols where the maxima
and minima occur
Speed of Transverse Waves on a Rope
The velocity of transverse waves on a perfectly flexible rope is given by
Tv
where T is the tension in the rope and where μ is its linear mass density (mass per unit length)
Common Test Questions
A common question is to be given an equation for a wave and to be asked what is amplitude
velocity etc Another common type of question is to be given facts about a wave like its
direction velocity and magnitude at a given place and time and to be asked which of several
equations correctly describe the wave
5
Serway Chapter 17
Sound Speed in Solids Liquids and Gases
The speed of sound in a liquid or solid of bulk modulus B and volume mass density ρ
Bv
The speed of sound in air is
331 1 273v m s T C
The speed of sound in air at room temperature is about 343 ms Notice it doesnrsquot depend on
pressure or frequency
Power Intensity and Loudness
The intensity of a sound wave is defined to be the power per unit area ie wattsm2 in the wave
For spherical waves traveling away from a small source emitting waves with average power Рav
the intensity falls off with distance from the source r according to the inverse-square law
av24
Ir
Hence the intensity I2 at distance r2 is related to the intensity I1 at a different distance r1 by
21 2
22 1
I r
I r
Experiments on human hearing have shown that we hear intensity differences logarithmically so
the decibel loudness scale for sound intensity was invented The loudness β of a sound in
decibels is related to its intensity I in Wm2 by the formula
1010logo
I
I
Where Io is a sound intensity near the threshold of hearing defined to be Io = 10-12 Wm2 Note
that on this scale a sound is made 10 times more intense by adding 10 decibels Remember that
intensity is proportional to velocity squared Amplitude is the A is the previous chapterrsquos
equation for a traveling wave If you make the A 10 times as much the intensity will increase by
100 times (20 dB)
Doppler Effect
If sound waves are traveling through a medium and if either the receiver of the waves or the
source of waves is moving then the frequency received is related to the frequency emitted by
6
0
s
v vf f
v v
where fprime is the frequency detected by the observer f is the frequency emitted by the source v is
the speed of the waves vo is the speed of the observer and vs is the speed of the source This
formula assumes that the source and receiver are either moving directly toward each other or
directly away from each other To know which signs to use remember that when observer and
source approach each other the observed frequency is higher while if they move away from each
other it is lower Just examine the signs in the formula and make the answer come out right
For electromagnetic waves (light radio waves X-rays) traveling in vacuum Einsteinrsquos theory of
relativity (and careful experiments) show that the Doppler shift is given by
r
r
c vf f
c v
where vr is the relative speed between the source and the observer
For all kinds of waves (sound light etc) if the relative speed of the source and the observer is
small compared to the speed of the waves then there is a simple approximation to the Doppler
effect For example if the relative speed is 1 of the wave speed then the frequency shifts by
1 Remember that this is only an approximation
Shock Waves
When an object moves through a medium at a speed greater than the speed of waves a V-shaped
shock wave is produced The V-shaped wake behind a speeding boat is a good example of this
effect and the cone of sonic-boom behind a supersonic aircraft is another The angle that V-line
makes with the direction of travel of the source is given by
sins
v
v
where v is the wave speed and vs is the source speed
7
Serway Chapter 18
Principle of Linear Superposition
We say that a system obeys the principle of linear superposition if two or more different motions
of the system can simply be added together to find the net motion of the system Light waves
obey this principle as they propagate through the air as can be seen by shining two flashlights so
that their beams cross The beams propagate along without affecting each other (Light sabers are
a spectacular but unfortunately fictional example of systems that do not obey the principle of
superposition) Wave pulses on an ideal rope also obey this principle two different pulses pass
through each other without change Standing waves are an example of this effect being simply
the linear superposition of two traveling waves of the same frequency but moving in opposite
directions Light waves in matter do not always obey this principle For instance two powerful
laser beams could be made to cross in a piece of glass in such a way that their combined heating
effect in the crossing region could melt the glass and scatter the beams in complicated ways This
is an example of a nonlinear effect
Interference
When two or more waves are present in the same medium at the same time their net effect may
often be obtained simply by adding them at each point in the medium according to the principle
of linear superposition (Note this wonrsquot work if the medium is nonlinear) When this addition
makes the total amplitude be greater than the individual amplitudes of the various waves we say
that the interference is constructive When the addition produces cancellation and an amplitude
less than the amplitudes of the separate waves we have destructive interference
Standing Waves
A standing wave is the superposition of two identical traveling waves moving in opposite
directions Nodes are places where the two waves perfectly destructively interfere to produce
zero amplitude at all times Anti-nodes are places where the two waves perfectly constructively
interfere to produce an amplitude maximum The distance between nodes is λ2 Standing waves
on a string fixed at both ends have nodes at each end of the string Standing waves in an air
column enclosed in a tube have displacement anti-nodes at open ends of the tube and
displacement nodes at closed ends The frequency of the standing wave with the lowest possible
frequency is called the fundamental frequency Standing waves on strings or in air columns all
have frequencies which are integer multiples of the fundamental frequency and are called
8
harmonics (The fundamental is called the ldquofirst harmonicrdquo)
Beats
Beats are heard when two waves with slightly different frequencies f1 and f2 are combined The
waves constructively interfere for a number of cycles then destructively interfere for a number
of cycles We hear a periodic ldquowah-wahrdquo frequency equal to the difference of the two wave
frequencies
1 2bf f f
Musical Instruments
Musical instruments produce tones by exciting standing waves on strings (violins piano) and in
tubes (trumpet organ) The fundamental frequency of the standing wave is called the pitch of the
tone The pitch of concert A is 440 Hz by definition Two tones are an octave apart if one pitch
has twice the frequency of the other In written music there are 12 intervals in each octave with
the ratio between successive intervals equal to 2112 = 105946 The ratios for each tone in an
octave starting at A and ending at the next higher A are
A A B C C D D E F F G G A
1 10595 11225 11892 12599 13348 14142 14983 15874 16818 17818 18877 2
A musical tone is actually a superposition of the fundamental frequency and the higher
harmonics The tone quality of a musical instrument is determined by the amplitudes of the
various harmonics that it produces A violin and a trumpet can play the same pitch but they
donrsquot sound at all alike to our ears The difference between them is in the various amplitudes of
their harmonics
9
Serway Chapter 19
Temperature
Formally temperature is what is measured by a thermometer Roughly high temperature is what
we call hot and low temperature is what we call cold On the atomic level temperature refers to
the kinetic energy of the molecules A collection of molecules is called ldquohotrdquo if the molecules
have rapid random motion while a collection of molecules is called ldquocoldrdquo if the random motion
is slow When two bodies are placed in close contact with each other they exchange molecular
kinetic energy until they come to the same temperature This is the microscopic picture of the
Zeroth Law of Thermodynamics
Absolute Zero
Absolute zero is the lowest possible temperature that any object can have This is the temperature
at which all of the energy than can be removed an object has been removed (This removable
energy we call thermal energy) There is still motion at absolute zero Electrons continue to orbit
around atomic nuclei and even atoms continue to move about with a small amount of kinetic
energy but this small energy cannot be removed from the object For example at absolute zero
helium is a liquid whose atoms still move and slide past each other
Temperature Scales
Kelvin Scale Absolute zero is at T = 0 K water freezes at T = 27315 K room temperature is
around T = 295 K and water boils at T = 373 K Note we donrsquot use a deg symbol Kelvin is
prefered SI Unit
Celsius Scale Absolute zero is at T = -273degC water freezes at T = 0degC room temperature is
around T = 22degC and water boils at T = 100degC
Fahrenheit Scale Absolute zero is at T = -459degF water freezes at TF = 32degF room temperature
is around T = 72degF and water boils at TF = 212degF TF =18TC + 32 Notice that temperature
differences are the same for the Kelvin and Celsius scales
Thermal Expansion
When materials are heated they usually expand and when they are cooled they usually contract
(Water near freezing is a spectacular counterexample it works the other way around) The
coefficient of linear expansion is defined by the relation
1
i
L
L T
10
where Li is the initial length of a rod of the material and ΔL is the change in its length due to a
small temperature change ΔT The coefficient of volume expansion is defined similarly
1
i
V
V T
where Vi is the initial volume of a piece of material and where ΔV is the change in its volume due
to a small temperature change ΔT
Avogadrorsquos Number (N or NA)
One mole of any substance corresponds to 6022 times 1023 molecules The atomic mass given on the
periodic table (p A32-33 in the text) is the mass in grams of one mole of atoms For example
the atomic mass of oxygen is 15999 gmol The mass of a single atom is given by the same
number in units of atomic mass units (u) For example the mass of an oxygen atom is 15999 u
(on the average)
Ideal Gas Law (an example of an equation of state)
When the molecules of a gas are sufficiently inert and widely separated that interactions between
them are negligible we say that it is an ideal gas The pressure P volume V and temperature T
(in kelvins) of such a gas are State Variables and are related by the ideal gas law
Bor PV=NkPV nRT T
where n is the number of moles of the gas where R is the gas constant
8314 Jmol KR
where N is the number of molecules and where kB is Boltzmannrsquos constant
231380 10 J KBk
It works well for air at atmosphere pressure and even better for partial vaccuums The relative
ease of measuring pressure and the linear relationship between pressure and temperature (if V
and n are held fixed) makes an ideal gas an ideal thermometer Thermometers can be based on
properties of solids or liquids but the behavior of these materials with temperature is more
complicated
11
Serway Chapter 20
Heat
Heat is energy that flows between a system and its environment because of a tempera-
ture difference between them The units of heat are Joules as expected for an energy
Unfortunately there are several competing units of energy They are related by
1 cal 4186 J 1 Cal 4186 J 1 Btu 1054 J
Heat Capacity
There is often a simple linear relation between the heat that flows in or out of part of a
system and the temperature change that results from this energy transfer When this
linear relation holds it is convenient to define the heat capacity C and the specific
heat c as follows
For the entire object the heat Q it absorbs is given by Q = C(Tf - Ti)
For a particular material the specific heat is defined by c = Cm which is the heat
capacity per unit mass so that
Q = mc(Tf ndash Ti)
Note C has units of energy (J or Cal)(Kelvin kg)
Heats of Transformation or Latent Heat Q = plusmn mL
When a substance changes phase from solid to liquid or from liquid to gas it absorbs
heat without a change in temperature The latent heat or heat of transformation is
usually given per unit mass of the substance For example for water the heat of fusion
(melting) is L = 333 kJkg while the heat of vaporization (boiling) is L = 2260 kJkg
Note that heat for boiling is considerably bigger than melting for water You have to
be careful with signs heat is given off (negative) if you go down in temperature and
condense steam
Work
In general the small amount of work done on a system as a force Fon is exerted on it
through a vector displacement dx is given by
on xdW d F
12
But if the displacement is done very slowly (as we always assume in thermodynamics)
then the force exerted on the system and the force exerted by the system are in
balance so the force exerted by the system is ndash Fon In thermodynamics it is more
convenient to talk about the force exerted by the system so we change the above
formula for the work done on the system to
xdW d F
where F is the force exerted by the system This has confused students for more than a
century now but this is the way your book and many other books do it so you are
stuck You will need to memorize the minus sign in this definition of the work to be
able to use your textbook
There are many chances to get signs wrong in this and the next two chapters (Mosiah
2 )
When an external agent changes the volume of a gas at pressure P by a small amount
dV the (small amount) of work done on the system is given by
dW PdV
Notice that this minus sign is just what we need to make dW be positive if the external
agent compresses the gas for then dV is negative If on the other hand the external
agent gives way allowing the gas to expand against it then dV is positive and we say
that the work done on the gas is negative
The work done on the system (eg by the gas in a cylinder) in a thermodynamic
process is the area under the curve in a PV diagram It is positive for compressions
and negative for expansions If the volume of gas remains constant in a process then
no work is done by the gas
Cyclic processes are important For cyclic processes represented by PV diagrams the
magnitude of the net work during one cycle is simply the area enclosed by the cycle
on the diagram Be careful to keep track of signs when you are calculating that
enclosed area In cyclic process Q = ndashW for a cycle Put a PV diagram in your notes
1
Path A-B B-C C-D D-A A to A net
Q
W
ΔU
ΔS
Internal Energy
The energy stored in a substance is called its internal energy Eint This energy may be
stored as random kinetic energy or as potential energy in each molecule (stretched
chemical bonds electrons in excited states etc) For ideal gases all states with the
same temperature will have the same Eint
First Law of Thermodynamics
The change ΔEint in the internal energy of a system is given by
intE Q W
where Q is the heat absorbed by the system and where W is the work done on the
system Hence if a system absorbs heat (and if Wge0) the internal energy increases
Likewise if the system does work (W on the system is negative) and if Qge0 the
internal energy decreases Potential Pitfall Many times people talk about work done
by the system It is the minus of W on the system Donrsquot get tripped up
Processes
Adiabatic process No heat is exchanged Q = 0 requiring that the system be insulated
from the environment A process may be approximately adiabatic if it happens so
rapidly that heat does not have time to enter or leave the system Work + or ndash is done
and ΔEint = W
Adiabatic free expansion A gas is allowed to expand into a vacuum without pushing
on anythingndashit just rushes into the vacuum In this process Q = W = 0 so the internal
energy and temperature does not change (Note the difference between an adiabatic
process and a free expansion is that NO work is done in the adiabatic free expansion)
Isobaric process The pressure is held fixed ΔP = 0 For example usually the
pressure increases when a gas is heated but if it were allowed to expand during the
2
heating process in just the right way its pressure could remain fixed In isobaric
processes both Q and W are non-zero (Work is eacy = ndashP(Vf ndash Vi)
Isovolumetric process The volume is held fixed ΔV = 0 The work done by the gas is
then zero and so we have ΔEint = Q
Isothermal process The temperature is held fixed ΔT = 0 There is no change in Eint
so Q = ndashW for isothermal processes For an ideal gas PV = nRT so PV = constant in
an isothermal process The work done on the gas is then given by
lnf
i
Vi
Vf
VW PdV nRT
V
Heat Conduction
The quantity P is defined to be the rate at which heat flows through an object and is a
power having units of watts It is analogous to electric current which is the rate at
which charge flows through an object If the flow of heat through a slab of length L
and cross-sectional area A is steady in time then P is given by the equation
h cT TdQkA
dt L
P =
where Th and Tc are the (hot and cold) temperatures of the two ends of the slab The
heat flows of course because of this temperature difference The quantity k is called
the thermal conductivity and is a constant that is characteristic of the material It is
analogous to the electrical conductivity h cT TL is sometimes called the temperature
gradient and is written dTd dTdx
R-Values
It is common to have the heat-conducting properties of materials described by their R-
values especially for insulating materials like fiberglass batting The connection
between k and R is R = Lk where L is the material thickness In this country R-values
always have units of 2ft F hourBtu
Polyurethane foam 6 (1 in thick) Air space 1 (35 in thick) Fiber glass batting 11
(35 in thick) Wood about 1 (1 in thick) Glass 09 (0125 in thick)
The heat flow rate through a slab of area A is given by
3
h cT TA
R
P
in units of Btuhour Note that A must be in square feet and the temperatures must be
in degrees Fahrenheit
Convection
Convection is the transfer of thermal energy by flow of material For instance a home
furnace doesnrsquot heat a house by waiting for the heat from the burner to slowly conduct
throughout the house instead it quickly pumps warm air to all of the rooms
Generally convection is a much faster way to transfer heat than conduction
Radiation
Electromagnetic radiation can also transfer heat When you warm yourself near a
campfire which has burned itself down into a bed of glowing embers you are
receiving radiant heat from the infrared portion of the electromagnetic spectrum The
rate at which an object emits radiant heat is given by Stefanrsquos law
4AeTP
where P is the radiated power in watts σ is a constant
8 2 45696 10 W m K
A is the surface area of the object in m2 and T is the temperature in kelvins The
constant e is called the emissivity and it varies from substance to substance A perfect
absorber (think black velvet) has e = 1 while a perfect reflector (think mirror) has e =
0 Hence black objects radiate very well while shiny ones do not Also an object that
is hotter than its surroundings radiates more energy than it absorbs whereas an object
that is cooler than its surroundings absorbs more energy than it radiates
Terminology
Transfer variables vs state variables
Energy transfer by heat as well as work done depends on the initial final and
intermediate states of the system They are transfer variables But their sum (Q + W =
4
Eint) is a state variable
Figure 205
5
Serway Chapter 21
Kinetic Theory
The ideal gas law works for all atoms and molecules at low pressure It is rather
amazing that it does Kinetic theory explains why The properties of an ideal gas can
be understood by thinking of it as N rapidly moving particles of mass m As these
particles collide with the container walls momentum is imparted to the walls which
we call the force of gas pressure In this picture the pressure is related to the average
of the square of the particle velocity 2v by
22 1( )
3 2
NP mv
V
Using the ideal gas law we obtain the average translational kinetic energy per
molecule
21 3
2 2 Bmv k T
The rms speed is then given by
2rms
3 3Bk T RTv v
m M
where M is the molecular mass in kgmol
Degrees of Freedom
Roughly speaking a degree of freedom is a way in which a molecule can store energy
For instance since there are three different directions in space along which a molecule
can move there are three degrees of freedom for the translational kinetic energy
There are also three different axes of rotation about which a polyatomic molecule can
spin so we say there are three degrees of freedom for the rotational kinetic energy
There are even degrees of freedom associated with the various ways in which a
molecule can vibrate and with the different energy levels in which the electrons of
the molecule can exist
Internal Energy and Degrees of Freedom The internal energy of an ideal gas made
up of molecules with J degrees of freedom is given by
int 2 2 B
J JE nRT Nk T
6
Heat Capacities of an Ideal Gas The heat capacity of a gas is described by means of
molar heat capacities CV and CP These are the heat capacities per mole and the
subscript V on CV means that the volume is being held constant while for CP the
pressure is held constant For example to raise the temperature of n moles of a gas
whose pressure is held constant by 10 K we would have to supply an amount of heat Q
= nCP (10) K
Molar Specific Heat of an Ideal Gas at Constant Volume
VQ nC T
3monatomic
2VC R
5diatomic
2VC R
5polyatomic
2VC
Real gases deviate from these formulas because in addition to the translational and ro-
tational degrees of freedom they also have vibrational and electronic degrees of
freedom These are unimportant at low temperatures due to quantum mechanical
effects but become increasingly important at higher temperatures The rough rule is
No of degrees of freedom
2VC R
Molar Specific Heat of an Ideal Gas at Constant Pressure
PQ nC T
P VC C R
The internal energy of an ideal gas depends only on the temperature
int VE nC T
Adiabatic Processes in an Ideal Gas
7
An adiabatic process is one in which no heat is exchanged between the system and the
environment When an ideal gas expands or contracts adiabatically not only does its
pressure change as expected from the ideal gas law but its temperature changes as
well Under these conditions the final pressure Pf can be computed from the initial
pressure Pi and from the final and initial volumes Vf and Vi by
or constantf f i iP V PV PV
where γ = CPCV The quantity γ is called the adiabatic exponent Note that this
doesnrsquot mean that the ideal gas law no longer holds it does and in fact it can be
combined with the adiabatic law for pressure given above to obtain the adiabatic law
for temperatures
1 constantTV
Compressions in sound waves are adiabatic because they happen too rapidly for any
appreciable amount of heat to flow This is why the adiabatic exponent appears in the
formula for the speed of sound in an ideal gas
RTv
M
Note that v depends only on T and not on P Because it depends only on the
temperature the speed of sound is the same in Provo as at sea level in spite of the
lower pressure here due to the difference in elevation
Equipartition of Energy
Every kind of molecule has a certain number of degrees of freedom which are
independent ways in which it can store energy Each such degree of freedom has
associated with it ndash on average ndash an energy of 12 Bk T per molecule (or 1
2 RT per mole)
(Note since a molecule has so many possible degrees of freedom it would seem that
there should be a lot of 12 sBk T to spread around But because energy is quantized
some of these degrees of freedom are not ldquoactiverdquo until the temperature becomes high
enough that 12 Bk T is as big as the lowest quantum of energy
8
Serway Chapter 22
Second Law of Thermodynamics
There are several equivalent forms of this important law
Kelvin It is not possible to change heat completely into work with no other change
taking place Or in other words there are no perfect heat engines
Clausius It is not possible for heat to flow from one body to another body at a higher
temperature with no other change taking place Or in other words there are no
perfect refrigerators
Entropy In any thermodynamic process that proceeds from one equilibrium state to
another the entropy of the system + environment either remains unchanged or
increases The total entropy never decreases This law is a bit of an oddity among the
laws of physics because it is not absolute Things are forbidden by the second law not
because it is impossible for them to happen but because it is extremely unlikely for
them to happen (See below for more information about entropy)
Reversible and Irreversible Processes
A reversible process is one which occurs so slowly that it is in thermal equilibrium (or
very nearly so) at all times A hallmark of such processes is that a motion picture of
them looks perfectly normal whether run forward or backward Imagine for instance
the slow expansion of a gas at constant temperature in a cylinder whose volume is being
increased by a slowly moving piston Run the movie backwards and what do you see
You see the slow compression of a gas at constant temperature which looks perfectly
normal
An irreversible process is one which occurs in such a way that thermal equilibrium is
not maintained throughout the process The mark of this kind of process is that a motion
picture of it looks very odd when run backward Imagine the sudden expansion of a gas
into a previously evacuated chamber because a hole was punched in the wall between a
pressurized chamber and the evacuated one Run the movie backward and what do you
see You see the gas in the soon-to-be-evacuated chamber gather itself together and
stream through a tiny hole into a chamber in which there is already plenty of gas If you
have ever seen this happen get in touch with the support group for those who have
witnessed the spontaneous combustion of large mammals by calling 1-800-PYROCOW
9
Heat Engines
Because of the vexing difference in sign between work done on and system and work done
by a system we will invent a new work variable Weng Heat engines do work and so the net
W for these engines is negative But in engineering applications hidden minus signs are
regarded as evil so for heat engines we donrsquot talk about W instead we talk about its
magnitude engW W So for heat engines the first law is
int engE Q W
But for heat pumps and refrigerators work is done on the system so we use the usual work
W when we talk about these systems
A heat engine is a machine that absorbs heat converts part of it to work and exhausts the
rest The heat must be absorbed at high temperature and exhausted at low temperature If the
absorbed heat is Qh the exhausted heat is Qc and the work done by the engine is Weng then
eng h cW Q Q
and the efficiency of the engine is defined to be
eng
h
We
Q
A perfect engine would convert the heat hQ completely into work Weng giving an effi-
ciency of e = 1 Energy conservation alone allows a perfect engine but the second law
requires e lt 1
Refrigerators and Heat Pumps
A refrigerator is a machine that absorbs heat at low temperature and exhausts it at high
temperature the ldquobackwardsrdquo heat transfer being driven by the work done on the machine
by some source of power A heat pump is a machine that either works like a refrigerator
keeping a place cold by transferring heat from this cold place to a higher temperature
environment (cooling mode like an air conditioner) or it functions as a heater
transferring heat into a warm place from a cooler one (heating mode like a window unit
that heats a house by extracting thermal energy from the cold outdoors) The coefficient
of performance of a refrigerator or of a heat pump in cooling mode is defined to be
COP cooling modecQ
W
10
For a heat pump in heating mode the coefficient of performance is
COP heating modehQ
W
Note that we donrsquot have to use the engineering work here because in these systems
work is done on the system and W is naturally positive
A perfect refrigerator would take in heat Qc and exhaust the same amount of heat without
doing any work giving an infinite coefficient of performance Energy conservation alone
allows a perfect refrigerator but the second law requires COP lt infin A perfect heat pump
in heating mode would transfer Qh into the house without doing any work and so would
also have an infinite coefficient of performance The second law forbids this too A good
coefficient of performance for a real device would be around 5 or 6
Carnot Cycle
The most efficient of all possible engines is one that uses the Carnot cycle This cycle
employs an ideal gas has no friction and operates very slowly so that the gas can be in
thermal equilibrium at all parts of the cycle This means of course that it canrsquot
possibly be built and even if it could be built it would not run fast enough to be useful
Nevertheless this cycle is very important because it gives an upper bound on the
efficiency of real engines There cannot possibly be an engine that is more efficient
than one based on the Carnot cycle This cycle consists of the following four steps
1 The ideal gas absorbs heat Qh at constant temperature Th while the gas increases its
volume The reason that heat is absorbed is that expansion tends to cool the gas but
thermal contact with the environment at Th keeps the temperature high by heat
conduction into the ideal gas
2 The ideal gas further increases its volume by an adiabatic expansion This expansion
causes the gas to cool so at the end of this part of the cycle the gas is at temperature Tc
3 The gas exhausts heat Qc at constant temperature Tc while the gas decreases in
volume The reason that heat is exhausted is that compression tends to heat the gas
but thermal contact with the environment at Tc keeps the temperature low by heat
conduction out of the ideal gas
4 The gas is adiabatically compressed back to its original volume (the volume it started
with in step 1) This compression heats the gas from Tc up to Th
11
The efficiency of a Carnot engine is given by the very simple formula
1 cC
h
Te
T
where the temperatures must in be Kelvin No real engine can be more efficient than
this
The coefficient of performance of a Carnot refrigerator or heat pump in cooling mode
is given by
COP cooling mode cC
h c
T
T T
and no real refrigerator can have a coefficient of performance greater than this
The coefficient of performance of a Carnot heat pump in heating mode is
COP heating mode hC
h c
T
T T
So why donrsquot we just use these wonderful Carnot engines and have perfect efficiency
To make the reversible steps in the cycle really reversible they would have to occur
infinitely slowly So the price you pay for making a perfect engine is that it takes
forever to get it to do any work
Entropy
The entropy of a system is defined in terms of its molecular makeup and measures
roughly the disorder of the system If the system is packed into a very small volume
then it is quite ordered and the entropy will be low If it occupies a large volume the
entropy is high (To see what this has to do with disorder note that socks in a drawer
occupy a small volume while socks on the bed in the corner by the door and
hanging from the chandelier occupy a large volume) If the system is very cold then
the molecules hardly move and may even reach out to each other and form a crystal
This is a highly ordered state and therefore has low entropy If the system is very hot
with rapidly speeding molecules crashing into the container walls and bouncing off
each other things are disordered and the entropy is high
It is possible to calculate the entropy of a system in terms of its macroscopic thermody-
namic properties ie pressure volume temperature number of moles etc The key to this
calculation is the concept of a reversible process A reversible process is one that is
12
carried out without friction and so slowly that the process can be reversed at any stage by
making an infinitesimal change in the environment of the system The slow expansion of the
gas at Th in the Carnot cycle with tiny amounts of heat being transferred to the environment
is reversible If we turned around and began slowly to compress the gas would just slowly
exhaust heat to the environment in the exact reverse way that it absorbed it during expansion
Most processes however are irreversible For example if a gas-filled box were suddenly
increased in size so that the particles were free to wander into the void created by the sudden
expansion then the gas would eventually fill the new volume uniformly at the same
temperature as before the expansion (The temperature is unchanged in this imaginary
process because the kinetic energy of the molecules would be unaffected by such an
instantaneous expansion of the container walls) This imaginary but highly thought-
stimulating process is called a free expansion and it is impossible to reverse it During the
expansion we didnrsquot push on any of the molecules so reversing this process would mean
making them go back into their original volume without pushing on them they simply will
not cooperate to this extent Another way to see that just pushing them back where they came
from does not reverse the free expansion is to think about what would happen if we just
compressed either adiabatically or isothermally An adiabatic compression back to the
original volume would heat the gas above its original temperature and an isothermal
compression would require that heat be exhausted to the environment But the free expansion
involved neither temperature changes nor heat exchanges so neither of these two processes
is the reverse of the free expansion It is simply impossible to reverse this rapid expansion in
a way that takes us back to the initial state
It is possible to calculate the change of entropy for both reversible and irreversible processes
Letrsquos consider a reversible process first In a reversible process the entropy change is given
by the formula
dQS
T
where dQ is the amount of heat added to the system during a small step of the process
The total energy change during the process may then simply be calculated by integration
f f
f i i i
dQS S S dS
T
13
(Just as in the case of energy we are mainly interested in differences rather than in
absolute magnitudes)
This integration method does not work for irreversible processes For instance in the
rapid free expansion discussed above no heat is added to the system but its disorder
obviously goes up We need to find some other way to calculate the entropy The key
is the fact that the entropy of a system depends only on its current state and not at all
on how it arrived there So to calculate the entropy change in an irreversible process
first find out what the initial and final conditions of the process are Then invent a
reversible process that takes the system from the initial state to the final state Since
the entropy depends only on the state of system and not on the process the entropy
change for the reversible process is the same as that for the irreversible process
Entropy is thus like pressure or temperature if the pressure changes from Pl to to P2
then the pressure difference is P2 - Pl regardless of how the pressure change was
made This seems obvious for pressure but not for entropy only because you arent
comfortable with entropy yet
Entropy of an Ideal Gas
For n moles of an ideal gas the difference in entropy between a state with temperature
T and volume V and some standard state with temperature To and volume Vo is given
by
ln lno o Vo o
T VS T V S T V nC nR
T V
Entropy in a Phase Change
Calculating entropy change in a phase change like melting or boiling is easy It is
QT Where Q is in the latent heat for example the flows in to cause the melting and
T is the temperature of the phase change There is NO integral to do
14
Serway Chapter 35
Angle of Reflection
If plane waves are incident on a reflecting surface with the propagation direction of
the waves making angle θ1 with the normal direction to the surface then the reflected
angle θ1prime relative to the surface normal is simply
1 1
ie the incident angles and reflected angles are the same
Refraction
If plane waves traveling through medium 1 are incident on a plane interface between
medium 1 and medium 2 then the angle of incidence of the incoming wave θ1 and
the angle of refraction of the transmitted wave θ2 are related by Snellrsquos law
2 2
1 1
sin
sin
v
v
where v1 and v2 are the wave speeds in medium 1 and medium 2 The angle of
incidence and the angle of refraction are both measured between the wave propagation
direction and the normal to the interface In terms of indices of refraction in the case
of light waves Snellrsquos law takes the more familiar form
1 1 2 2sin sinn n
where ni = cvi
Total Internal Reflection
If a wave is incident from a medium of low wave speed into a medium of high wave
speed the law of refraction requires that the angle of refraction be greater than the
angle of incidence If the angle of refraction is required to be greater than 90deg then no
refracted wave can exist and total internal reflection occurs The critical incident
angle θc beyond which total internal reflection occurs is given by
2
1
sin c
n
n
15
Dispersion of Light
In addition to the speed of light varying from material to material it also varies with
wavelength within each material This means that the index of refraction is generally a
function of wavelength
c
nv
Since the wave speed is not constant such a medium is dispersive meaning in this
context that refraction actually disperses white light into its various colors because
Snellrsquos law gives a different angle for each wavelength In most materials the
variation with wavelength is quite small but this small effect is responsible for some
of the most spectacular color effects we ever see including rainbows a flashing
crystal chandelier and the colored fire of a diamond solitaire by candlelight
16
Serway Chapter 36
Real and Virtual Images
When light rays are focused at a certain plane producing an image if a sheet of white
paper is placed there we call the image a real image The images produced by film
projectors and overhead projectors are examples of real images
When light rays appear to come from a certain location but no image is produced
when a screen is placed there we say that there is a virtual image at that location For
instance when you look in a mirror it appears that someone is behind the mirror but
a screen placed back there in the dark would show nothing Your image in the mirror
is a virtual image
Ray Tracing
There are lots of rules about how to find the images in optical systems but the best
way to keep things straight is to learn how to draw the principal rays for curved
mirrors and lenses The rays for convex and concave mirrors are shown in Fig 3615
The rays for converging and diverging thin lenses are shown in Fig 3627 You should
memorize the principal rays and know how to use them to locate images
Curved Mirrors
The focal length of a curved mirror with radius of curvature R is given by
2
Rf
The relation between object distance p image distance q and focal length f is
1 1 1
p q f
If the mirror is a diverging mirror f should be negative and if q should turn out to be
negative the image is virtual
Lateral Magnification
The lateral magnification in an optical system is defined by the ratio of the image size
to the object size
17
Image height
Object height
qM
p
for curved mirrors
As usual there are sign conventions here too but this formula just gives the
magnitude It is better to keep keep track of upright images versus inverted images by
means of ray diagrams rather than by memorizing sign conventions
Thin Lenses
The focal length of a thin lens is related to the radii of curvature of the two faces R1
and R2 of the lens by
1 2
1 1 11n
f R R
Note that this formula differs from Eq (3611) in the text by not having a minus sign
between the two R-terms We like this form better because for a simple converging
lens like a magnifying glass we just use positive values of R for both surfaces If one
of the faces is concave producing divergence use a negative value for R And if a
surface is flat use R = infin If the face is flat the radius is infinite
The relation between the image object and focal distances for a thin lens is the same
as that for a curved mirror
1 1 1
q p f
Use a negative focal length if the lens is diverging
The lateral magnification for a thin lens is the same as for a curved mirror
Image height
Object height
qM
p
Camera
The lens system in a camera projects a real image of an object onto the film (or CCD
array in a digital camera) The position of the image is adjusted to be on the film by
moving the lens into or out of the camera
18
Eye
The eye is like a camera in that a real image is formed on the retina Unlike a camera
the image position is adjusted by changing the focal length of the lens This is done by
the ciliary muscle which squeezes the lens changing its shape
Near Point The near point is the closest distance from the eye for which the lens can
focus an image on the retina It is usually 18-25 cm for young persons
Far Point The far point is the greatest distance from the eye for which the lens can
focus an image on the retina For a person with normal vision the far point is at
infinity
Nearsightedness A person is nearsighted if their far point is at some finite distance
less than infinity This condition can be corrected with a lens that takes an object at
infinity and produces a virtual image at the personrsquos far point
Farsightedness and Presbyopia A person is farsighted if their near point is too far
away for comfortable near work like reading or knitting This can be corrected by a
lens which takes an object at a normal near point distance of 18-25 cm and produces a
virtual image at the personrsquos natural near point Presbyopia involves a similar
problem which nearly all people experience as they age The ciliary muscle becomes
too weak and the lens becomes too stiff to allow the eye to provide for both near and
far vision The solution for this problem is either reading glasses or bifocal lenses
Reading classes are just weak magnifying glasses mounted on eyeglass frames
Bifocal lenses are split into upper and lower halves The lower half is a lens which
gives the proper correction for near work and the upper half is a different lens for
proper focusing at infinity
Angular Size
When an object is brought closer to the eye it appears to be larger because the image
on the retina is larger The size of this image is directly proportional to the objectrsquos
angular size which is the angle subtended by the object measured from the center of
the lens of the eye In optical instruments which are to be used with the eye the
angular size of the final image is whatrsquos important because it determines how large
the image will appear to the viewer
19
Simple Magnifier
A simple magnifier is a single converging lens or magnifying glass It takes an object
closer to the eye than a normal near point and produces a virtual image at or beyond
this near point The angular magnification is defined to be the ratio of the angular size
when viewed through the lens to the angular size of the object when viewed at the
normal near point (without aid of the lens)
Microscope
This instrument has two lenses (1) The objective is near the object being viewed and
produces a greatly magnified real image (2) The eyepiece is a simple magnifier which
the viewer uses to closely examine the image from (1)
Telescope
This instrument also has two lenses (1) The objective at the front of the telescope
takes light from a distant object and produces a real inverted image (which is rather
small) near its focal point (2) This small real image is then examined by the eyepiece
functioning as a simple magnifier to produce a virtual image with a larger angular
size
20
Serway Chapter 37
Two-Slit Interference
If light is incident on two closely spaced narrow slits a pattern of light and dark
stripes is produced beyond the slits The bright stripes or fringes are caused by
constructive interference of the two waves coming from the slits Constructive
interference occurs whenever two waves arrive at a location in phase with each other
This occurs when the distance x1 from slit 1 to a point P on the screen and the
distance x2 from slit 2 to point P differ by in integral number of wavelengths
1 2 where 0 1 2x x m m
where λ is the wavelength of the light When x1 and x2 are much larger than the slit
spacing d this condition reduces to
sind m
where θ is the angle between the direction of the incident light and the direction of the
light arriving at the screen
Thin Films
When light is partially reflected and partially transmitted by a thin film of transparent
material it is possible to have interference between the wave reflected from the front
of the film and light reflected from the back of the film (The colored reflections from
the thin film of oil on the water in a rain-soaked parking lot are an example of this
effect) It is difficult to write down formulas that will work in all cases so we will
just review the important principles here
1 If the two reflected waves are in phase with each other the film has enhanced
reflection (constructive interference) but if the two reflected waves are out of phase
with each other reflection is diminished (destructive interference) Phase shifts occur
due to reflection and due to the extra path length through the film of the wave
reflected from the back of the film
2 The phase change due to reflection is determined by the difference in index of
refraction between the two media involved in the reflection If the wave is incident
21
from a medium with a low index of refraction into a medium with a high index of
refraction a phase change of 180deg occurs and the reflected wave is inverted If
incident from high to low no phase shift occurs and the reflected wave is non-
inverted
3 The extra path length through the film of the wave reflected from the back of the
film is equal to 2t where t is the thickness of the film (The incident light is assumed
to be normal to the surface of the film) The number of wavelengths contained in the
extra path length is equal to 2tλn where λn = λn is the wavelength of the light in the
film
4 Rules for reflection from thin films
If one ray is inverted and the other is not then we have
12 constructive
2 nt m
2 destructivent m
If either both rays are inverted or both are non-inverted we have
2 constructivent m
12 destructive
2 nt m
22
Serway Chapter 38
Diffraction Grating
A diffraction grating is simply a fancy version of two-slit interference with the two
slits replaced by thousands of slits Just as in the two-slit case the bright fringes
occur at angles given by
sin md
but in this case d the distance between neighboring slits is made to be very small
The effect of having many slits instead of two is to make each bright fringe highly
localized with wide dark regions between neighboring maxima
Single Slit Diffraction
When light passes through an opening in an opaque screen an interference pattern is
produced beyond the opening To understand why we may replace the single opening
by many small coherent sources of light These many sources interfere with each
other producing a pattern known as a diffraction pattern (Note that many authors do
not distinguish between interference and diffraction treating them as interchangeable
terms) If the opening is a slit of width a then the diffraction pattern far from the slit
will have a bright central maximum with a succession of minima and weaker maxima
on either side The angle between the incident direction and the minima is given by
sin ma
where m = plusmn1 plusmn2
If the opening is circular with diameter D the angle between the incident direction
and the first minimum is given by
sin 122D
Optical Resolution and Rayleighrsquos Criterion
Two point sources can just be resolved (distinguished from each other) if the peak of
the diffraction image of the first source overlies the first minimum of the diffraction
image of the second source For circular holes of the kind usually encountered in
23
optical devices this condition is approximately satisfied when the angular separation
between the two sources as viewed from the optical instrument is greater than or equal
to the critical angle
min 122D
where λ is the wavelength of the light and where D is the diameter of the aperture in
the instrument
Polarization
We say that an electromagnetic wave is polarized if its electric field vector doesnrsquot
change direction in a random fashion The simplest kind of polarization is linear
polarization in which the electric field vector oscillates back and forth along the
same axis in space Polarized light can be produced from normal unpolarized light by
selective absorption (as in Polaroid sunglasses) by passing light through crystals that
have different indices of refraction for different polarizations (double refraction) by
scattering (the blue sky is polarized) and by reflection (glare) Polarization by
reflection occurs when light reflects from a shiny insulating (non-metallic) surface
The amount of polarization is greatest for reflection at Brewsterrsquos angle
2
1
tan P
n
n
where θP is the incidence angle of light from medium 1 onto medium 2 and where n1
and n2 are the indices of refraction for the media (In our everyday experience n1 = 1
since the light comes in through the air and n2 is the index of refraction of the shiny
insulating material producing the glare eg water glass plastic paint etc)
Malusrsquos Law
The intensity of transmitted polarized light through a perfect polarizer is related to the
incident intensity of polarized light by Malusrsquos law
2cosoI I
where Io is the intensity of the incident light and where θ is the angle between the
electric field vector in the incident wave and the transmission axis the polarizer
24
Serway Chapter 39
Principles of Relativity
All of the weirdness of relativity flows from two simple principles
(1) The laws of physics must be the same in all inertial (non-accelerating) reference
frames
(2) The speed of light in vacuum has the same value in all inertial reference frames
Note that (1) seems reasonable but (2) is very odd It says that if two space ships are
approaching each other at nearly the speed of light and a laser pulse is shot from ship
1 toward ship 2 then when the people on ship 2 measure the speed of the pulse as it
goes by it is moving at 3 times 108 ms the same as if the ships were stationary
Simultaneity
If observer 1 sees two events in her own frame as simultaneous at two different
locations a moving observer 2 will see these two events happening at different times
Relativistic gamma
The factor γ (gamma) appears regularly in the formulas of relativity
2
2
1
1 vc
where v is the relative speed between two inertial frames
Time Dilation
If observer 1 sees two events at the same location in space separated by time Δtp in his
own frame then observer 2 moving at speed v relative to observer 1 will see these
two events separated by a longer time Δt
pt t
Moving clocks run slow
Length Contraction
If observer 1 measures the length of an object along the x-direction at some instant of
time in her frame to be Lp then observer 2 moving in the x-direction at speed v will
measure the length L of the object to be shorter
pLL
25
Moving meter sticks are short
Velocity Addition
If a particle is moving at velocity vac relative to frame c and if frame c is moving at
speed vcb with respect to frame b (with both velocities directed along the same line)
what would be the velocity of the particle in frame b In classical physics the answer is
ab ac cbv v v
but if any of these velocities are an appreciable fraction of the speed of light the
answer changes
21ac cb
abac cb
v Vv
v v c
In using this equation make sure that vac has the same sign in the numerator and in the
denominator and that vbc also has the same sign in both places Note that this equation
is different (and in our opinion easier to understand) than the equations in the text
Momentum and Energy
The momentum and total energy of a particle moving at velocity v (its speed is v v )
are given by
22
2 2
22
1 1 vvc c
m mcm E mc
vp v
Rest Energy
When a particle is at rest (v = 0 and γ = 1) then E = mc2 is called the rest energy It
includes the potential energy of the particle so that if we change the potential energy
of a particle we change its rest mass
Kinetic Energy The kinetic energy is the difference between the total energy and the
rest energy
2 2K mc mc
For v ltlt c this reduces to K = mv22
General Relativity
Einsteinrsquos principle of equivalence in the theory of general relativity is In the vicinity
of a point a gravitational field is equivalent to an accelerated frame of reference in
26
the absence of gravitational effects For example there is no difference whatever in
the physics experienced by an astronaut on earth and one who is in interstellar space
far away from any stars or planets but who has just turned on her spaceshiprsquos engines
and is accelerating at 98 ms2
27
Serway Chapter 40
Photons
We now know that even though light behaves like a wave it is better described as
consisting of small packets of energy called photons The energy of a photon is related
to its frequency by
E hf
where h is Planckrsquos constant
346626 10 J sh
Photons also carry momentum given by
hf hp
c
Since the amount of energy in each photon is fixed the difference between dim light
and intense light (of the same wavelength) is that dim light consists of fewer photons
than intense light
Photoelectric Effect
In the photoelectric effect light shining onto a metal surface gives the electrons in the
metal enough energy to escape and be detected It requires a certain amount of energy
(called the work function typically = 3 ndash 5 eV) just to kick the electron out of the
metal so the light must deliver at least this much energy to an electron to produce the
effect It is observed that red light no matter how intense never produces electrons
But ultraviolet light even if quite dim will eject electrons from the metal Since red
light consists of 2 eV photons and ultraviolet light has photons with energies around
4-6 eV the photon idea explains the behavior of the photoelectric effect The
maximum energy that an ejected electron can have is
maxK hf
where f is the photon frequency The minimum frequency that light can have and cause
electrons to be ejected is called the cutoff energy and is found by setting Kmax = 0
cf h
28
Compton Effect
When high-frequency light interacts with free electrons the scattered light does not
have the same wavelength as the incident light contrary to what classical
electromagnetism would predict But the photon picture in which photons have
momentum and energy allows us to treat scattering as a collision between the photon
and the electron In this collision the electron and the scattered photon both have a
different momentum and energy than they did before And since p = hλ for a photon
if the momentum is different the wavelength will also be different
after before 1 cose
h
m c
where the angle θ is the angle between the incoming photon direction and the
direction of the scattered photon
Particles are Waves
Since photons behave like particles sometimes it is not surprising that elementary
particles can behave like waves sometimes The wavelength of a particle with
momentum p is given by
h
p
Wave-particle Duality
Both photons and elementary particles have a dual nature sometimes they behave like
particles and sometimes they behave like waves It is hard for us to comprehend the
nature of such an object by making mental pictures but experiments definitively show
that this is the case Since we have no direct experience with photons and elementary
particles (because their energies are so much smaller than the energies of the everyday
objects we encounter) it is perhaps not surprising that we have a hard time forming a
mental picture of how they behave
Electron Interference
Since an electron is both a particle and a wave just like a photon it should be able to
produce an interference pattern This is observed If an electron beam is shot at two
closely-spaced slits and if the electrons that pass through the slits are detected
downstream it is found that there are some locations where electrons are never detected
29
and others where lots of electrons are detected The pattern is exactly the same as the
one observed for light waves provided that we use the electron wavelength λ = hp in
place of the wavelength of light This pattern is observed even though each electron is
detected as a single dot on the screen Only after many such dots are collected does the
pattern emerge And if we try to understand how this effect could possibly work by
looking closely at each slit to see which one the electron came through the pattern
disappears the act of measurement destroys the interference This means that each
single electron somehow comes through both slits (as a wave would)
Uncertainty Principle
In classical physics we always imagine that the positions and momenta of moving
particles have definite values It might be hard to measure them but surely at each
instant of time a particle should be precisely located at some point in space and have a
similarly precise momentum This turns out not to be true Instead both position and
momentum are required to be uncertain with their uncertainties Δx and Δp satisfying
the Heisenberg uncertainty relation
2x p
So if the particle were known to be precisely at some particular location (so that Δx =
0) we couldnrsquot know anything about its momentum (Δp = infin) and if its momentum
were exactly known we couldnrsquot know its position
There is a similar relation involving the particlersquos energy E and the time interval Δt
over which this energy is measured
2E t
This means that energy is not actually conserved if we are considering very small time
intervals and this brief non-conservation of energy has been observed
30
Serway Chapter 41
Wave Function ψ and Probability
Quantum mechanics does not predict exactly what an electron or a photon will do Instead
it specifies the wave function or probability amplitude ψ of an electron or a photon
This wave function is a complex-valued function of space and time whose squared
magnitude is the probability density P for finding a particle at a particular place in
space at a certain time
2P
where is the complex conjugate of ψ
For example the wave function of an electron with perfectly specified momentum p
would have a wavelength given by
h
p
and its probability amplitude would be proportional to
2i x ipxe e
The corresponding probability density would then be
21ipx ip ipxP e e e
which means that the electron is equally probable to be anywhere along the x axis
This is in accord with the uncertainty principle since we specified the momentum
precisely we canrsquot have any idea about the position of the electron
As another example you have probably seen ldquofuzzy ballrdquo drawings of electron
orbitals in a chemistry book These fuzzy balls are meant to indicate the distribution
of the probability density 2
P in the orbital
Particle in a Box
A simple example in which we can calculate the wave function is the case of a particle of
mass m constrained to be inside a 1-dimensional box between x = 0 and x = L In this
case the general wave function is a linear superposition of wave functions ψn of the form
sin niE tn
n xx t A e
L
31
where A is a positive constant where n = 1 2 3 and where the energy associated
with each of the quantum states ψn is given by
22
28n
hE n
mL
This wave function is zero at x = 0 and x = L which means that the particle will never
be found at the walls of the box The wave function has maximum values in the
interior and at these places the particle is most likely to be found and it also has
places where it is zero and at these places the particle will also never be found as
expressed by the formula
2sinn x
PL
The particle in the box is interfering with itself producing a probability interference
pattern across the box just like the interference patterns we studied with light and
sound
32
Serway Chapter 42
Atoms
Many people picture an atom as a miniature solar system where electrons orbit around
a massive nucleus at the center This picture is misleading because in atoms the wave
nature of electrons dominates The electrons form 3-dimensional standing waves
(called orbitals) centered at the nucleus And if you ask what is it thatrsquos waving the
answer is ψ the probability amplitude (see Chapter 41)
Standing waves on a string can occur only for certain frequencies (the fundamental
and higher harmonics) Similarly atomic orbitals occur only for certain energies For
the hydrogen atom the energies of the orbitals have the a particularly simple form
2
1136 eVnE
n
where n = 1 2 3 is called the principal quantum number For other atoms the
determination of the orbital energies requires numerical calculation by computers
Atomic Spectra
If an electron is somehow given extra extra energy (we say that it is excited) so that it
occupies a higher orbital it will eventually ldquofallrdquo back down to a lower orbital Each
time an electron falls to a lower orbital it loses the difference in energy between the
two orbitals in the form of a photon Since the orbital energies are discrete so are the
energy differences and so are the wavelengths of the emitted light The entire set of
these discrete wavelengths is called the atomic spectrum and it is unique to each type
of atom For hydrogen the spectrum can be simply written as
H 2 2
1 1 1
f i
Rn n
where RH = 10973732 times 107 m-1 For other atoms the spectrum cannot be expressed as
a formula
Orbital Angular Momentum Electrons in atoms also have quantized values of
angular momentum The orbital quantum number ℓ specifies the value of this
quantized angular momentum through the formula
1L
33
If we want to know the value of the angular momentum along some direction in space
say the z direction the answer is not L but rather
zL m
where mℓ is another quantum number which runs from
1 1m
This quantum number is important when an atom sits in a magnetic field
Spin Angular Momentum It has been found experimentally that electrons and other
charged particles also carry internal angular momentum which we call spin
Electrons have an intrinsic spin angular momentum s along a specified axis that is
extremely quantized it can only take on 2 possible values
1
2zs s
We thus say that electrons have spin s = 12 or that electrons are rdquospin one-half parti-
cles Since s is an angular momentum it obeys the same rule as that for orbital angular
momentum namely that if its value along some axis is s then its total magnitude is
given by
31
2S s s
I know this seems weird but quantum mechanics is weird The only excuse for this
bizarre way of looking at the world is that it predicts what happens in experiments
Exclusion Principle The answer to the question of how many electrons (or any other
spin one-half particle) can be in one particular quantum state was discovered by
Wolfgang Pauli and is called the exclusion principle
ldquoNo two electrons can ever be in the same quantum state therefore no two electrons
in the same atom can have the same set of quantum numbersrdquo
This is the reason that we have atoms with different properties instead of every atom
simple having all of its electrons in the ground state All of the variety we see around
us in the world is the result of chemical differences and these differences would not
exist unless electrons obeyed this important principle The entire structure of the
periodic table (see pages 1377-1379 in Serway) is an expression of this principle
34
Serway Chapter 44
Nuclear Properties
The nucleus is composed of protons with charge +e and mass mp = 1007226 u and of
neutrons with zero charge and mass mn = 1008665 u where u is one atomic mass unit
-271 u=1660540 10 kg
The atomic number Z counts the number of protons in a nucleus while the neutron
number N counts the number of neutrons The mass number A is the sum of the two
A N Z
Protons and nuetrons have about the same mass and some times called baryons (heavy
ones) or nucleons (partcles in the nucleus) A is sometimes called the baryon
number The chemical elements are distinguished by Z ie hydrogen has Z = 1 iron
has Z = 26 uranium has Z = 92 etc But for a given Z there might be several nuclei
with different numbers of neutrons These nuclear siblings are called isotopes
These numbers are used to label nuclei according to the pattern
5626 ie FeA
Z X
denotes the isotope of iron with N = 56 ndash 26 = 30 neutrons
The nucleus is roughly spherical with a radius given approximately by
1 3 150 0where 12 10 mr r A r
Nuclear Stability
Because the positively charged protons electrically repel each other with an enormous
force at distances as small as 10-15 m there must be some really strong force that that
overcomes electrical repulsion to hold protons and neutrons together This force is
called with some lack of imagination the strong force It is a very short-range force
(it only acts over a distance of about 2 x 10-15 m) and attracts protons to protons
neutrons to neutrons and neutrons to protons But in spite of this strong nuclear force
the coulomb repulsion of the protons is still present so anything that might keep the
protons from being right next to each other would help keep the nucleus from
35
exploding This role is played by the neutrons and for nuclei with Z le 20 the stable
nuclei roughly have N = Z
For Z gt 20 the coulomb repulsion force is more powerful and more neutrons are
needed to dilute it up to about N = 15Z around Z = 80 For Z gt 83 no amount of
neutrons can help and these nuclei are unstable (radioactive) There is an ldquoisland of
(relative) stabilityrdquo around Z=90-92 (thorium and uranium respectively) These
elements have one or two isotopes with half-lifes of billions of years so there are
substantial amounts of such elements on Earth This fact makes it possible to have
practical fission devices
Radioactive Decay
There are three types of radioactive decay
Alpha decay the nucleus kicks out a helium nucleus (N = 2 Z = 2 A = 4) So if the
original nucleus is called X and the new nucleus is called Y then the decay would look
like this
4 42 2X Y+ HeA A
Z Z
Beta decay the nucleus either kicks out an electron (endash) or its positively-charged
antimatter twin the positron (e+) plus either an electron neutrino v or an electron
anti-neutrino v
1X Y+e electron decayA AZ Z v
1X Y+e positron decayA AZ Z v
A neutrino is a particle with no charge hardly any mass (much less than the electron
mass) and interacts so weakly with matter that most neutrinos upon encountering the
planet earth just pass right through it as if it werenrsquot there
Gamma decay the nucleons in the nucleus X are in an excited energy state X
(perhaps as a result of having undergone alpha or beta decay) and they drop down to a
lower energy state shedding the energy as a high frequency photon
X XA AZ Z
36
This process is exactly analogous to the way that the electrons in atoms emit photons
Decay Rate and Half Life
There is no way to predict exactly when an unstable or excited nucleus will decay but
there is an average rate at which this decay occurs called the decay constant λ The
meaning of this constant is that if there are a large number N of nuclei in a sample
then the number of decays per second that will be observed (called the decay rate R)
is R = λN In mathematical language
dMR N
dt
This simple differential equation has for its solution
0tN t N e
where N0 is the number of nuclei in the sample at time t = 0
The half-life is the time it takes for half of the nuclei in the sample to decay and is
related to the decay constant by
1 2
ln 2 0693T
Disintegration Energy
When a nucleus decays it is making a transition to an overall state of lower energy
which means according to Einsteinrsquos famous formula E = mc2 that the sum of the
masses after the decay must be less than the mass before with the lost mass appearing
as kinetic energy among the decay products For example in alpha decay this kinetic
energy called the disintegration energy Q is given by
2X YQ M M M c
37
Serway Chapter 45
Nuclear energy
There are two ways to extract energy from the nucleus fission and fusion For nuclei
with Z greater than 26 breaking the nucleus apart into pieces leads to a lower
2mc energy than the original nucleus so energy can be extracted by fission For
nuclei with Z less than 26 a lower energy is achieved by combining nuclei so energy
can be extracted by fusion
Fission
Since neutrons have no charge they are not repelled from nuclei as protons are for
this reason their behavior is the key to understanding how fission works
Fast neutrons mostly bounce off other nuclei with each collision slowing the neutron
down This elastic energy loss is most effective if the other nuclei have low mass (like
hydrogen) and these materials are called moderators because of their ability to slow
down fast neutrons
The reason that slowing neutrons is important is that slow neutrons are much more
likely to be absorbed by a nucleus which then leads to nuclear reactions of various
kinds For a few very large nuclei like uranium-235 and some plutonium isotopes
absorption of a slow neutron causes the nucleus to split into two large fragments plus
2 or 3 fast neutrons (energy is released in the process too) Thus one slow neutron
can produce energy plus 2 or 3 more neutrons which if moderated from fast to slow
can split 2 or 3 more nuclei leading to a runaway chain reaction If the reaction is
allowed to proceed unchecked it produces a large explosion If a neutron absorbing
material is added to the mix (like the cadmium in reactor control rods) it is possible to
keep the reaction under control and to extract the released energy as heat to drive
steam turbines and produce electricity
Fusion
Fusion involves mashing two nuclei together and since they are both charged and repel
each other this reaction is much harder to make go The nuclei must have enough energy to
overcome the coulomb repulsion which is why this reaction requires a high temperature
(hundreds of millions of degrees K) like that in the sun (which is burning hydrogen to
helium via fusion) or like that in the center of the fission explosion that is used to detonate
38
a hydrogen bomb
This reaction is of interest for power production in spite of this difficult temperature
requirement because of the abundance of fusion fuel on the planet There are about
012 g of deuterium ( 21H ) in every gallon of water on earth and it only costs about 4
cents to extract it The fusion energy available from this minuscule amount of
deuterium would run a 1000-Megawatt power plant for 10 seconds Water is so
abundant on earth that if fusion were to work we would have an essentially
inexhaustible source of energy
So why donrsquot we have fusion power plants Well the fuel is cheap but the match is
incredibly expensive The only way we know to control this difficult high-temperature
reaction is with large and expensive pieces of equipment involving either large
magnetic fields and complex high-power electromagnetic antennas or with gigantic
(football-fieldsized) laser facilities involving more than a hundred of the highest-
energy lasers ever built Power plants based on these current methods for controlling
fusion are unattractive to the fiscally-minded people who run the electric power
industry Hopefully better designs will be discovered as experiments continue
39
c actinium 68 Er erbium 101 Md mendelevium 104 Rf rutherfordium l aluminum 63 Eu europium 80 Hg mercury 62 Sm samarium m americium 100 Fm fermium 42 Mo molybdenum 21 Sc scandium b antimony 9 F fluorine 60 Nd neodymium 106 Sg seaborgium r argon 87 Fr francium 10 Ne neon 34 Se selenium s arsenic 64 Gd gadolinium 93 Np neptunium 14 Si silicon t astatine 31 Ga gallium 28 Ni nickel 47 Ag silver a barium 32 Ge germanium 41 Nb niobium 11 Na sodium k berkelium 79 Au gold 7 N nitrogen 38 Sr strontium e beryllium 72 Hf hafnium 102 No nobelium 16 S sulfur i bismuth 108 Hs hassium 76 Os osmium 73 Ta tantalum h bohrium 2 He helium 8 O oxygen 43 Tc technetium
boron 67 Ho holmium 46 Pd palladium 52 Te tellurium r bromine 1 H hydrogen 15 P phosphorus 65 Tb terbium d cadmium 49 In indium 78 Pt platinum 81 Tl thallium a calcium 53 I iodine 94 Pu plutonium 90 Th thorium f californium 77 Ir iridium 84 Po polonium 69 Tm thulium
carbon 26 Fe iron 19 K potassium 50 Sn tin e cerium 36 Kr krypton 59 Pr praseodymium 22 Ti titanium s cesium 57 La lanthanum 61 Pm promethium 74 W tungsten l chlorine 103 Lr lawrencium 91 Pa protactinium 92 U uranium r chromium 82 Pb lead 88 Ra radium 23 V vanadium o cobalt 3 Li lithium 86 Rn radon 54 Xe xenon u copper 71 Lu lutetium 75 Re rhenium 70 Yb ytterbium m curium 12 Mg magnesium 45 Rh rhodium 39 Y yttrium b dubnium 25 Mn manganese 37 Rb rubidium 30 Zn zinc y dysprosium 109 Mt meitnerium 44 Ru ruthenium 40 Zr zirconium s einsteinium
40
UNITS The SI units are given in parentheses Other commonly used units are given in terms of the SI units Symbols conform with the recommendations of the American National Standards Institute (ANSI) and the American Institute of Physics (AIP) Commonly used multiples of SI units
prefix symbol factor tera T 1012 giga G 109
mega M 106 kilo k 103
centi c 10-2 milli m 10-3
micro μ 10-6 nano n 10-9 pico p 10-12
femto f 10-15
length meter (m) angstrom 1 Ǻ = 10-10 m inch 1 in = 254 x 10-2 m foot 1 ft = 03048 m mile 1 mi = 1609 m
mass kilogram (kg) atomic mass unit 1 u = 1661 x 10-27 kgslug 1 slug = 1459 kg
force newton (N = kg bull ms2) dyne 1 dyn = 10-5 N pound 1 lb = 4448 N
pressure Pascal (Pa = kgm bull s2) atmosphere 1 atm = 1013 x 105 Pa poundssquare inch 1 psi = 6895 Pa cm of mercury 1 cm Hg = 1333 Pa bar 1 bar = 1000 x 105 Pa torr 1 torr = 1333 Pa
time second (s) minute 1 min = 60 s hour 1 h =3600 s
frequency hertz (Hz = s-1) radianssecond 1 rads = 12π Hz
energy joule (J = kg bull m2s2) erg 1 erg = 10-7 J electron volt 1 eV = 1602 x 10-19 J calorie 1 cal = 4187 J kilowatt-hour 1 kWh = 36 x 106 J British thermal unit 1 Btu = 1055 J
power watt (W = kg bull m2s3) horsepower 1 hp = 7457 W
charge coulomb (C = A bull s)
electric potential volt (V = kg bull m2 s3 bull A)
current ampere (A)
resistance ohm (Ω = kg bull m2s3 bull A2)
capacitance farad (F = s4 bull A2kg bull m2)
magnetic field tesla (T = kgs2 bull A) gauss 1 G = 10-4 T
magnetic flux weber (Wb = kg m2s2 bull A) maxwell 1 Mx = 10-8 Wb
magnetic inductance henry (H = kg bull m2s2 bullA2)
temperature kelvin (K) degrees Celsius 0degC = 27315 K
angle radian (rad) degree 1 deg = π180 rad revolution 1 rev = 2π rad
41
Some Physical Constantsa Quantity Symbol Valueb Atomic mass unit μ 1660 538 73 (13) x 10-27 kg
931494 013 (37) MeVc2 Avogadros number NA 6022 141 99 (47) x 1023 particlesmol Avogadrorsquos number 6022 x 1023 mol Bohr magneton
2B e
em 9274 008 99 (37) x 10-24 JT
Bohr magneton μB 927 x 10-24 JT Bohr radius
0
2
2e em e k
a 5291 772 083 (19) x 10-11 m
Boltzmanns constant B A
RNk 1380 650 3 (24) x 10-23 JK
Boltzmannrsquos constant kB 1380 x 10-23 JK Compton wavelength
C eh
m c 2426 310 215 (18) x 10-12 m
Deuteron mass md 3343 583 09 (26) x 10-27 kg 2013 553 212 71 (35) u
electron charge e 1602 x 10-19 C Electron mass me 9109 381 88 (72) x 10-31 kg
5485 799 110 (12) x 10-4 u 0510 998 902 (21) MeVc2
electron mass 911 x 10-31 kg Electron volt eV 1602 176 462 (63) x 10-19 J Elementary charge e 1602 176 462 (63) x 10-19 C Gas constant R 8314 472 (15) JKmol Gravitational constant G 6673 (10) x 10-11 Nm2kg2 Neutron mass mn 1674 927 16 (13) x 10-27 kg
1008 664 915 78 (55) u 939565 330 (38) MeVc2
neutron mass 1675 x 10-27 kg Nuclear magneton
2n p
em 5050 783 17 (20) x 10-27 JT
permeability constant μ0 1257 x 10-6 Hm permittivity constant ε0 8854 x 10-12 Fm Plancks constant h
2h
6626 068 76 (52) x 10-34 Js 1054 571 596 (82) x 10-34 Js
Planckrsquos constant h ħ
6626 x 10-34 Js 1055 x 10-34 Js
Proton mass mp 1672621 58 (13) x 10-27 kg 1007 276 466 88 (13) u 938271 998 (38) MeVc2
proton mass 1673 x 10-27 kg Rydberg constant RH 1097 373 156 854 9 (83) x 107 m-1 Speed of light in vacuum c 2997 92458 x 108 ms (exact) speed of light c 300 x 108 ms a
These constants are the values recommended in 1998 by CODATA based on a least-squares adjustment of data from different measurements For a more complete list see P J Mohr and B N Taylor Rev Mod Phys 72351 2000 b The numbers in parentheses for the values above represent the uncertainties of the last two digits
NOTE The ones in red are the ones from Appendix 13 The ones in yellow are the ones that are from the Serway book table but that were already on the Appendix 13 table
42
INDEX
Absolute zero 9
Adiabatic 17
Adiabatic exponent 18
Adiabatic process 17
Alpha decay 46
Amplitude 4
Angle of reflection 25
Angular frequency 3
Angular magnification 30
Angular momentum atomic 43
Angular size 29
Archimedes Principle 1
Atomic spectra 43
Atoms 43
Avogadrorsquos number 10
Beats 8
Bernoullirsquos Equation 2
Beta decay 46
Bifocals 29
Brewsterrsquos angle 34
British Thermal Unit Btu 11
Buoyancy 2
calorie 11
Calorie 11
Camera 28
Carnot cycle 21
Carnot efficiency 22
Celsius scale 9
Ciliary muscle 29
Coefficient of performance 21
Compton effect 39
Constructive interference 7
Continuity equation of 2
Contraction length 35
Convection 14
Curved mirrors 27
Decay constant 46
Decay rate radioactive 46
Decibel scale 5
Degrees of freedom 16
Density 1
Destructive interference 7
Diffraction single slit 33
Diffraction grating 33
Dilation time 35
Disintegration energy 47
Dispersion of light 26
Doppler effect 6
Double slit interference 31
Efficiency 20
Electron interference 39
Emissivity 15
Energy nuclear decay 47
Energy relativistic 36
43
Engineering work 20
Entropy 22
Entropy ideal gas 24
Equipartition of energy 18
Equivalence principle 36
Exclusion principle 44
Expansion thermal 9
Expansion coefficient linear 9
Eye 29
Fahrenheit scale 9
Far point 29
Farsightedness 29
First Law of Thermodynamics 12
Fission 48
Flux volume 2
Free expansion 23
Fringes 31
Fusion 48
Fusion heat of 11
Gamma relativistic 35
Gamma decay 46
General relativity 36
Half-life 46
Heat 11
Heat capacity 11
Heat conduction 13
Heat engine 20
Heat of fusion 11
Heat of vaporization 11
Heat pump 20
Heats of transformation 11
Hydrostatics 1
Ideal Gas Law 10
Images real and virtual 27
Intensity sound 5
Interference 7
Interference two-slit 31
Internal energy 12
Internal energy degrees of freedom 16
Irreversible process 19 23
Isotopes 45
Joule 11
Kelvin scale 9
Kinetic energy relativity 36
Kinetic theory 16
Latent heat 11
Length contraction 35
Linear expansion coefficient 9
Linear polarization 34
Linear superposition 7
Longitudinal wave 3
Loudness 5
Magnification lateral 27
Magnifying glass 30
Malusrsquos law 34
Microscope 30
Momentum relativistic 36
Muscle ciliary 29
44
Musical instruments 8
Musical scale 8
Near point 29
Nearsightedness 29
Nonlinear 7
Nuclear energy 48
Nuclear properties 45
Nuclear stability 45
Octave 8
Optical resolution 33
Orbital quantum number ℓ 43
Orbitals 43
Particle in a Box 41
Particles are waves 39
Pascalrsquos Principle 1
Period 3
Photoelectric effect 38
Photons 38
Pitch 8
Polarization 34
Power sound 5
Presbyopia 29
Pressure 1
Principal quantum number n 43
Principle of equivalence 36
Principle of linear superposition 7
Probability amplitude ψ 41
Processes thermodynamic 13
R-value 14
Radiation thermal 14
Radioactive decay 46
Radioactive decay rate 46
Ray tracing 27
Rayleighrsquos criterion 33
Reading glasses 29
Real image 27
Refraction 25
Refrigerator 20
Relativistic gamma 35
Relativity principles 35
Resolved for light sources 33
Rest energy 36
Reversible process 19 22
Rope wave speed 4
Second Law of Thermodynamics 19
Shock waves 6
Simple magnifier 30
Simultaneity 35
Single slit diffraction 33
Snellrsquos law 25
Sound speed 5
Specific heat 11
Spin Angular momentum 44
Standing waves 7
Stefanrsquos law 14
Strong force 45
Telescope 30
Temperature 9
45
Temperature Scales 9
Thermal conductivity 14
Thermal energy 12
Thermal expansion 9
Thin film interference 31
Thin lenses 28
Time dilation 35
Tone musical 8
Total internal reflection 25
Transverse wave 3
Traveling Waves 3
Two-slit interference 31
Uncertainty principle 40
Vaporization heat of 11
Velocity addition relativity 36
Virtual image 27
Volume flux 2
Wave function 41
Wave Function ψ and Probability 41
Wave speed 3
Wave-particle duality 39
Wavelength 3
Wavenumber 3
Wien Displacement Law 16
Work 11
2
where ρ is the density of the fluid V is the submerged volume of the object and g = 980 ms2 is
the acceleration of gravity If the object is floating on the liquid then the submerged volume V is
only that portion which is below the liquid surface
Equation of Continuity
For a fluid whose density is constant the rate of fluid flow or volume flux Φ measured in units
of m3s for a cross-section of fluid streamlines having area A and flow speed v is given by
Av
Bernoullirsquos Equation
The principle of conservation of energy for steady fluid flow states Along each streamline of the
flow the following quantity has a constant value
21
2P v gy
where y is measured upward against gravity Note that y in this equation and h in the hydrostatic
pressure equation have opposite signs This law works for gases and liquids The demos are
mostly for gases and are set up so that ρ does not vary
3
Serway Chapter 16
Longitudinal and Transverse Waves
A longitudinal wave is one in which the direction of vibration of the medium and the direction of
propagation of the wave are in the same direction Sound waves in the air in water and in solids
are longitudinal waves
A transverse wave is one in which the direction of vibration of the medium is perpendicular to
the direction of propagation of the wave Ocean waves waves on ropes and electromagnetic
waves are transverse waves
Wavelength and Wavenumber
The wavelength λ is the period in space of the wave ie it is the distance from one wave crest to
the next The wavenumber k (units reciprocal length) is related to the wavelength by the formula
2k
Frequency and Angular Frequency
The period T is the period in time of the wave ie it is the time between the arrival of one wave
crest and the next at some point in space The frequency f is the number of crests that arrive per
second which is just 1T and the angular frequency ω is 2π times the frequency Both have units
of per second (reciprocal time)
1 22f f
T T
Wave Speed
The wave speed of a wave is defined to be the the speed at which the wave crests or troughs
move through the medium This speed is related to the frequency and the wavelength by the
formulas
v fk
where k is the wavenumber of the wave v is also called the phase velocity
Traveling Waves
Traveling waves are oscillations in which the wave crests and troughs glide smoothly through the
medium Waves on oceans and lakes are traveling waves The mathematical form of a harmonic
(sinusoidal) traveling wave is
4
siny x t A kx t
where A is the amplitude of the wave As usual ω and k are related by the relation ω = kv where
v is the wave speed the speed at which the crests travel the minus sign means that the wave is
traveling to the right towards higher values of x φ is the phase and constrols where the maxima
and minima occur
Speed of Transverse Waves on a Rope
The velocity of transverse waves on a perfectly flexible rope is given by
Tv
where T is the tension in the rope and where μ is its linear mass density (mass per unit length)
Common Test Questions
A common question is to be given an equation for a wave and to be asked what is amplitude
velocity etc Another common type of question is to be given facts about a wave like its
direction velocity and magnitude at a given place and time and to be asked which of several
equations correctly describe the wave
5
Serway Chapter 17
Sound Speed in Solids Liquids and Gases
The speed of sound in a liquid or solid of bulk modulus B and volume mass density ρ
Bv
The speed of sound in air is
331 1 273v m s T C
The speed of sound in air at room temperature is about 343 ms Notice it doesnrsquot depend on
pressure or frequency
Power Intensity and Loudness
The intensity of a sound wave is defined to be the power per unit area ie wattsm2 in the wave
For spherical waves traveling away from a small source emitting waves with average power Рav
the intensity falls off with distance from the source r according to the inverse-square law
av24
Ir
Hence the intensity I2 at distance r2 is related to the intensity I1 at a different distance r1 by
21 2
22 1
I r
I r
Experiments on human hearing have shown that we hear intensity differences logarithmically so
the decibel loudness scale for sound intensity was invented The loudness β of a sound in
decibels is related to its intensity I in Wm2 by the formula
1010logo
I
I
Where Io is a sound intensity near the threshold of hearing defined to be Io = 10-12 Wm2 Note
that on this scale a sound is made 10 times more intense by adding 10 decibels Remember that
intensity is proportional to velocity squared Amplitude is the A is the previous chapterrsquos
equation for a traveling wave If you make the A 10 times as much the intensity will increase by
100 times (20 dB)
Doppler Effect
If sound waves are traveling through a medium and if either the receiver of the waves or the
source of waves is moving then the frequency received is related to the frequency emitted by
6
0
s
v vf f
v v
where fprime is the frequency detected by the observer f is the frequency emitted by the source v is
the speed of the waves vo is the speed of the observer and vs is the speed of the source This
formula assumes that the source and receiver are either moving directly toward each other or
directly away from each other To know which signs to use remember that when observer and
source approach each other the observed frequency is higher while if they move away from each
other it is lower Just examine the signs in the formula and make the answer come out right
For electromagnetic waves (light radio waves X-rays) traveling in vacuum Einsteinrsquos theory of
relativity (and careful experiments) show that the Doppler shift is given by
r
r
c vf f
c v
where vr is the relative speed between the source and the observer
For all kinds of waves (sound light etc) if the relative speed of the source and the observer is
small compared to the speed of the waves then there is a simple approximation to the Doppler
effect For example if the relative speed is 1 of the wave speed then the frequency shifts by
1 Remember that this is only an approximation
Shock Waves
When an object moves through a medium at a speed greater than the speed of waves a V-shaped
shock wave is produced The V-shaped wake behind a speeding boat is a good example of this
effect and the cone of sonic-boom behind a supersonic aircraft is another The angle that V-line
makes with the direction of travel of the source is given by
sins
v
v
where v is the wave speed and vs is the source speed
7
Serway Chapter 18
Principle of Linear Superposition
We say that a system obeys the principle of linear superposition if two or more different motions
of the system can simply be added together to find the net motion of the system Light waves
obey this principle as they propagate through the air as can be seen by shining two flashlights so
that their beams cross The beams propagate along without affecting each other (Light sabers are
a spectacular but unfortunately fictional example of systems that do not obey the principle of
superposition) Wave pulses on an ideal rope also obey this principle two different pulses pass
through each other without change Standing waves are an example of this effect being simply
the linear superposition of two traveling waves of the same frequency but moving in opposite
directions Light waves in matter do not always obey this principle For instance two powerful
laser beams could be made to cross in a piece of glass in such a way that their combined heating
effect in the crossing region could melt the glass and scatter the beams in complicated ways This
is an example of a nonlinear effect
Interference
When two or more waves are present in the same medium at the same time their net effect may
often be obtained simply by adding them at each point in the medium according to the principle
of linear superposition (Note this wonrsquot work if the medium is nonlinear) When this addition
makes the total amplitude be greater than the individual amplitudes of the various waves we say
that the interference is constructive When the addition produces cancellation and an amplitude
less than the amplitudes of the separate waves we have destructive interference
Standing Waves
A standing wave is the superposition of two identical traveling waves moving in opposite
directions Nodes are places where the two waves perfectly destructively interfere to produce
zero amplitude at all times Anti-nodes are places where the two waves perfectly constructively
interfere to produce an amplitude maximum The distance between nodes is λ2 Standing waves
on a string fixed at both ends have nodes at each end of the string Standing waves in an air
column enclosed in a tube have displacement anti-nodes at open ends of the tube and
displacement nodes at closed ends The frequency of the standing wave with the lowest possible
frequency is called the fundamental frequency Standing waves on strings or in air columns all
have frequencies which are integer multiples of the fundamental frequency and are called
8
harmonics (The fundamental is called the ldquofirst harmonicrdquo)
Beats
Beats are heard when two waves with slightly different frequencies f1 and f2 are combined The
waves constructively interfere for a number of cycles then destructively interfere for a number
of cycles We hear a periodic ldquowah-wahrdquo frequency equal to the difference of the two wave
frequencies
1 2bf f f
Musical Instruments
Musical instruments produce tones by exciting standing waves on strings (violins piano) and in
tubes (trumpet organ) The fundamental frequency of the standing wave is called the pitch of the
tone The pitch of concert A is 440 Hz by definition Two tones are an octave apart if one pitch
has twice the frequency of the other In written music there are 12 intervals in each octave with
the ratio between successive intervals equal to 2112 = 105946 The ratios for each tone in an
octave starting at A and ending at the next higher A are
A A B C C D D E F F G G A
1 10595 11225 11892 12599 13348 14142 14983 15874 16818 17818 18877 2
A musical tone is actually a superposition of the fundamental frequency and the higher
harmonics The tone quality of a musical instrument is determined by the amplitudes of the
various harmonics that it produces A violin and a trumpet can play the same pitch but they
donrsquot sound at all alike to our ears The difference between them is in the various amplitudes of
their harmonics
9
Serway Chapter 19
Temperature
Formally temperature is what is measured by a thermometer Roughly high temperature is what
we call hot and low temperature is what we call cold On the atomic level temperature refers to
the kinetic energy of the molecules A collection of molecules is called ldquohotrdquo if the molecules
have rapid random motion while a collection of molecules is called ldquocoldrdquo if the random motion
is slow When two bodies are placed in close contact with each other they exchange molecular
kinetic energy until they come to the same temperature This is the microscopic picture of the
Zeroth Law of Thermodynamics
Absolute Zero
Absolute zero is the lowest possible temperature that any object can have This is the temperature
at which all of the energy than can be removed an object has been removed (This removable
energy we call thermal energy) There is still motion at absolute zero Electrons continue to orbit
around atomic nuclei and even atoms continue to move about with a small amount of kinetic
energy but this small energy cannot be removed from the object For example at absolute zero
helium is a liquid whose atoms still move and slide past each other
Temperature Scales
Kelvin Scale Absolute zero is at T = 0 K water freezes at T = 27315 K room temperature is
around T = 295 K and water boils at T = 373 K Note we donrsquot use a deg symbol Kelvin is
prefered SI Unit
Celsius Scale Absolute zero is at T = -273degC water freezes at T = 0degC room temperature is
around T = 22degC and water boils at T = 100degC
Fahrenheit Scale Absolute zero is at T = -459degF water freezes at TF = 32degF room temperature
is around T = 72degF and water boils at TF = 212degF TF =18TC + 32 Notice that temperature
differences are the same for the Kelvin and Celsius scales
Thermal Expansion
When materials are heated they usually expand and when they are cooled they usually contract
(Water near freezing is a spectacular counterexample it works the other way around) The
coefficient of linear expansion is defined by the relation
1
i
L
L T
10
where Li is the initial length of a rod of the material and ΔL is the change in its length due to a
small temperature change ΔT The coefficient of volume expansion is defined similarly
1
i
V
V T
where Vi is the initial volume of a piece of material and where ΔV is the change in its volume due
to a small temperature change ΔT
Avogadrorsquos Number (N or NA)
One mole of any substance corresponds to 6022 times 1023 molecules The atomic mass given on the
periodic table (p A32-33 in the text) is the mass in grams of one mole of atoms For example
the atomic mass of oxygen is 15999 gmol The mass of a single atom is given by the same
number in units of atomic mass units (u) For example the mass of an oxygen atom is 15999 u
(on the average)
Ideal Gas Law (an example of an equation of state)
When the molecules of a gas are sufficiently inert and widely separated that interactions between
them are negligible we say that it is an ideal gas The pressure P volume V and temperature T
(in kelvins) of such a gas are State Variables and are related by the ideal gas law
Bor PV=NkPV nRT T
where n is the number of moles of the gas where R is the gas constant
8314 Jmol KR
where N is the number of molecules and where kB is Boltzmannrsquos constant
231380 10 J KBk
It works well for air at atmosphere pressure and even better for partial vaccuums The relative
ease of measuring pressure and the linear relationship between pressure and temperature (if V
and n are held fixed) makes an ideal gas an ideal thermometer Thermometers can be based on
properties of solids or liquids but the behavior of these materials with temperature is more
complicated
11
Serway Chapter 20
Heat
Heat is energy that flows between a system and its environment because of a tempera-
ture difference between them The units of heat are Joules as expected for an energy
Unfortunately there are several competing units of energy They are related by
1 cal 4186 J 1 Cal 4186 J 1 Btu 1054 J
Heat Capacity
There is often a simple linear relation between the heat that flows in or out of part of a
system and the temperature change that results from this energy transfer When this
linear relation holds it is convenient to define the heat capacity C and the specific
heat c as follows
For the entire object the heat Q it absorbs is given by Q = C(Tf - Ti)
For a particular material the specific heat is defined by c = Cm which is the heat
capacity per unit mass so that
Q = mc(Tf ndash Ti)
Note C has units of energy (J or Cal)(Kelvin kg)
Heats of Transformation or Latent Heat Q = plusmn mL
When a substance changes phase from solid to liquid or from liquid to gas it absorbs
heat without a change in temperature The latent heat or heat of transformation is
usually given per unit mass of the substance For example for water the heat of fusion
(melting) is L = 333 kJkg while the heat of vaporization (boiling) is L = 2260 kJkg
Note that heat for boiling is considerably bigger than melting for water You have to
be careful with signs heat is given off (negative) if you go down in temperature and
condense steam
Work
In general the small amount of work done on a system as a force Fon is exerted on it
through a vector displacement dx is given by
on xdW d F
12
But if the displacement is done very slowly (as we always assume in thermodynamics)
then the force exerted on the system and the force exerted by the system are in
balance so the force exerted by the system is ndash Fon In thermodynamics it is more
convenient to talk about the force exerted by the system so we change the above
formula for the work done on the system to
xdW d F
where F is the force exerted by the system This has confused students for more than a
century now but this is the way your book and many other books do it so you are
stuck You will need to memorize the minus sign in this definition of the work to be
able to use your textbook
There are many chances to get signs wrong in this and the next two chapters (Mosiah
2 )
When an external agent changes the volume of a gas at pressure P by a small amount
dV the (small amount) of work done on the system is given by
dW PdV
Notice that this minus sign is just what we need to make dW be positive if the external
agent compresses the gas for then dV is negative If on the other hand the external
agent gives way allowing the gas to expand against it then dV is positive and we say
that the work done on the gas is negative
The work done on the system (eg by the gas in a cylinder) in a thermodynamic
process is the area under the curve in a PV diagram It is positive for compressions
and negative for expansions If the volume of gas remains constant in a process then
no work is done by the gas
Cyclic processes are important For cyclic processes represented by PV diagrams the
magnitude of the net work during one cycle is simply the area enclosed by the cycle
on the diagram Be careful to keep track of signs when you are calculating that
enclosed area In cyclic process Q = ndashW for a cycle Put a PV diagram in your notes
1
Path A-B B-C C-D D-A A to A net
Q
W
ΔU
ΔS
Internal Energy
The energy stored in a substance is called its internal energy Eint This energy may be
stored as random kinetic energy or as potential energy in each molecule (stretched
chemical bonds electrons in excited states etc) For ideal gases all states with the
same temperature will have the same Eint
First Law of Thermodynamics
The change ΔEint in the internal energy of a system is given by
intE Q W
where Q is the heat absorbed by the system and where W is the work done on the
system Hence if a system absorbs heat (and if Wge0) the internal energy increases
Likewise if the system does work (W on the system is negative) and if Qge0 the
internal energy decreases Potential Pitfall Many times people talk about work done
by the system It is the minus of W on the system Donrsquot get tripped up
Processes
Adiabatic process No heat is exchanged Q = 0 requiring that the system be insulated
from the environment A process may be approximately adiabatic if it happens so
rapidly that heat does not have time to enter or leave the system Work + or ndash is done
and ΔEint = W
Adiabatic free expansion A gas is allowed to expand into a vacuum without pushing
on anythingndashit just rushes into the vacuum In this process Q = W = 0 so the internal
energy and temperature does not change (Note the difference between an adiabatic
process and a free expansion is that NO work is done in the adiabatic free expansion)
Isobaric process The pressure is held fixed ΔP = 0 For example usually the
pressure increases when a gas is heated but if it were allowed to expand during the
2
heating process in just the right way its pressure could remain fixed In isobaric
processes both Q and W are non-zero (Work is eacy = ndashP(Vf ndash Vi)
Isovolumetric process The volume is held fixed ΔV = 0 The work done by the gas is
then zero and so we have ΔEint = Q
Isothermal process The temperature is held fixed ΔT = 0 There is no change in Eint
so Q = ndashW for isothermal processes For an ideal gas PV = nRT so PV = constant in
an isothermal process The work done on the gas is then given by
lnf
i
Vi
Vf
VW PdV nRT
V
Heat Conduction
The quantity P is defined to be the rate at which heat flows through an object and is a
power having units of watts It is analogous to electric current which is the rate at
which charge flows through an object If the flow of heat through a slab of length L
and cross-sectional area A is steady in time then P is given by the equation
h cT TdQkA
dt L
P =
where Th and Tc are the (hot and cold) temperatures of the two ends of the slab The
heat flows of course because of this temperature difference The quantity k is called
the thermal conductivity and is a constant that is characteristic of the material It is
analogous to the electrical conductivity h cT TL is sometimes called the temperature
gradient and is written dTd dTdx
R-Values
It is common to have the heat-conducting properties of materials described by their R-
values especially for insulating materials like fiberglass batting The connection
between k and R is R = Lk where L is the material thickness In this country R-values
always have units of 2ft F hourBtu
Polyurethane foam 6 (1 in thick) Air space 1 (35 in thick) Fiber glass batting 11
(35 in thick) Wood about 1 (1 in thick) Glass 09 (0125 in thick)
The heat flow rate through a slab of area A is given by
3
h cT TA
R
P
in units of Btuhour Note that A must be in square feet and the temperatures must be
in degrees Fahrenheit
Convection
Convection is the transfer of thermal energy by flow of material For instance a home
furnace doesnrsquot heat a house by waiting for the heat from the burner to slowly conduct
throughout the house instead it quickly pumps warm air to all of the rooms
Generally convection is a much faster way to transfer heat than conduction
Radiation
Electromagnetic radiation can also transfer heat When you warm yourself near a
campfire which has burned itself down into a bed of glowing embers you are
receiving radiant heat from the infrared portion of the electromagnetic spectrum The
rate at which an object emits radiant heat is given by Stefanrsquos law
4AeTP
where P is the radiated power in watts σ is a constant
8 2 45696 10 W m K
A is the surface area of the object in m2 and T is the temperature in kelvins The
constant e is called the emissivity and it varies from substance to substance A perfect
absorber (think black velvet) has e = 1 while a perfect reflector (think mirror) has e =
0 Hence black objects radiate very well while shiny ones do not Also an object that
is hotter than its surroundings radiates more energy than it absorbs whereas an object
that is cooler than its surroundings absorbs more energy than it radiates
Terminology
Transfer variables vs state variables
Energy transfer by heat as well as work done depends on the initial final and
intermediate states of the system They are transfer variables But their sum (Q + W =
4
Eint) is a state variable
Figure 205
5
Serway Chapter 21
Kinetic Theory
The ideal gas law works for all atoms and molecules at low pressure It is rather
amazing that it does Kinetic theory explains why The properties of an ideal gas can
be understood by thinking of it as N rapidly moving particles of mass m As these
particles collide with the container walls momentum is imparted to the walls which
we call the force of gas pressure In this picture the pressure is related to the average
of the square of the particle velocity 2v by
22 1( )
3 2
NP mv
V
Using the ideal gas law we obtain the average translational kinetic energy per
molecule
21 3
2 2 Bmv k T
The rms speed is then given by
2rms
3 3Bk T RTv v
m M
where M is the molecular mass in kgmol
Degrees of Freedom
Roughly speaking a degree of freedom is a way in which a molecule can store energy
For instance since there are three different directions in space along which a molecule
can move there are three degrees of freedom for the translational kinetic energy
There are also three different axes of rotation about which a polyatomic molecule can
spin so we say there are three degrees of freedom for the rotational kinetic energy
There are even degrees of freedom associated with the various ways in which a
molecule can vibrate and with the different energy levels in which the electrons of
the molecule can exist
Internal Energy and Degrees of Freedom The internal energy of an ideal gas made
up of molecules with J degrees of freedom is given by
int 2 2 B
J JE nRT Nk T
6
Heat Capacities of an Ideal Gas The heat capacity of a gas is described by means of
molar heat capacities CV and CP These are the heat capacities per mole and the
subscript V on CV means that the volume is being held constant while for CP the
pressure is held constant For example to raise the temperature of n moles of a gas
whose pressure is held constant by 10 K we would have to supply an amount of heat Q
= nCP (10) K
Molar Specific Heat of an Ideal Gas at Constant Volume
VQ nC T
3monatomic
2VC R
5diatomic
2VC R
5polyatomic
2VC
Real gases deviate from these formulas because in addition to the translational and ro-
tational degrees of freedom they also have vibrational and electronic degrees of
freedom These are unimportant at low temperatures due to quantum mechanical
effects but become increasingly important at higher temperatures The rough rule is
No of degrees of freedom
2VC R
Molar Specific Heat of an Ideal Gas at Constant Pressure
PQ nC T
P VC C R
The internal energy of an ideal gas depends only on the temperature
int VE nC T
Adiabatic Processes in an Ideal Gas
7
An adiabatic process is one in which no heat is exchanged between the system and the
environment When an ideal gas expands or contracts adiabatically not only does its
pressure change as expected from the ideal gas law but its temperature changes as
well Under these conditions the final pressure Pf can be computed from the initial
pressure Pi and from the final and initial volumes Vf and Vi by
or constantf f i iP V PV PV
where γ = CPCV The quantity γ is called the adiabatic exponent Note that this
doesnrsquot mean that the ideal gas law no longer holds it does and in fact it can be
combined with the adiabatic law for pressure given above to obtain the adiabatic law
for temperatures
1 constantTV
Compressions in sound waves are adiabatic because they happen too rapidly for any
appreciable amount of heat to flow This is why the adiabatic exponent appears in the
formula for the speed of sound in an ideal gas
RTv
M
Note that v depends only on T and not on P Because it depends only on the
temperature the speed of sound is the same in Provo as at sea level in spite of the
lower pressure here due to the difference in elevation
Equipartition of Energy
Every kind of molecule has a certain number of degrees of freedom which are
independent ways in which it can store energy Each such degree of freedom has
associated with it ndash on average ndash an energy of 12 Bk T per molecule (or 1
2 RT per mole)
(Note since a molecule has so many possible degrees of freedom it would seem that
there should be a lot of 12 sBk T to spread around But because energy is quantized
some of these degrees of freedom are not ldquoactiverdquo until the temperature becomes high
enough that 12 Bk T is as big as the lowest quantum of energy
8
Serway Chapter 22
Second Law of Thermodynamics
There are several equivalent forms of this important law
Kelvin It is not possible to change heat completely into work with no other change
taking place Or in other words there are no perfect heat engines
Clausius It is not possible for heat to flow from one body to another body at a higher
temperature with no other change taking place Or in other words there are no
perfect refrigerators
Entropy In any thermodynamic process that proceeds from one equilibrium state to
another the entropy of the system + environment either remains unchanged or
increases The total entropy never decreases This law is a bit of an oddity among the
laws of physics because it is not absolute Things are forbidden by the second law not
because it is impossible for them to happen but because it is extremely unlikely for
them to happen (See below for more information about entropy)
Reversible and Irreversible Processes
A reversible process is one which occurs so slowly that it is in thermal equilibrium (or
very nearly so) at all times A hallmark of such processes is that a motion picture of
them looks perfectly normal whether run forward or backward Imagine for instance
the slow expansion of a gas at constant temperature in a cylinder whose volume is being
increased by a slowly moving piston Run the movie backwards and what do you see
You see the slow compression of a gas at constant temperature which looks perfectly
normal
An irreversible process is one which occurs in such a way that thermal equilibrium is
not maintained throughout the process The mark of this kind of process is that a motion
picture of it looks very odd when run backward Imagine the sudden expansion of a gas
into a previously evacuated chamber because a hole was punched in the wall between a
pressurized chamber and the evacuated one Run the movie backward and what do you
see You see the gas in the soon-to-be-evacuated chamber gather itself together and
stream through a tiny hole into a chamber in which there is already plenty of gas If you
have ever seen this happen get in touch with the support group for those who have
witnessed the spontaneous combustion of large mammals by calling 1-800-PYROCOW
9
Heat Engines
Because of the vexing difference in sign between work done on and system and work done
by a system we will invent a new work variable Weng Heat engines do work and so the net
W for these engines is negative But in engineering applications hidden minus signs are
regarded as evil so for heat engines we donrsquot talk about W instead we talk about its
magnitude engW W So for heat engines the first law is
int engE Q W
But for heat pumps and refrigerators work is done on the system so we use the usual work
W when we talk about these systems
A heat engine is a machine that absorbs heat converts part of it to work and exhausts the
rest The heat must be absorbed at high temperature and exhausted at low temperature If the
absorbed heat is Qh the exhausted heat is Qc and the work done by the engine is Weng then
eng h cW Q Q
and the efficiency of the engine is defined to be
eng
h
We
Q
A perfect engine would convert the heat hQ completely into work Weng giving an effi-
ciency of e = 1 Energy conservation alone allows a perfect engine but the second law
requires e lt 1
Refrigerators and Heat Pumps
A refrigerator is a machine that absorbs heat at low temperature and exhausts it at high
temperature the ldquobackwardsrdquo heat transfer being driven by the work done on the machine
by some source of power A heat pump is a machine that either works like a refrigerator
keeping a place cold by transferring heat from this cold place to a higher temperature
environment (cooling mode like an air conditioner) or it functions as a heater
transferring heat into a warm place from a cooler one (heating mode like a window unit
that heats a house by extracting thermal energy from the cold outdoors) The coefficient
of performance of a refrigerator or of a heat pump in cooling mode is defined to be
COP cooling modecQ
W
10
For a heat pump in heating mode the coefficient of performance is
COP heating modehQ
W
Note that we donrsquot have to use the engineering work here because in these systems
work is done on the system and W is naturally positive
A perfect refrigerator would take in heat Qc and exhaust the same amount of heat without
doing any work giving an infinite coefficient of performance Energy conservation alone
allows a perfect refrigerator but the second law requires COP lt infin A perfect heat pump
in heating mode would transfer Qh into the house without doing any work and so would
also have an infinite coefficient of performance The second law forbids this too A good
coefficient of performance for a real device would be around 5 or 6
Carnot Cycle
The most efficient of all possible engines is one that uses the Carnot cycle This cycle
employs an ideal gas has no friction and operates very slowly so that the gas can be in
thermal equilibrium at all parts of the cycle This means of course that it canrsquot
possibly be built and even if it could be built it would not run fast enough to be useful
Nevertheless this cycle is very important because it gives an upper bound on the
efficiency of real engines There cannot possibly be an engine that is more efficient
than one based on the Carnot cycle This cycle consists of the following four steps
1 The ideal gas absorbs heat Qh at constant temperature Th while the gas increases its
volume The reason that heat is absorbed is that expansion tends to cool the gas but
thermal contact with the environment at Th keeps the temperature high by heat
conduction into the ideal gas
2 The ideal gas further increases its volume by an adiabatic expansion This expansion
causes the gas to cool so at the end of this part of the cycle the gas is at temperature Tc
3 The gas exhausts heat Qc at constant temperature Tc while the gas decreases in
volume The reason that heat is exhausted is that compression tends to heat the gas
but thermal contact with the environment at Tc keeps the temperature low by heat
conduction out of the ideal gas
4 The gas is adiabatically compressed back to its original volume (the volume it started
with in step 1) This compression heats the gas from Tc up to Th
11
The efficiency of a Carnot engine is given by the very simple formula
1 cC
h
Te
T
where the temperatures must in be Kelvin No real engine can be more efficient than
this
The coefficient of performance of a Carnot refrigerator or heat pump in cooling mode
is given by
COP cooling mode cC
h c
T
T T
and no real refrigerator can have a coefficient of performance greater than this
The coefficient of performance of a Carnot heat pump in heating mode is
COP heating mode hC
h c
T
T T
So why donrsquot we just use these wonderful Carnot engines and have perfect efficiency
To make the reversible steps in the cycle really reversible they would have to occur
infinitely slowly So the price you pay for making a perfect engine is that it takes
forever to get it to do any work
Entropy
The entropy of a system is defined in terms of its molecular makeup and measures
roughly the disorder of the system If the system is packed into a very small volume
then it is quite ordered and the entropy will be low If it occupies a large volume the
entropy is high (To see what this has to do with disorder note that socks in a drawer
occupy a small volume while socks on the bed in the corner by the door and
hanging from the chandelier occupy a large volume) If the system is very cold then
the molecules hardly move and may even reach out to each other and form a crystal
This is a highly ordered state and therefore has low entropy If the system is very hot
with rapidly speeding molecules crashing into the container walls and bouncing off
each other things are disordered and the entropy is high
It is possible to calculate the entropy of a system in terms of its macroscopic thermody-
namic properties ie pressure volume temperature number of moles etc The key to this
calculation is the concept of a reversible process A reversible process is one that is
12
carried out without friction and so slowly that the process can be reversed at any stage by
making an infinitesimal change in the environment of the system The slow expansion of the
gas at Th in the Carnot cycle with tiny amounts of heat being transferred to the environment
is reversible If we turned around and began slowly to compress the gas would just slowly
exhaust heat to the environment in the exact reverse way that it absorbed it during expansion
Most processes however are irreversible For example if a gas-filled box were suddenly
increased in size so that the particles were free to wander into the void created by the sudden
expansion then the gas would eventually fill the new volume uniformly at the same
temperature as before the expansion (The temperature is unchanged in this imaginary
process because the kinetic energy of the molecules would be unaffected by such an
instantaneous expansion of the container walls) This imaginary but highly thought-
stimulating process is called a free expansion and it is impossible to reverse it During the
expansion we didnrsquot push on any of the molecules so reversing this process would mean
making them go back into their original volume without pushing on them they simply will
not cooperate to this extent Another way to see that just pushing them back where they came
from does not reverse the free expansion is to think about what would happen if we just
compressed either adiabatically or isothermally An adiabatic compression back to the
original volume would heat the gas above its original temperature and an isothermal
compression would require that heat be exhausted to the environment But the free expansion
involved neither temperature changes nor heat exchanges so neither of these two processes
is the reverse of the free expansion It is simply impossible to reverse this rapid expansion in
a way that takes us back to the initial state
It is possible to calculate the change of entropy for both reversible and irreversible processes
Letrsquos consider a reversible process first In a reversible process the entropy change is given
by the formula
dQS
T
where dQ is the amount of heat added to the system during a small step of the process
The total energy change during the process may then simply be calculated by integration
f f
f i i i
dQS S S dS
T
13
(Just as in the case of energy we are mainly interested in differences rather than in
absolute magnitudes)
This integration method does not work for irreversible processes For instance in the
rapid free expansion discussed above no heat is added to the system but its disorder
obviously goes up We need to find some other way to calculate the entropy The key
is the fact that the entropy of a system depends only on its current state and not at all
on how it arrived there So to calculate the entropy change in an irreversible process
first find out what the initial and final conditions of the process are Then invent a
reversible process that takes the system from the initial state to the final state Since
the entropy depends only on the state of system and not on the process the entropy
change for the reversible process is the same as that for the irreversible process
Entropy is thus like pressure or temperature if the pressure changes from Pl to to P2
then the pressure difference is P2 - Pl regardless of how the pressure change was
made This seems obvious for pressure but not for entropy only because you arent
comfortable with entropy yet
Entropy of an Ideal Gas
For n moles of an ideal gas the difference in entropy between a state with temperature
T and volume V and some standard state with temperature To and volume Vo is given
by
ln lno o Vo o
T VS T V S T V nC nR
T V
Entropy in a Phase Change
Calculating entropy change in a phase change like melting or boiling is easy It is
QT Where Q is in the latent heat for example the flows in to cause the melting and
T is the temperature of the phase change There is NO integral to do
14
Serway Chapter 35
Angle of Reflection
If plane waves are incident on a reflecting surface with the propagation direction of
the waves making angle θ1 with the normal direction to the surface then the reflected
angle θ1prime relative to the surface normal is simply
1 1
ie the incident angles and reflected angles are the same
Refraction
If plane waves traveling through medium 1 are incident on a plane interface between
medium 1 and medium 2 then the angle of incidence of the incoming wave θ1 and
the angle of refraction of the transmitted wave θ2 are related by Snellrsquos law
2 2
1 1
sin
sin
v
v
where v1 and v2 are the wave speeds in medium 1 and medium 2 The angle of
incidence and the angle of refraction are both measured between the wave propagation
direction and the normal to the interface In terms of indices of refraction in the case
of light waves Snellrsquos law takes the more familiar form
1 1 2 2sin sinn n
where ni = cvi
Total Internal Reflection
If a wave is incident from a medium of low wave speed into a medium of high wave
speed the law of refraction requires that the angle of refraction be greater than the
angle of incidence If the angle of refraction is required to be greater than 90deg then no
refracted wave can exist and total internal reflection occurs The critical incident
angle θc beyond which total internal reflection occurs is given by
2
1
sin c
n
n
15
Dispersion of Light
In addition to the speed of light varying from material to material it also varies with
wavelength within each material This means that the index of refraction is generally a
function of wavelength
c
nv
Since the wave speed is not constant such a medium is dispersive meaning in this
context that refraction actually disperses white light into its various colors because
Snellrsquos law gives a different angle for each wavelength In most materials the
variation with wavelength is quite small but this small effect is responsible for some
of the most spectacular color effects we ever see including rainbows a flashing
crystal chandelier and the colored fire of a diamond solitaire by candlelight
16
Serway Chapter 36
Real and Virtual Images
When light rays are focused at a certain plane producing an image if a sheet of white
paper is placed there we call the image a real image The images produced by film
projectors and overhead projectors are examples of real images
When light rays appear to come from a certain location but no image is produced
when a screen is placed there we say that there is a virtual image at that location For
instance when you look in a mirror it appears that someone is behind the mirror but
a screen placed back there in the dark would show nothing Your image in the mirror
is a virtual image
Ray Tracing
There are lots of rules about how to find the images in optical systems but the best
way to keep things straight is to learn how to draw the principal rays for curved
mirrors and lenses The rays for convex and concave mirrors are shown in Fig 3615
The rays for converging and diverging thin lenses are shown in Fig 3627 You should
memorize the principal rays and know how to use them to locate images
Curved Mirrors
The focal length of a curved mirror with radius of curvature R is given by
2
Rf
The relation between object distance p image distance q and focal length f is
1 1 1
p q f
If the mirror is a diverging mirror f should be negative and if q should turn out to be
negative the image is virtual
Lateral Magnification
The lateral magnification in an optical system is defined by the ratio of the image size
to the object size
17
Image height
Object height
qM
p
for curved mirrors
As usual there are sign conventions here too but this formula just gives the
magnitude It is better to keep keep track of upright images versus inverted images by
means of ray diagrams rather than by memorizing sign conventions
Thin Lenses
The focal length of a thin lens is related to the radii of curvature of the two faces R1
and R2 of the lens by
1 2
1 1 11n
f R R
Note that this formula differs from Eq (3611) in the text by not having a minus sign
between the two R-terms We like this form better because for a simple converging
lens like a magnifying glass we just use positive values of R for both surfaces If one
of the faces is concave producing divergence use a negative value for R And if a
surface is flat use R = infin If the face is flat the radius is infinite
The relation between the image object and focal distances for a thin lens is the same
as that for a curved mirror
1 1 1
q p f
Use a negative focal length if the lens is diverging
The lateral magnification for a thin lens is the same as for a curved mirror
Image height
Object height
qM
p
Camera
The lens system in a camera projects a real image of an object onto the film (or CCD
array in a digital camera) The position of the image is adjusted to be on the film by
moving the lens into or out of the camera
18
Eye
The eye is like a camera in that a real image is formed on the retina Unlike a camera
the image position is adjusted by changing the focal length of the lens This is done by
the ciliary muscle which squeezes the lens changing its shape
Near Point The near point is the closest distance from the eye for which the lens can
focus an image on the retina It is usually 18-25 cm for young persons
Far Point The far point is the greatest distance from the eye for which the lens can
focus an image on the retina For a person with normal vision the far point is at
infinity
Nearsightedness A person is nearsighted if their far point is at some finite distance
less than infinity This condition can be corrected with a lens that takes an object at
infinity and produces a virtual image at the personrsquos far point
Farsightedness and Presbyopia A person is farsighted if their near point is too far
away for comfortable near work like reading or knitting This can be corrected by a
lens which takes an object at a normal near point distance of 18-25 cm and produces a
virtual image at the personrsquos natural near point Presbyopia involves a similar
problem which nearly all people experience as they age The ciliary muscle becomes
too weak and the lens becomes too stiff to allow the eye to provide for both near and
far vision The solution for this problem is either reading glasses or bifocal lenses
Reading classes are just weak magnifying glasses mounted on eyeglass frames
Bifocal lenses are split into upper and lower halves The lower half is a lens which
gives the proper correction for near work and the upper half is a different lens for
proper focusing at infinity
Angular Size
When an object is brought closer to the eye it appears to be larger because the image
on the retina is larger The size of this image is directly proportional to the objectrsquos
angular size which is the angle subtended by the object measured from the center of
the lens of the eye In optical instruments which are to be used with the eye the
angular size of the final image is whatrsquos important because it determines how large
the image will appear to the viewer
19
Simple Magnifier
A simple magnifier is a single converging lens or magnifying glass It takes an object
closer to the eye than a normal near point and produces a virtual image at or beyond
this near point The angular magnification is defined to be the ratio of the angular size
when viewed through the lens to the angular size of the object when viewed at the
normal near point (without aid of the lens)
Microscope
This instrument has two lenses (1) The objective is near the object being viewed and
produces a greatly magnified real image (2) The eyepiece is a simple magnifier which
the viewer uses to closely examine the image from (1)
Telescope
This instrument also has two lenses (1) The objective at the front of the telescope
takes light from a distant object and produces a real inverted image (which is rather
small) near its focal point (2) This small real image is then examined by the eyepiece
functioning as a simple magnifier to produce a virtual image with a larger angular
size
20
Serway Chapter 37
Two-Slit Interference
If light is incident on two closely spaced narrow slits a pattern of light and dark
stripes is produced beyond the slits The bright stripes or fringes are caused by
constructive interference of the two waves coming from the slits Constructive
interference occurs whenever two waves arrive at a location in phase with each other
This occurs when the distance x1 from slit 1 to a point P on the screen and the
distance x2 from slit 2 to point P differ by in integral number of wavelengths
1 2 where 0 1 2x x m m
where λ is the wavelength of the light When x1 and x2 are much larger than the slit
spacing d this condition reduces to
sind m
where θ is the angle between the direction of the incident light and the direction of the
light arriving at the screen
Thin Films
When light is partially reflected and partially transmitted by a thin film of transparent
material it is possible to have interference between the wave reflected from the front
of the film and light reflected from the back of the film (The colored reflections from
the thin film of oil on the water in a rain-soaked parking lot are an example of this
effect) It is difficult to write down formulas that will work in all cases so we will
just review the important principles here
1 If the two reflected waves are in phase with each other the film has enhanced
reflection (constructive interference) but if the two reflected waves are out of phase
with each other reflection is diminished (destructive interference) Phase shifts occur
due to reflection and due to the extra path length through the film of the wave
reflected from the back of the film
2 The phase change due to reflection is determined by the difference in index of
refraction between the two media involved in the reflection If the wave is incident
21
from a medium with a low index of refraction into a medium with a high index of
refraction a phase change of 180deg occurs and the reflected wave is inverted If
incident from high to low no phase shift occurs and the reflected wave is non-
inverted
3 The extra path length through the film of the wave reflected from the back of the
film is equal to 2t where t is the thickness of the film (The incident light is assumed
to be normal to the surface of the film) The number of wavelengths contained in the
extra path length is equal to 2tλn where λn = λn is the wavelength of the light in the
film
4 Rules for reflection from thin films
If one ray is inverted and the other is not then we have
12 constructive
2 nt m
2 destructivent m
If either both rays are inverted or both are non-inverted we have
2 constructivent m
12 destructive
2 nt m
22
Serway Chapter 38
Diffraction Grating
A diffraction grating is simply a fancy version of two-slit interference with the two
slits replaced by thousands of slits Just as in the two-slit case the bright fringes
occur at angles given by
sin md
but in this case d the distance between neighboring slits is made to be very small
The effect of having many slits instead of two is to make each bright fringe highly
localized with wide dark regions between neighboring maxima
Single Slit Diffraction
When light passes through an opening in an opaque screen an interference pattern is
produced beyond the opening To understand why we may replace the single opening
by many small coherent sources of light These many sources interfere with each
other producing a pattern known as a diffraction pattern (Note that many authors do
not distinguish between interference and diffraction treating them as interchangeable
terms) If the opening is a slit of width a then the diffraction pattern far from the slit
will have a bright central maximum with a succession of minima and weaker maxima
on either side The angle between the incident direction and the minima is given by
sin ma
where m = plusmn1 plusmn2
If the opening is circular with diameter D the angle between the incident direction
and the first minimum is given by
sin 122D
Optical Resolution and Rayleighrsquos Criterion
Two point sources can just be resolved (distinguished from each other) if the peak of
the diffraction image of the first source overlies the first minimum of the diffraction
image of the second source For circular holes of the kind usually encountered in
23
optical devices this condition is approximately satisfied when the angular separation
between the two sources as viewed from the optical instrument is greater than or equal
to the critical angle
min 122D
where λ is the wavelength of the light and where D is the diameter of the aperture in
the instrument
Polarization
We say that an electromagnetic wave is polarized if its electric field vector doesnrsquot
change direction in a random fashion The simplest kind of polarization is linear
polarization in which the electric field vector oscillates back and forth along the
same axis in space Polarized light can be produced from normal unpolarized light by
selective absorption (as in Polaroid sunglasses) by passing light through crystals that
have different indices of refraction for different polarizations (double refraction) by
scattering (the blue sky is polarized) and by reflection (glare) Polarization by
reflection occurs when light reflects from a shiny insulating (non-metallic) surface
The amount of polarization is greatest for reflection at Brewsterrsquos angle
2
1
tan P
n
n
where θP is the incidence angle of light from medium 1 onto medium 2 and where n1
and n2 are the indices of refraction for the media (In our everyday experience n1 = 1
since the light comes in through the air and n2 is the index of refraction of the shiny
insulating material producing the glare eg water glass plastic paint etc)
Malusrsquos Law
The intensity of transmitted polarized light through a perfect polarizer is related to the
incident intensity of polarized light by Malusrsquos law
2cosoI I
where Io is the intensity of the incident light and where θ is the angle between the
electric field vector in the incident wave and the transmission axis the polarizer
24
Serway Chapter 39
Principles of Relativity
All of the weirdness of relativity flows from two simple principles
(1) The laws of physics must be the same in all inertial (non-accelerating) reference
frames
(2) The speed of light in vacuum has the same value in all inertial reference frames
Note that (1) seems reasonable but (2) is very odd It says that if two space ships are
approaching each other at nearly the speed of light and a laser pulse is shot from ship
1 toward ship 2 then when the people on ship 2 measure the speed of the pulse as it
goes by it is moving at 3 times 108 ms the same as if the ships were stationary
Simultaneity
If observer 1 sees two events in her own frame as simultaneous at two different
locations a moving observer 2 will see these two events happening at different times
Relativistic gamma
The factor γ (gamma) appears regularly in the formulas of relativity
2
2
1
1 vc
where v is the relative speed between two inertial frames
Time Dilation
If observer 1 sees two events at the same location in space separated by time Δtp in his
own frame then observer 2 moving at speed v relative to observer 1 will see these
two events separated by a longer time Δt
pt t
Moving clocks run slow
Length Contraction
If observer 1 measures the length of an object along the x-direction at some instant of
time in her frame to be Lp then observer 2 moving in the x-direction at speed v will
measure the length L of the object to be shorter
pLL
25
Moving meter sticks are short
Velocity Addition
If a particle is moving at velocity vac relative to frame c and if frame c is moving at
speed vcb with respect to frame b (with both velocities directed along the same line)
what would be the velocity of the particle in frame b In classical physics the answer is
ab ac cbv v v
but if any of these velocities are an appreciable fraction of the speed of light the
answer changes
21ac cb
abac cb
v Vv
v v c
In using this equation make sure that vac has the same sign in the numerator and in the
denominator and that vbc also has the same sign in both places Note that this equation
is different (and in our opinion easier to understand) than the equations in the text
Momentum and Energy
The momentum and total energy of a particle moving at velocity v (its speed is v v )
are given by
22
2 2
22
1 1 vvc c
m mcm E mc
vp v
Rest Energy
When a particle is at rest (v = 0 and γ = 1) then E = mc2 is called the rest energy It
includes the potential energy of the particle so that if we change the potential energy
of a particle we change its rest mass
Kinetic Energy The kinetic energy is the difference between the total energy and the
rest energy
2 2K mc mc
For v ltlt c this reduces to K = mv22
General Relativity
Einsteinrsquos principle of equivalence in the theory of general relativity is In the vicinity
of a point a gravitational field is equivalent to an accelerated frame of reference in
26
the absence of gravitational effects For example there is no difference whatever in
the physics experienced by an astronaut on earth and one who is in interstellar space
far away from any stars or planets but who has just turned on her spaceshiprsquos engines
and is accelerating at 98 ms2
27
Serway Chapter 40
Photons
We now know that even though light behaves like a wave it is better described as
consisting of small packets of energy called photons The energy of a photon is related
to its frequency by
E hf
where h is Planckrsquos constant
346626 10 J sh
Photons also carry momentum given by
hf hp
c
Since the amount of energy in each photon is fixed the difference between dim light
and intense light (of the same wavelength) is that dim light consists of fewer photons
than intense light
Photoelectric Effect
In the photoelectric effect light shining onto a metal surface gives the electrons in the
metal enough energy to escape and be detected It requires a certain amount of energy
(called the work function typically = 3 ndash 5 eV) just to kick the electron out of the
metal so the light must deliver at least this much energy to an electron to produce the
effect It is observed that red light no matter how intense never produces electrons
But ultraviolet light even if quite dim will eject electrons from the metal Since red
light consists of 2 eV photons and ultraviolet light has photons with energies around
4-6 eV the photon idea explains the behavior of the photoelectric effect The
maximum energy that an ejected electron can have is
maxK hf
where f is the photon frequency The minimum frequency that light can have and cause
electrons to be ejected is called the cutoff energy and is found by setting Kmax = 0
cf h
28
Compton Effect
When high-frequency light interacts with free electrons the scattered light does not
have the same wavelength as the incident light contrary to what classical
electromagnetism would predict But the photon picture in which photons have
momentum and energy allows us to treat scattering as a collision between the photon
and the electron In this collision the electron and the scattered photon both have a
different momentum and energy than they did before And since p = hλ for a photon
if the momentum is different the wavelength will also be different
after before 1 cose
h
m c
where the angle θ is the angle between the incoming photon direction and the
direction of the scattered photon
Particles are Waves
Since photons behave like particles sometimes it is not surprising that elementary
particles can behave like waves sometimes The wavelength of a particle with
momentum p is given by
h
p
Wave-particle Duality
Both photons and elementary particles have a dual nature sometimes they behave like
particles and sometimes they behave like waves It is hard for us to comprehend the
nature of such an object by making mental pictures but experiments definitively show
that this is the case Since we have no direct experience with photons and elementary
particles (because their energies are so much smaller than the energies of the everyday
objects we encounter) it is perhaps not surprising that we have a hard time forming a
mental picture of how they behave
Electron Interference
Since an electron is both a particle and a wave just like a photon it should be able to
produce an interference pattern This is observed If an electron beam is shot at two
closely-spaced slits and if the electrons that pass through the slits are detected
downstream it is found that there are some locations where electrons are never detected
29
and others where lots of electrons are detected The pattern is exactly the same as the
one observed for light waves provided that we use the electron wavelength λ = hp in
place of the wavelength of light This pattern is observed even though each electron is
detected as a single dot on the screen Only after many such dots are collected does the
pattern emerge And if we try to understand how this effect could possibly work by
looking closely at each slit to see which one the electron came through the pattern
disappears the act of measurement destroys the interference This means that each
single electron somehow comes through both slits (as a wave would)
Uncertainty Principle
In classical physics we always imagine that the positions and momenta of moving
particles have definite values It might be hard to measure them but surely at each
instant of time a particle should be precisely located at some point in space and have a
similarly precise momentum This turns out not to be true Instead both position and
momentum are required to be uncertain with their uncertainties Δx and Δp satisfying
the Heisenberg uncertainty relation
2x p
So if the particle were known to be precisely at some particular location (so that Δx =
0) we couldnrsquot know anything about its momentum (Δp = infin) and if its momentum
were exactly known we couldnrsquot know its position
There is a similar relation involving the particlersquos energy E and the time interval Δt
over which this energy is measured
2E t
This means that energy is not actually conserved if we are considering very small time
intervals and this brief non-conservation of energy has been observed
30
Serway Chapter 41
Wave Function ψ and Probability
Quantum mechanics does not predict exactly what an electron or a photon will do Instead
it specifies the wave function or probability amplitude ψ of an electron or a photon
This wave function is a complex-valued function of space and time whose squared
magnitude is the probability density P for finding a particle at a particular place in
space at a certain time
2P
where is the complex conjugate of ψ
For example the wave function of an electron with perfectly specified momentum p
would have a wavelength given by
h
p
and its probability amplitude would be proportional to
2i x ipxe e
The corresponding probability density would then be
21ipx ip ipxP e e e
which means that the electron is equally probable to be anywhere along the x axis
This is in accord with the uncertainty principle since we specified the momentum
precisely we canrsquot have any idea about the position of the electron
As another example you have probably seen ldquofuzzy ballrdquo drawings of electron
orbitals in a chemistry book These fuzzy balls are meant to indicate the distribution
of the probability density 2
P in the orbital
Particle in a Box
A simple example in which we can calculate the wave function is the case of a particle of
mass m constrained to be inside a 1-dimensional box between x = 0 and x = L In this
case the general wave function is a linear superposition of wave functions ψn of the form
sin niE tn
n xx t A e
L
31
where A is a positive constant where n = 1 2 3 and where the energy associated
with each of the quantum states ψn is given by
22
28n
hE n
mL
This wave function is zero at x = 0 and x = L which means that the particle will never
be found at the walls of the box The wave function has maximum values in the
interior and at these places the particle is most likely to be found and it also has
places where it is zero and at these places the particle will also never be found as
expressed by the formula
2sinn x
PL
The particle in the box is interfering with itself producing a probability interference
pattern across the box just like the interference patterns we studied with light and
sound
32
Serway Chapter 42
Atoms
Many people picture an atom as a miniature solar system where electrons orbit around
a massive nucleus at the center This picture is misleading because in atoms the wave
nature of electrons dominates The electrons form 3-dimensional standing waves
(called orbitals) centered at the nucleus And if you ask what is it thatrsquos waving the
answer is ψ the probability amplitude (see Chapter 41)
Standing waves on a string can occur only for certain frequencies (the fundamental
and higher harmonics) Similarly atomic orbitals occur only for certain energies For
the hydrogen atom the energies of the orbitals have the a particularly simple form
2
1136 eVnE
n
where n = 1 2 3 is called the principal quantum number For other atoms the
determination of the orbital energies requires numerical calculation by computers
Atomic Spectra
If an electron is somehow given extra extra energy (we say that it is excited) so that it
occupies a higher orbital it will eventually ldquofallrdquo back down to a lower orbital Each
time an electron falls to a lower orbital it loses the difference in energy between the
two orbitals in the form of a photon Since the orbital energies are discrete so are the
energy differences and so are the wavelengths of the emitted light The entire set of
these discrete wavelengths is called the atomic spectrum and it is unique to each type
of atom For hydrogen the spectrum can be simply written as
H 2 2
1 1 1
f i
Rn n
where RH = 10973732 times 107 m-1 For other atoms the spectrum cannot be expressed as
a formula
Orbital Angular Momentum Electrons in atoms also have quantized values of
angular momentum The orbital quantum number ℓ specifies the value of this
quantized angular momentum through the formula
1L
33
If we want to know the value of the angular momentum along some direction in space
say the z direction the answer is not L but rather
zL m
where mℓ is another quantum number which runs from
1 1m
This quantum number is important when an atom sits in a magnetic field
Spin Angular Momentum It has been found experimentally that electrons and other
charged particles also carry internal angular momentum which we call spin
Electrons have an intrinsic spin angular momentum s along a specified axis that is
extremely quantized it can only take on 2 possible values
1
2zs s
We thus say that electrons have spin s = 12 or that electrons are rdquospin one-half parti-
cles Since s is an angular momentum it obeys the same rule as that for orbital angular
momentum namely that if its value along some axis is s then its total magnitude is
given by
31
2S s s
I know this seems weird but quantum mechanics is weird The only excuse for this
bizarre way of looking at the world is that it predicts what happens in experiments
Exclusion Principle The answer to the question of how many electrons (or any other
spin one-half particle) can be in one particular quantum state was discovered by
Wolfgang Pauli and is called the exclusion principle
ldquoNo two electrons can ever be in the same quantum state therefore no two electrons
in the same atom can have the same set of quantum numbersrdquo
This is the reason that we have atoms with different properties instead of every atom
simple having all of its electrons in the ground state All of the variety we see around
us in the world is the result of chemical differences and these differences would not
exist unless electrons obeyed this important principle The entire structure of the
periodic table (see pages 1377-1379 in Serway) is an expression of this principle
34
Serway Chapter 44
Nuclear Properties
The nucleus is composed of protons with charge +e and mass mp = 1007226 u and of
neutrons with zero charge and mass mn = 1008665 u where u is one atomic mass unit
-271 u=1660540 10 kg
The atomic number Z counts the number of protons in a nucleus while the neutron
number N counts the number of neutrons The mass number A is the sum of the two
A N Z
Protons and nuetrons have about the same mass and some times called baryons (heavy
ones) or nucleons (partcles in the nucleus) A is sometimes called the baryon
number The chemical elements are distinguished by Z ie hydrogen has Z = 1 iron
has Z = 26 uranium has Z = 92 etc But for a given Z there might be several nuclei
with different numbers of neutrons These nuclear siblings are called isotopes
These numbers are used to label nuclei according to the pattern
5626 ie FeA
Z X
denotes the isotope of iron with N = 56 ndash 26 = 30 neutrons
The nucleus is roughly spherical with a radius given approximately by
1 3 150 0where 12 10 mr r A r
Nuclear Stability
Because the positively charged protons electrically repel each other with an enormous
force at distances as small as 10-15 m there must be some really strong force that that
overcomes electrical repulsion to hold protons and neutrons together This force is
called with some lack of imagination the strong force It is a very short-range force
(it only acts over a distance of about 2 x 10-15 m) and attracts protons to protons
neutrons to neutrons and neutrons to protons But in spite of this strong nuclear force
the coulomb repulsion of the protons is still present so anything that might keep the
protons from being right next to each other would help keep the nucleus from
35
exploding This role is played by the neutrons and for nuclei with Z le 20 the stable
nuclei roughly have N = Z
For Z gt 20 the coulomb repulsion force is more powerful and more neutrons are
needed to dilute it up to about N = 15Z around Z = 80 For Z gt 83 no amount of
neutrons can help and these nuclei are unstable (radioactive) There is an ldquoisland of
(relative) stabilityrdquo around Z=90-92 (thorium and uranium respectively) These
elements have one or two isotopes with half-lifes of billions of years so there are
substantial amounts of such elements on Earth This fact makes it possible to have
practical fission devices
Radioactive Decay
There are three types of radioactive decay
Alpha decay the nucleus kicks out a helium nucleus (N = 2 Z = 2 A = 4) So if the
original nucleus is called X and the new nucleus is called Y then the decay would look
like this
4 42 2X Y+ HeA A
Z Z
Beta decay the nucleus either kicks out an electron (endash) or its positively-charged
antimatter twin the positron (e+) plus either an electron neutrino v or an electron
anti-neutrino v
1X Y+e electron decayA AZ Z v
1X Y+e positron decayA AZ Z v
A neutrino is a particle with no charge hardly any mass (much less than the electron
mass) and interacts so weakly with matter that most neutrinos upon encountering the
planet earth just pass right through it as if it werenrsquot there
Gamma decay the nucleons in the nucleus X are in an excited energy state X
(perhaps as a result of having undergone alpha or beta decay) and they drop down to a
lower energy state shedding the energy as a high frequency photon
X XA AZ Z
36
This process is exactly analogous to the way that the electrons in atoms emit photons
Decay Rate and Half Life
There is no way to predict exactly when an unstable or excited nucleus will decay but
there is an average rate at which this decay occurs called the decay constant λ The
meaning of this constant is that if there are a large number N of nuclei in a sample
then the number of decays per second that will be observed (called the decay rate R)
is R = λN In mathematical language
dMR N
dt
This simple differential equation has for its solution
0tN t N e
where N0 is the number of nuclei in the sample at time t = 0
The half-life is the time it takes for half of the nuclei in the sample to decay and is
related to the decay constant by
1 2
ln 2 0693T
Disintegration Energy
When a nucleus decays it is making a transition to an overall state of lower energy
which means according to Einsteinrsquos famous formula E = mc2 that the sum of the
masses after the decay must be less than the mass before with the lost mass appearing
as kinetic energy among the decay products For example in alpha decay this kinetic
energy called the disintegration energy Q is given by
2X YQ M M M c
37
Serway Chapter 45
Nuclear energy
There are two ways to extract energy from the nucleus fission and fusion For nuclei
with Z greater than 26 breaking the nucleus apart into pieces leads to a lower
2mc energy than the original nucleus so energy can be extracted by fission For
nuclei with Z less than 26 a lower energy is achieved by combining nuclei so energy
can be extracted by fusion
Fission
Since neutrons have no charge they are not repelled from nuclei as protons are for
this reason their behavior is the key to understanding how fission works
Fast neutrons mostly bounce off other nuclei with each collision slowing the neutron
down This elastic energy loss is most effective if the other nuclei have low mass (like
hydrogen) and these materials are called moderators because of their ability to slow
down fast neutrons
The reason that slowing neutrons is important is that slow neutrons are much more
likely to be absorbed by a nucleus which then leads to nuclear reactions of various
kinds For a few very large nuclei like uranium-235 and some plutonium isotopes
absorption of a slow neutron causes the nucleus to split into two large fragments plus
2 or 3 fast neutrons (energy is released in the process too) Thus one slow neutron
can produce energy plus 2 or 3 more neutrons which if moderated from fast to slow
can split 2 or 3 more nuclei leading to a runaway chain reaction If the reaction is
allowed to proceed unchecked it produces a large explosion If a neutron absorbing
material is added to the mix (like the cadmium in reactor control rods) it is possible to
keep the reaction under control and to extract the released energy as heat to drive
steam turbines and produce electricity
Fusion
Fusion involves mashing two nuclei together and since they are both charged and repel
each other this reaction is much harder to make go The nuclei must have enough energy to
overcome the coulomb repulsion which is why this reaction requires a high temperature
(hundreds of millions of degrees K) like that in the sun (which is burning hydrogen to
helium via fusion) or like that in the center of the fission explosion that is used to detonate
38
a hydrogen bomb
This reaction is of interest for power production in spite of this difficult temperature
requirement because of the abundance of fusion fuel on the planet There are about
012 g of deuterium ( 21H ) in every gallon of water on earth and it only costs about 4
cents to extract it The fusion energy available from this minuscule amount of
deuterium would run a 1000-Megawatt power plant for 10 seconds Water is so
abundant on earth that if fusion were to work we would have an essentially
inexhaustible source of energy
So why donrsquot we have fusion power plants Well the fuel is cheap but the match is
incredibly expensive The only way we know to control this difficult high-temperature
reaction is with large and expensive pieces of equipment involving either large
magnetic fields and complex high-power electromagnetic antennas or with gigantic
(football-fieldsized) laser facilities involving more than a hundred of the highest-
energy lasers ever built Power plants based on these current methods for controlling
fusion are unattractive to the fiscally-minded people who run the electric power
industry Hopefully better designs will be discovered as experiments continue
39
c actinium 68 Er erbium 101 Md mendelevium 104 Rf rutherfordium l aluminum 63 Eu europium 80 Hg mercury 62 Sm samarium m americium 100 Fm fermium 42 Mo molybdenum 21 Sc scandium b antimony 9 F fluorine 60 Nd neodymium 106 Sg seaborgium r argon 87 Fr francium 10 Ne neon 34 Se selenium s arsenic 64 Gd gadolinium 93 Np neptunium 14 Si silicon t astatine 31 Ga gallium 28 Ni nickel 47 Ag silver a barium 32 Ge germanium 41 Nb niobium 11 Na sodium k berkelium 79 Au gold 7 N nitrogen 38 Sr strontium e beryllium 72 Hf hafnium 102 No nobelium 16 S sulfur i bismuth 108 Hs hassium 76 Os osmium 73 Ta tantalum h bohrium 2 He helium 8 O oxygen 43 Tc technetium
boron 67 Ho holmium 46 Pd palladium 52 Te tellurium r bromine 1 H hydrogen 15 P phosphorus 65 Tb terbium d cadmium 49 In indium 78 Pt platinum 81 Tl thallium a calcium 53 I iodine 94 Pu plutonium 90 Th thorium f californium 77 Ir iridium 84 Po polonium 69 Tm thulium
carbon 26 Fe iron 19 K potassium 50 Sn tin e cerium 36 Kr krypton 59 Pr praseodymium 22 Ti titanium s cesium 57 La lanthanum 61 Pm promethium 74 W tungsten l chlorine 103 Lr lawrencium 91 Pa protactinium 92 U uranium r chromium 82 Pb lead 88 Ra radium 23 V vanadium o cobalt 3 Li lithium 86 Rn radon 54 Xe xenon u copper 71 Lu lutetium 75 Re rhenium 70 Yb ytterbium m curium 12 Mg magnesium 45 Rh rhodium 39 Y yttrium b dubnium 25 Mn manganese 37 Rb rubidium 30 Zn zinc y dysprosium 109 Mt meitnerium 44 Ru ruthenium 40 Zr zirconium s einsteinium
40
UNITS The SI units are given in parentheses Other commonly used units are given in terms of the SI units Symbols conform with the recommendations of the American National Standards Institute (ANSI) and the American Institute of Physics (AIP) Commonly used multiples of SI units
prefix symbol factor tera T 1012 giga G 109
mega M 106 kilo k 103
centi c 10-2 milli m 10-3
micro μ 10-6 nano n 10-9 pico p 10-12
femto f 10-15
length meter (m) angstrom 1 Ǻ = 10-10 m inch 1 in = 254 x 10-2 m foot 1 ft = 03048 m mile 1 mi = 1609 m
mass kilogram (kg) atomic mass unit 1 u = 1661 x 10-27 kgslug 1 slug = 1459 kg
force newton (N = kg bull ms2) dyne 1 dyn = 10-5 N pound 1 lb = 4448 N
pressure Pascal (Pa = kgm bull s2) atmosphere 1 atm = 1013 x 105 Pa poundssquare inch 1 psi = 6895 Pa cm of mercury 1 cm Hg = 1333 Pa bar 1 bar = 1000 x 105 Pa torr 1 torr = 1333 Pa
time second (s) minute 1 min = 60 s hour 1 h =3600 s
frequency hertz (Hz = s-1) radianssecond 1 rads = 12π Hz
energy joule (J = kg bull m2s2) erg 1 erg = 10-7 J electron volt 1 eV = 1602 x 10-19 J calorie 1 cal = 4187 J kilowatt-hour 1 kWh = 36 x 106 J British thermal unit 1 Btu = 1055 J
power watt (W = kg bull m2s3) horsepower 1 hp = 7457 W
charge coulomb (C = A bull s)
electric potential volt (V = kg bull m2 s3 bull A)
current ampere (A)
resistance ohm (Ω = kg bull m2s3 bull A2)
capacitance farad (F = s4 bull A2kg bull m2)
magnetic field tesla (T = kgs2 bull A) gauss 1 G = 10-4 T
magnetic flux weber (Wb = kg m2s2 bull A) maxwell 1 Mx = 10-8 Wb
magnetic inductance henry (H = kg bull m2s2 bullA2)
temperature kelvin (K) degrees Celsius 0degC = 27315 K
angle radian (rad) degree 1 deg = π180 rad revolution 1 rev = 2π rad
41
Some Physical Constantsa Quantity Symbol Valueb Atomic mass unit μ 1660 538 73 (13) x 10-27 kg
931494 013 (37) MeVc2 Avogadros number NA 6022 141 99 (47) x 1023 particlesmol Avogadrorsquos number 6022 x 1023 mol Bohr magneton
2B e
em 9274 008 99 (37) x 10-24 JT
Bohr magneton μB 927 x 10-24 JT Bohr radius
0
2
2e em e k
a 5291 772 083 (19) x 10-11 m
Boltzmanns constant B A
RNk 1380 650 3 (24) x 10-23 JK
Boltzmannrsquos constant kB 1380 x 10-23 JK Compton wavelength
C eh
m c 2426 310 215 (18) x 10-12 m
Deuteron mass md 3343 583 09 (26) x 10-27 kg 2013 553 212 71 (35) u
electron charge e 1602 x 10-19 C Electron mass me 9109 381 88 (72) x 10-31 kg
5485 799 110 (12) x 10-4 u 0510 998 902 (21) MeVc2
electron mass 911 x 10-31 kg Electron volt eV 1602 176 462 (63) x 10-19 J Elementary charge e 1602 176 462 (63) x 10-19 C Gas constant R 8314 472 (15) JKmol Gravitational constant G 6673 (10) x 10-11 Nm2kg2 Neutron mass mn 1674 927 16 (13) x 10-27 kg
1008 664 915 78 (55) u 939565 330 (38) MeVc2
neutron mass 1675 x 10-27 kg Nuclear magneton
2n p
em 5050 783 17 (20) x 10-27 JT
permeability constant μ0 1257 x 10-6 Hm permittivity constant ε0 8854 x 10-12 Fm Plancks constant h
2h
6626 068 76 (52) x 10-34 Js 1054 571 596 (82) x 10-34 Js
Planckrsquos constant h ħ
6626 x 10-34 Js 1055 x 10-34 Js
Proton mass mp 1672621 58 (13) x 10-27 kg 1007 276 466 88 (13) u 938271 998 (38) MeVc2
proton mass 1673 x 10-27 kg Rydberg constant RH 1097 373 156 854 9 (83) x 107 m-1 Speed of light in vacuum c 2997 92458 x 108 ms (exact) speed of light c 300 x 108 ms a
These constants are the values recommended in 1998 by CODATA based on a least-squares adjustment of data from different measurements For a more complete list see P J Mohr and B N Taylor Rev Mod Phys 72351 2000 b The numbers in parentheses for the values above represent the uncertainties of the last two digits
NOTE The ones in red are the ones from Appendix 13 The ones in yellow are the ones that are from the Serway book table but that were already on the Appendix 13 table
42
INDEX
Absolute zero 9
Adiabatic 17
Adiabatic exponent 18
Adiabatic process 17
Alpha decay 46
Amplitude 4
Angle of reflection 25
Angular frequency 3
Angular magnification 30
Angular momentum atomic 43
Angular size 29
Archimedes Principle 1
Atomic spectra 43
Atoms 43
Avogadrorsquos number 10
Beats 8
Bernoullirsquos Equation 2
Beta decay 46
Bifocals 29
Brewsterrsquos angle 34
British Thermal Unit Btu 11
Buoyancy 2
calorie 11
Calorie 11
Camera 28
Carnot cycle 21
Carnot efficiency 22
Celsius scale 9
Ciliary muscle 29
Coefficient of performance 21
Compton effect 39
Constructive interference 7
Continuity equation of 2
Contraction length 35
Convection 14
Curved mirrors 27
Decay constant 46
Decay rate radioactive 46
Decibel scale 5
Degrees of freedom 16
Density 1
Destructive interference 7
Diffraction single slit 33
Diffraction grating 33
Dilation time 35
Disintegration energy 47
Dispersion of light 26
Doppler effect 6
Double slit interference 31
Efficiency 20
Electron interference 39
Emissivity 15
Energy nuclear decay 47
Energy relativistic 36
43
Engineering work 20
Entropy 22
Entropy ideal gas 24
Equipartition of energy 18
Equivalence principle 36
Exclusion principle 44
Expansion thermal 9
Expansion coefficient linear 9
Eye 29
Fahrenheit scale 9
Far point 29
Farsightedness 29
First Law of Thermodynamics 12
Fission 48
Flux volume 2
Free expansion 23
Fringes 31
Fusion 48
Fusion heat of 11
Gamma relativistic 35
Gamma decay 46
General relativity 36
Half-life 46
Heat 11
Heat capacity 11
Heat conduction 13
Heat engine 20
Heat of fusion 11
Heat of vaporization 11
Heat pump 20
Heats of transformation 11
Hydrostatics 1
Ideal Gas Law 10
Images real and virtual 27
Intensity sound 5
Interference 7
Interference two-slit 31
Internal energy 12
Internal energy degrees of freedom 16
Irreversible process 19 23
Isotopes 45
Joule 11
Kelvin scale 9
Kinetic energy relativity 36
Kinetic theory 16
Latent heat 11
Length contraction 35
Linear expansion coefficient 9
Linear polarization 34
Linear superposition 7
Longitudinal wave 3
Loudness 5
Magnification lateral 27
Magnifying glass 30
Malusrsquos law 34
Microscope 30
Momentum relativistic 36
Muscle ciliary 29
44
Musical instruments 8
Musical scale 8
Near point 29
Nearsightedness 29
Nonlinear 7
Nuclear energy 48
Nuclear properties 45
Nuclear stability 45
Octave 8
Optical resolution 33
Orbital quantum number ℓ 43
Orbitals 43
Particle in a Box 41
Particles are waves 39
Pascalrsquos Principle 1
Period 3
Photoelectric effect 38
Photons 38
Pitch 8
Polarization 34
Power sound 5
Presbyopia 29
Pressure 1
Principal quantum number n 43
Principle of equivalence 36
Principle of linear superposition 7
Probability amplitude ψ 41
Processes thermodynamic 13
R-value 14
Radiation thermal 14
Radioactive decay 46
Radioactive decay rate 46
Ray tracing 27
Rayleighrsquos criterion 33
Reading glasses 29
Real image 27
Refraction 25
Refrigerator 20
Relativistic gamma 35
Relativity principles 35
Resolved for light sources 33
Rest energy 36
Reversible process 19 22
Rope wave speed 4
Second Law of Thermodynamics 19
Shock waves 6
Simple magnifier 30
Simultaneity 35
Single slit diffraction 33
Snellrsquos law 25
Sound speed 5
Specific heat 11
Spin Angular momentum 44
Standing waves 7
Stefanrsquos law 14
Strong force 45
Telescope 30
Temperature 9
45
Temperature Scales 9
Thermal conductivity 14
Thermal energy 12
Thermal expansion 9
Thin film interference 31
Thin lenses 28
Time dilation 35
Tone musical 8
Total internal reflection 25
Transverse wave 3
Traveling Waves 3
Two-slit interference 31
Uncertainty principle 40
Vaporization heat of 11
Velocity addition relativity 36
Virtual image 27
Volume flux 2
Wave function 41
Wave Function ψ and Probability 41
Wave speed 3
Wave-particle duality 39
Wavelength 3
Wavenumber 3
Wien Displacement Law 16
Work 11
3
Serway Chapter 16
Longitudinal and Transverse Waves
A longitudinal wave is one in which the direction of vibration of the medium and the direction of
propagation of the wave are in the same direction Sound waves in the air in water and in solids
are longitudinal waves
A transverse wave is one in which the direction of vibration of the medium is perpendicular to
the direction of propagation of the wave Ocean waves waves on ropes and electromagnetic
waves are transverse waves
Wavelength and Wavenumber
The wavelength λ is the period in space of the wave ie it is the distance from one wave crest to
the next The wavenumber k (units reciprocal length) is related to the wavelength by the formula
2k
Frequency and Angular Frequency
The period T is the period in time of the wave ie it is the time between the arrival of one wave
crest and the next at some point in space The frequency f is the number of crests that arrive per
second which is just 1T and the angular frequency ω is 2π times the frequency Both have units
of per second (reciprocal time)
1 22f f
T T
Wave Speed
The wave speed of a wave is defined to be the the speed at which the wave crests or troughs
move through the medium This speed is related to the frequency and the wavelength by the
formulas
v fk
where k is the wavenumber of the wave v is also called the phase velocity
Traveling Waves
Traveling waves are oscillations in which the wave crests and troughs glide smoothly through the
medium Waves on oceans and lakes are traveling waves The mathematical form of a harmonic
(sinusoidal) traveling wave is
4
siny x t A kx t
where A is the amplitude of the wave As usual ω and k are related by the relation ω = kv where
v is the wave speed the speed at which the crests travel the minus sign means that the wave is
traveling to the right towards higher values of x φ is the phase and constrols where the maxima
and minima occur
Speed of Transverse Waves on a Rope
The velocity of transverse waves on a perfectly flexible rope is given by
Tv
where T is the tension in the rope and where μ is its linear mass density (mass per unit length)
Common Test Questions
A common question is to be given an equation for a wave and to be asked what is amplitude
velocity etc Another common type of question is to be given facts about a wave like its
direction velocity and magnitude at a given place and time and to be asked which of several
equations correctly describe the wave
5
Serway Chapter 17
Sound Speed in Solids Liquids and Gases
The speed of sound in a liquid or solid of bulk modulus B and volume mass density ρ
Bv
The speed of sound in air is
331 1 273v m s T C
The speed of sound in air at room temperature is about 343 ms Notice it doesnrsquot depend on
pressure or frequency
Power Intensity and Loudness
The intensity of a sound wave is defined to be the power per unit area ie wattsm2 in the wave
For spherical waves traveling away from a small source emitting waves with average power Рav
the intensity falls off with distance from the source r according to the inverse-square law
av24
Ir
Hence the intensity I2 at distance r2 is related to the intensity I1 at a different distance r1 by
21 2
22 1
I r
I r
Experiments on human hearing have shown that we hear intensity differences logarithmically so
the decibel loudness scale for sound intensity was invented The loudness β of a sound in
decibels is related to its intensity I in Wm2 by the formula
1010logo
I
I
Where Io is a sound intensity near the threshold of hearing defined to be Io = 10-12 Wm2 Note
that on this scale a sound is made 10 times more intense by adding 10 decibels Remember that
intensity is proportional to velocity squared Amplitude is the A is the previous chapterrsquos
equation for a traveling wave If you make the A 10 times as much the intensity will increase by
100 times (20 dB)
Doppler Effect
If sound waves are traveling through a medium and if either the receiver of the waves or the
source of waves is moving then the frequency received is related to the frequency emitted by
6
0
s
v vf f
v v
where fprime is the frequency detected by the observer f is the frequency emitted by the source v is
the speed of the waves vo is the speed of the observer and vs is the speed of the source This
formula assumes that the source and receiver are either moving directly toward each other or
directly away from each other To know which signs to use remember that when observer and
source approach each other the observed frequency is higher while if they move away from each
other it is lower Just examine the signs in the formula and make the answer come out right
For electromagnetic waves (light radio waves X-rays) traveling in vacuum Einsteinrsquos theory of
relativity (and careful experiments) show that the Doppler shift is given by
r
r
c vf f
c v
where vr is the relative speed between the source and the observer
For all kinds of waves (sound light etc) if the relative speed of the source and the observer is
small compared to the speed of the waves then there is a simple approximation to the Doppler
effect For example if the relative speed is 1 of the wave speed then the frequency shifts by
1 Remember that this is only an approximation
Shock Waves
When an object moves through a medium at a speed greater than the speed of waves a V-shaped
shock wave is produced The V-shaped wake behind a speeding boat is a good example of this
effect and the cone of sonic-boom behind a supersonic aircraft is another The angle that V-line
makes with the direction of travel of the source is given by
sins
v
v
where v is the wave speed and vs is the source speed
7
Serway Chapter 18
Principle of Linear Superposition
We say that a system obeys the principle of linear superposition if two or more different motions
of the system can simply be added together to find the net motion of the system Light waves
obey this principle as they propagate through the air as can be seen by shining two flashlights so
that their beams cross The beams propagate along without affecting each other (Light sabers are
a spectacular but unfortunately fictional example of systems that do not obey the principle of
superposition) Wave pulses on an ideal rope also obey this principle two different pulses pass
through each other without change Standing waves are an example of this effect being simply
the linear superposition of two traveling waves of the same frequency but moving in opposite
directions Light waves in matter do not always obey this principle For instance two powerful
laser beams could be made to cross in a piece of glass in such a way that their combined heating
effect in the crossing region could melt the glass and scatter the beams in complicated ways This
is an example of a nonlinear effect
Interference
When two or more waves are present in the same medium at the same time their net effect may
often be obtained simply by adding them at each point in the medium according to the principle
of linear superposition (Note this wonrsquot work if the medium is nonlinear) When this addition
makes the total amplitude be greater than the individual amplitudes of the various waves we say
that the interference is constructive When the addition produces cancellation and an amplitude
less than the amplitudes of the separate waves we have destructive interference
Standing Waves
A standing wave is the superposition of two identical traveling waves moving in opposite
directions Nodes are places where the two waves perfectly destructively interfere to produce
zero amplitude at all times Anti-nodes are places where the two waves perfectly constructively
interfere to produce an amplitude maximum The distance between nodes is λ2 Standing waves
on a string fixed at both ends have nodes at each end of the string Standing waves in an air
column enclosed in a tube have displacement anti-nodes at open ends of the tube and
displacement nodes at closed ends The frequency of the standing wave with the lowest possible
frequency is called the fundamental frequency Standing waves on strings or in air columns all
have frequencies which are integer multiples of the fundamental frequency and are called
8
harmonics (The fundamental is called the ldquofirst harmonicrdquo)
Beats
Beats are heard when two waves with slightly different frequencies f1 and f2 are combined The
waves constructively interfere for a number of cycles then destructively interfere for a number
of cycles We hear a periodic ldquowah-wahrdquo frequency equal to the difference of the two wave
frequencies
1 2bf f f
Musical Instruments
Musical instruments produce tones by exciting standing waves on strings (violins piano) and in
tubes (trumpet organ) The fundamental frequency of the standing wave is called the pitch of the
tone The pitch of concert A is 440 Hz by definition Two tones are an octave apart if one pitch
has twice the frequency of the other In written music there are 12 intervals in each octave with
the ratio between successive intervals equal to 2112 = 105946 The ratios for each tone in an
octave starting at A and ending at the next higher A are
A A B C C D D E F F G G A
1 10595 11225 11892 12599 13348 14142 14983 15874 16818 17818 18877 2
A musical tone is actually a superposition of the fundamental frequency and the higher
harmonics The tone quality of a musical instrument is determined by the amplitudes of the
various harmonics that it produces A violin and a trumpet can play the same pitch but they
donrsquot sound at all alike to our ears The difference between them is in the various amplitudes of
their harmonics
9
Serway Chapter 19
Temperature
Formally temperature is what is measured by a thermometer Roughly high temperature is what
we call hot and low temperature is what we call cold On the atomic level temperature refers to
the kinetic energy of the molecules A collection of molecules is called ldquohotrdquo if the molecules
have rapid random motion while a collection of molecules is called ldquocoldrdquo if the random motion
is slow When two bodies are placed in close contact with each other they exchange molecular
kinetic energy until they come to the same temperature This is the microscopic picture of the
Zeroth Law of Thermodynamics
Absolute Zero
Absolute zero is the lowest possible temperature that any object can have This is the temperature
at which all of the energy than can be removed an object has been removed (This removable
energy we call thermal energy) There is still motion at absolute zero Electrons continue to orbit
around atomic nuclei and even atoms continue to move about with a small amount of kinetic
energy but this small energy cannot be removed from the object For example at absolute zero
helium is a liquid whose atoms still move and slide past each other
Temperature Scales
Kelvin Scale Absolute zero is at T = 0 K water freezes at T = 27315 K room temperature is
around T = 295 K and water boils at T = 373 K Note we donrsquot use a deg symbol Kelvin is
prefered SI Unit
Celsius Scale Absolute zero is at T = -273degC water freezes at T = 0degC room temperature is
around T = 22degC and water boils at T = 100degC
Fahrenheit Scale Absolute zero is at T = -459degF water freezes at TF = 32degF room temperature
is around T = 72degF and water boils at TF = 212degF TF =18TC + 32 Notice that temperature
differences are the same for the Kelvin and Celsius scales
Thermal Expansion
When materials are heated they usually expand and when they are cooled they usually contract
(Water near freezing is a spectacular counterexample it works the other way around) The
coefficient of linear expansion is defined by the relation
1
i
L
L T
10
where Li is the initial length of a rod of the material and ΔL is the change in its length due to a
small temperature change ΔT The coefficient of volume expansion is defined similarly
1
i
V
V T
where Vi is the initial volume of a piece of material and where ΔV is the change in its volume due
to a small temperature change ΔT
Avogadrorsquos Number (N or NA)
One mole of any substance corresponds to 6022 times 1023 molecules The atomic mass given on the
periodic table (p A32-33 in the text) is the mass in grams of one mole of atoms For example
the atomic mass of oxygen is 15999 gmol The mass of a single atom is given by the same
number in units of atomic mass units (u) For example the mass of an oxygen atom is 15999 u
(on the average)
Ideal Gas Law (an example of an equation of state)
When the molecules of a gas are sufficiently inert and widely separated that interactions between
them are negligible we say that it is an ideal gas The pressure P volume V and temperature T
(in kelvins) of such a gas are State Variables and are related by the ideal gas law
Bor PV=NkPV nRT T
where n is the number of moles of the gas where R is the gas constant
8314 Jmol KR
where N is the number of molecules and where kB is Boltzmannrsquos constant
231380 10 J KBk
It works well for air at atmosphere pressure and even better for partial vaccuums The relative
ease of measuring pressure and the linear relationship between pressure and temperature (if V
and n are held fixed) makes an ideal gas an ideal thermometer Thermometers can be based on
properties of solids or liquids but the behavior of these materials with temperature is more
complicated
11
Serway Chapter 20
Heat
Heat is energy that flows between a system and its environment because of a tempera-
ture difference between them The units of heat are Joules as expected for an energy
Unfortunately there are several competing units of energy They are related by
1 cal 4186 J 1 Cal 4186 J 1 Btu 1054 J
Heat Capacity
There is often a simple linear relation between the heat that flows in or out of part of a
system and the temperature change that results from this energy transfer When this
linear relation holds it is convenient to define the heat capacity C and the specific
heat c as follows
For the entire object the heat Q it absorbs is given by Q = C(Tf - Ti)
For a particular material the specific heat is defined by c = Cm which is the heat
capacity per unit mass so that
Q = mc(Tf ndash Ti)
Note C has units of energy (J or Cal)(Kelvin kg)
Heats of Transformation or Latent Heat Q = plusmn mL
When a substance changes phase from solid to liquid or from liquid to gas it absorbs
heat without a change in temperature The latent heat or heat of transformation is
usually given per unit mass of the substance For example for water the heat of fusion
(melting) is L = 333 kJkg while the heat of vaporization (boiling) is L = 2260 kJkg
Note that heat for boiling is considerably bigger than melting for water You have to
be careful with signs heat is given off (negative) if you go down in temperature and
condense steam
Work
In general the small amount of work done on a system as a force Fon is exerted on it
through a vector displacement dx is given by
on xdW d F
12
But if the displacement is done very slowly (as we always assume in thermodynamics)
then the force exerted on the system and the force exerted by the system are in
balance so the force exerted by the system is ndash Fon In thermodynamics it is more
convenient to talk about the force exerted by the system so we change the above
formula for the work done on the system to
xdW d F
where F is the force exerted by the system This has confused students for more than a
century now but this is the way your book and many other books do it so you are
stuck You will need to memorize the minus sign in this definition of the work to be
able to use your textbook
There are many chances to get signs wrong in this and the next two chapters (Mosiah
2 )
When an external agent changes the volume of a gas at pressure P by a small amount
dV the (small amount) of work done on the system is given by
dW PdV
Notice that this minus sign is just what we need to make dW be positive if the external
agent compresses the gas for then dV is negative If on the other hand the external
agent gives way allowing the gas to expand against it then dV is positive and we say
that the work done on the gas is negative
The work done on the system (eg by the gas in a cylinder) in a thermodynamic
process is the area under the curve in a PV diagram It is positive for compressions
and negative for expansions If the volume of gas remains constant in a process then
no work is done by the gas
Cyclic processes are important For cyclic processes represented by PV diagrams the
magnitude of the net work during one cycle is simply the area enclosed by the cycle
on the diagram Be careful to keep track of signs when you are calculating that
enclosed area In cyclic process Q = ndashW for a cycle Put a PV diagram in your notes
1
Path A-B B-C C-D D-A A to A net
Q
W
ΔU
ΔS
Internal Energy
The energy stored in a substance is called its internal energy Eint This energy may be
stored as random kinetic energy or as potential energy in each molecule (stretched
chemical bonds electrons in excited states etc) For ideal gases all states with the
same temperature will have the same Eint
First Law of Thermodynamics
The change ΔEint in the internal energy of a system is given by
intE Q W
where Q is the heat absorbed by the system and where W is the work done on the
system Hence if a system absorbs heat (and if Wge0) the internal energy increases
Likewise if the system does work (W on the system is negative) and if Qge0 the
internal energy decreases Potential Pitfall Many times people talk about work done
by the system It is the minus of W on the system Donrsquot get tripped up
Processes
Adiabatic process No heat is exchanged Q = 0 requiring that the system be insulated
from the environment A process may be approximately adiabatic if it happens so
rapidly that heat does not have time to enter or leave the system Work + or ndash is done
and ΔEint = W
Adiabatic free expansion A gas is allowed to expand into a vacuum without pushing
on anythingndashit just rushes into the vacuum In this process Q = W = 0 so the internal
energy and temperature does not change (Note the difference between an adiabatic
process and a free expansion is that NO work is done in the adiabatic free expansion)
Isobaric process The pressure is held fixed ΔP = 0 For example usually the
pressure increases when a gas is heated but if it were allowed to expand during the
2
heating process in just the right way its pressure could remain fixed In isobaric
processes both Q and W are non-zero (Work is eacy = ndashP(Vf ndash Vi)
Isovolumetric process The volume is held fixed ΔV = 0 The work done by the gas is
then zero and so we have ΔEint = Q
Isothermal process The temperature is held fixed ΔT = 0 There is no change in Eint
so Q = ndashW for isothermal processes For an ideal gas PV = nRT so PV = constant in
an isothermal process The work done on the gas is then given by
lnf
i
Vi
Vf
VW PdV nRT
V
Heat Conduction
The quantity P is defined to be the rate at which heat flows through an object and is a
power having units of watts It is analogous to electric current which is the rate at
which charge flows through an object If the flow of heat through a slab of length L
and cross-sectional area A is steady in time then P is given by the equation
h cT TdQkA
dt L
P =
where Th and Tc are the (hot and cold) temperatures of the two ends of the slab The
heat flows of course because of this temperature difference The quantity k is called
the thermal conductivity and is a constant that is characteristic of the material It is
analogous to the electrical conductivity h cT TL is sometimes called the temperature
gradient and is written dTd dTdx
R-Values
It is common to have the heat-conducting properties of materials described by their R-
values especially for insulating materials like fiberglass batting The connection
between k and R is R = Lk where L is the material thickness In this country R-values
always have units of 2ft F hourBtu
Polyurethane foam 6 (1 in thick) Air space 1 (35 in thick) Fiber glass batting 11
(35 in thick) Wood about 1 (1 in thick) Glass 09 (0125 in thick)
The heat flow rate through a slab of area A is given by
3
h cT TA
R
P
in units of Btuhour Note that A must be in square feet and the temperatures must be
in degrees Fahrenheit
Convection
Convection is the transfer of thermal energy by flow of material For instance a home
furnace doesnrsquot heat a house by waiting for the heat from the burner to slowly conduct
throughout the house instead it quickly pumps warm air to all of the rooms
Generally convection is a much faster way to transfer heat than conduction
Radiation
Electromagnetic radiation can also transfer heat When you warm yourself near a
campfire which has burned itself down into a bed of glowing embers you are
receiving radiant heat from the infrared portion of the electromagnetic spectrum The
rate at which an object emits radiant heat is given by Stefanrsquos law
4AeTP
where P is the radiated power in watts σ is a constant
8 2 45696 10 W m K
A is the surface area of the object in m2 and T is the temperature in kelvins The
constant e is called the emissivity and it varies from substance to substance A perfect
absorber (think black velvet) has e = 1 while a perfect reflector (think mirror) has e =
0 Hence black objects radiate very well while shiny ones do not Also an object that
is hotter than its surroundings radiates more energy than it absorbs whereas an object
that is cooler than its surroundings absorbs more energy than it radiates
Terminology
Transfer variables vs state variables
Energy transfer by heat as well as work done depends on the initial final and
intermediate states of the system They are transfer variables But their sum (Q + W =
4
Eint) is a state variable
Figure 205
5
Serway Chapter 21
Kinetic Theory
The ideal gas law works for all atoms and molecules at low pressure It is rather
amazing that it does Kinetic theory explains why The properties of an ideal gas can
be understood by thinking of it as N rapidly moving particles of mass m As these
particles collide with the container walls momentum is imparted to the walls which
we call the force of gas pressure In this picture the pressure is related to the average
of the square of the particle velocity 2v by
22 1( )
3 2
NP mv
V
Using the ideal gas law we obtain the average translational kinetic energy per
molecule
21 3
2 2 Bmv k T
The rms speed is then given by
2rms
3 3Bk T RTv v
m M
where M is the molecular mass in kgmol
Degrees of Freedom
Roughly speaking a degree of freedom is a way in which a molecule can store energy
For instance since there are three different directions in space along which a molecule
can move there are three degrees of freedom for the translational kinetic energy
There are also three different axes of rotation about which a polyatomic molecule can
spin so we say there are three degrees of freedom for the rotational kinetic energy
There are even degrees of freedom associated with the various ways in which a
molecule can vibrate and with the different energy levels in which the electrons of
the molecule can exist
Internal Energy and Degrees of Freedom The internal energy of an ideal gas made
up of molecules with J degrees of freedom is given by
int 2 2 B
J JE nRT Nk T
6
Heat Capacities of an Ideal Gas The heat capacity of a gas is described by means of
molar heat capacities CV and CP These are the heat capacities per mole and the
subscript V on CV means that the volume is being held constant while for CP the
pressure is held constant For example to raise the temperature of n moles of a gas
whose pressure is held constant by 10 K we would have to supply an amount of heat Q
= nCP (10) K
Molar Specific Heat of an Ideal Gas at Constant Volume
VQ nC T
3monatomic
2VC R
5diatomic
2VC R
5polyatomic
2VC
Real gases deviate from these formulas because in addition to the translational and ro-
tational degrees of freedom they also have vibrational and electronic degrees of
freedom These are unimportant at low temperatures due to quantum mechanical
effects but become increasingly important at higher temperatures The rough rule is
No of degrees of freedom
2VC R
Molar Specific Heat of an Ideal Gas at Constant Pressure
PQ nC T
P VC C R
The internal energy of an ideal gas depends only on the temperature
int VE nC T
Adiabatic Processes in an Ideal Gas
7
An adiabatic process is one in which no heat is exchanged between the system and the
environment When an ideal gas expands or contracts adiabatically not only does its
pressure change as expected from the ideal gas law but its temperature changes as
well Under these conditions the final pressure Pf can be computed from the initial
pressure Pi and from the final and initial volumes Vf and Vi by
or constantf f i iP V PV PV
where γ = CPCV The quantity γ is called the adiabatic exponent Note that this
doesnrsquot mean that the ideal gas law no longer holds it does and in fact it can be
combined with the adiabatic law for pressure given above to obtain the adiabatic law
for temperatures
1 constantTV
Compressions in sound waves are adiabatic because they happen too rapidly for any
appreciable amount of heat to flow This is why the adiabatic exponent appears in the
formula for the speed of sound in an ideal gas
RTv
M
Note that v depends only on T and not on P Because it depends only on the
temperature the speed of sound is the same in Provo as at sea level in spite of the
lower pressure here due to the difference in elevation
Equipartition of Energy
Every kind of molecule has a certain number of degrees of freedom which are
independent ways in which it can store energy Each such degree of freedom has
associated with it ndash on average ndash an energy of 12 Bk T per molecule (or 1
2 RT per mole)
(Note since a molecule has so many possible degrees of freedom it would seem that
there should be a lot of 12 sBk T to spread around But because energy is quantized
some of these degrees of freedom are not ldquoactiverdquo until the temperature becomes high
enough that 12 Bk T is as big as the lowest quantum of energy
8
Serway Chapter 22
Second Law of Thermodynamics
There are several equivalent forms of this important law
Kelvin It is not possible to change heat completely into work with no other change
taking place Or in other words there are no perfect heat engines
Clausius It is not possible for heat to flow from one body to another body at a higher
temperature with no other change taking place Or in other words there are no
perfect refrigerators
Entropy In any thermodynamic process that proceeds from one equilibrium state to
another the entropy of the system + environment either remains unchanged or
increases The total entropy never decreases This law is a bit of an oddity among the
laws of physics because it is not absolute Things are forbidden by the second law not
because it is impossible for them to happen but because it is extremely unlikely for
them to happen (See below for more information about entropy)
Reversible and Irreversible Processes
A reversible process is one which occurs so slowly that it is in thermal equilibrium (or
very nearly so) at all times A hallmark of such processes is that a motion picture of
them looks perfectly normal whether run forward or backward Imagine for instance
the slow expansion of a gas at constant temperature in a cylinder whose volume is being
increased by a slowly moving piston Run the movie backwards and what do you see
You see the slow compression of a gas at constant temperature which looks perfectly
normal
An irreversible process is one which occurs in such a way that thermal equilibrium is
not maintained throughout the process The mark of this kind of process is that a motion
picture of it looks very odd when run backward Imagine the sudden expansion of a gas
into a previously evacuated chamber because a hole was punched in the wall between a
pressurized chamber and the evacuated one Run the movie backward and what do you
see You see the gas in the soon-to-be-evacuated chamber gather itself together and
stream through a tiny hole into a chamber in which there is already plenty of gas If you
have ever seen this happen get in touch with the support group for those who have
witnessed the spontaneous combustion of large mammals by calling 1-800-PYROCOW
9
Heat Engines
Because of the vexing difference in sign between work done on and system and work done
by a system we will invent a new work variable Weng Heat engines do work and so the net
W for these engines is negative But in engineering applications hidden minus signs are
regarded as evil so for heat engines we donrsquot talk about W instead we talk about its
magnitude engW W So for heat engines the first law is
int engE Q W
But for heat pumps and refrigerators work is done on the system so we use the usual work
W when we talk about these systems
A heat engine is a machine that absorbs heat converts part of it to work and exhausts the
rest The heat must be absorbed at high temperature and exhausted at low temperature If the
absorbed heat is Qh the exhausted heat is Qc and the work done by the engine is Weng then
eng h cW Q Q
and the efficiency of the engine is defined to be
eng
h
We
Q
A perfect engine would convert the heat hQ completely into work Weng giving an effi-
ciency of e = 1 Energy conservation alone allows a perfect engine but the second law
requires e lt 1
Refrigerators and Heat Pumps
A refrigerator is a machine that absorbs heat at low temperature and exhausts it at high
temperature the ldquobackwardsrdquo heat transfer being driven by the work done on the machine
by some source of power A heat pump is a machine that either works like a refrigerator
keeping a place cold by transferring heat from this cold place to a higher temperature
environment (cooling mode like an air conditioner) or it functions as a heater
transferring heat into a warm place from a cooler one (heating mode like a window unit
that heats a house by extracting thermal energy from the cold outdoors) The coefficient
of performance of a refrigerator or of a heat pump in cooling mode is defined to be
COP cooling modecQ
W
10
For a heat pump in heating mode the coefficient of performance is
COP heating modehQ
W
Note that we donrsquot have to use the engineering work here because in these systems
work is done on the system and W is naturally positive
A perfect refrigerator would take in heat Qc and exhaust the same amount of heat without
doing any work giving an infinite coefficient of performance Energy conservation alone
allows a perfect refrigerator but the second law requires COP lt infin A perfect heat pump
in heating mode would transfer Qh into the house without doing any work and so would
also have an infinite coefficient of performance The second law forbids this too A good
coefficient of performance for a real device would be around 5 or 6
Carnot Cycle
The most efficient of all possible engines is one that uses the Carnot cycle This cycle
employs an ideal gas has no friction and operates very slowly so that the gas can be in
thermal equilibrium at all parts of the cycle This means of course that it canrsquot
possibly be built and even if it could be built it would not run fast enough to be useful
Nevertheless this cycle is very important because it gives an upper bound on the
efficiency of real engines There cannot possibly be an engine that is more efficient
than one based on the Carnot cycle This cycle consists of the following four steps
1 The ideal gas absorbs heat Qh at constant temperature Th while the gas increases its
volume The reason that heat is absorbed is that expansion tends to cool the gas but
thermal contact with the environment at Th keeps the temperature high by heat
conduction into the ideal gas
2 The ideal gas further increases its volume by an adiabatic expansion This expansion
causes the gas to cool so at the end of this part of the cycle the gas is at temperature Tc
3 The gas exhausts heat Qc at constant temperature Tc while the gas decreases in
volume The reason that heat is exhausted is that compression tends to heat the gas
but thermal contact with the environment at Tc keeps the temperature low by heat
conduction out of the ideal gas
4 The gas is adiabatically compressed back to its original volume (the volume it started
with in step 1) This compression heats the gas from Tc up to Th
11
The efficiency of a Carnot engine is given by the very simple formula
1 cC
h
Te
T
where the temperatures must in be Kelvin No real engine can be more efficient than
this
The coefficient of performance of a Carnot refrigerator or heat pump in cooling mode
is given by
COP cooling mode cC
h c
T
T T
and no real refrigerator can have a coefficient of performance greater than this
The coefficient of performance of a Carnot heat pump in heating mode is
COP heating mode hC
h c
T
T T
So why donrsquot we just use these wonderful Carnot engines and have perfect efficiency
To make the reversible steps in the cycle really reversible they would have to occur
infinitely slowly So the price you pay for making a perfect engine is that it takes
forever to get it to do any work
Entropy
The entropy of a system is defined in terms of its molecular makeup and measures
roughly the disorder of the system If the system is packed into a very small volume
then it is quite ordered and the entropy will be low If it occupies a large volume the
entropy is high (To see what this has to do with disorder note that socks in a drawer
occupy a small volume while socks on the bed in the corner by the door and
hanging from the chandelier occupy a large volume) If the system is very cold then
the molecules hardly move and may even reach out to each other and form a crystal
This is a highly ordered state and therefore has low entropy If the system is very hot
with rapidly speeding molecules crashing into the container walls and bouncing off
each other things are disordered and the entropy is high
It is possible to calculate the entropy of a system in terms of its macroscopic thermody-
namic properties ie pressure volume temperature number of moles etc The key to this
calculation is the concept of a reversible process A reversible process is one that is
12
carried out without friction and so slowly that the process can be reversed at any stage by
making an infinitesimal change in the environment of the system The slow expansion of the
gas at Th in the Carnot cycle with tiny amounts of heat being transferred to the environment
is reversible If we turned around and began slowly to compress the gas would just slowly
exhaust heat to the environment in the exact reverse way that it absorbed it during expansion
Most processes however are irreversible For example if a gas-filled box were suddenly
increased in size so that the particles were free to wander into the void created by the sudden
expansion then the gas would eventually fill the new volume uniformly at the same
temperature as before the expansion (The temperature is unchanged in this imaginary
process because the kinetic energy of the molecules would be unaffected by such an
instantaneous expansion of the container walls) This imaginary but highly thought-
stimulating process is called a free expansion and it is impossible to reverse it During the
expansion we didnrsquot push on any of the molecules so reversing this process would mean
making them go back into their original volume without pushing on them they simply will
not cooperate to this extent Another way to see that just pushing them back where they came
from does not reverse the free expansion is to think about what would happen if we just
compressed either adiabatically or isothermally An adiabatic compression back to the
original volume would heat the gas above its original temperature and an isothermal
compression would require that heat be exhausted to the environment But the free expansion
involved neither temperature changes nor heat exchanges so neither of these two processes
is the reverse of the free expansion It is simply impossible to reverse this rapid expansion in
a way that takes us back to the initial state
It is possible to calculate the change of entropy for both reversible and irreversible processes
Letrsquos consider a reversible process first In a reversible process the entropy change is given
by the formula
dQS
T
where dQ is the amount of heat added to the system during a small step of the process
The total energy change during the process may then simply be calculated by integration
f f
f i i i
dQS S S dS
T
13
(Just as in the case of energy we are mainly interested in differences rather than in
absolute magnitudes)
This integration method does not work for irreversible processes For instance in the
rapid free expansion discussed above no heat is added to the system but its disorder
obviously goes up We need to find some other way to calculate the entropy The key
is the fact that the entropy of a system depends only on its current state and not at all
on how it arrived there So to calculate the entropy change in an irreversible process
first find out what the initial and final conditions of the process are Then invent a
reversible process that takes the system from the initial state to the final state Since
the entropy depends only on the state of system and not on the process the entropy
change for the reversible process is the same as that for the irreversible process
Entropy is thus like pressure or temperature if the pressure changes from Pl to to P2
then the pressure difference is P2 - Pl regardless of how the pressure change was
made This seems obvious for pressure but not for entropy only because you arent
comfortable with entropy yet
Entropy of an Ideal Gas
For n moles of an ideal gas the difference in entropy between a state with temperature
T and volume V and some standard state with temperature To and volume Vo is given
by
ln lno o Vo o
T VS T V S T V nC nR
T V
Entropy in a Phase Change
Calculating entropy change in a phase change like melting or boiling is easy It is
QT Where Q is in the latent heat for example the flows in to cause the melting and
T is the temperature of the phase change There is NO integral to do
14
Serway Chapter 35
Angle of Reflection
If plane waves are incident on a reflecting surface with the propagation direction of
the waves making angle θ1 with the normal direction to the surface then the reflected
angle θ1prime relative to the surface normal is simply
1 1
ie the incident angles and reflected angles are the same
Refraction
If plane waves traveling through medium 1 are incident on a plane interface between
medium 1 and medium 2 then the angle of incidence of the incoming wave θ1 and
the angle of refraction of the transmitted wave θ2 are related by Snellrsquos law
2 2
1 1
sin
sin
v
v
where v1 and v2 are the wave speeds in medium 1 and medium 2 The angle of
incidence and the angle of refraction are both measured between the wave propagation
direction and the normal to the interface In terms of indices of refraction in the case
of light waves Snellrsquos law takes the more familiar form
1 1 2 2sin sinn n
where ni = cvi
Total Internal Reflection
If a wave is incident from a medium of low wave speed into a medium of high wave
speed the law of refraction requires that the angle of refraction be greater than the
angle of incidence If the angle of refraction is required to be greater than 90deg then no
refracted wave can exist and total internal reflection occurs The critical incident
angle θc beyond which total internal reflection occurs is given by
2
1
sin c
n
n
15
Dispersion of Light
In addition to the speed of light varying from material to material it also varies with
wavelength within each material This means that the index of refraction is generally a
function of wavelength
c
nv
Since the wave speed is not constant such a medium is dispersive meaning in this
context that refraction actually disperses white light into its various colors because
Snellrsquos law gives a different angle for each wavelength In most materials the
variation with wavelength is quite small but this small effect is responsible for some
of the most spectacular color effects we ever see including rainbows a flashing
crystal chandelier and the colored fire of a diamond solitaire by candlelight
16
Serway Chapter 36
Real and Virtual Images
When light rays are focused at a certain plane producing an image if a sheet of white
paper is placed there we call the image a real image The images produced by film
projectors and overhead projectors are examples of real images
When light rays appear to come from a certain location but no image is produced
when a screen is placed there we say that there is a virtual image at that location For
instance when you look in a mirror it appears that someone is behind the mirror but
a screen placed back there in the dark would show nothing Your image in the mirror
is a virtual image
Ray Tracing
There are lots of rules about how to find the images in optical systems but the best
way to keep things straight is to learn how to draw the principal rays for curved
mirrors and lenses The rays for convex and concave mirrors are shown in Fig 3615
The rays for converging and diverging thin lenses are shown in Fig 3627 You should
memorize the principal rays and know how to use them to locate images
Curved Mirrors
The focal length of a curved mirror with radius of curvature R is given by
2
Rf
The relation between object distance p image distance q and focal length f is
1 1 1
p q f
If the mirror is a diverging mirror f should be negative and if q should turn out to be
negative the image is virtual
Lateral Magnification
The lateral magnification in an optical system is defined by the ratio of the image size
to the object size
17
Image height
Object height
qM
p
for curved mirrors
As usual there are sign conventions here too but this formula just gives the
magnitude It is better to keep keep track of upright images versus inverted images by
means of ray diagrams rather than by memorizing sign conventions
Thin Lenses
The focal length of a thin lens is related to the radii of curvature of the two faces R1
and R2 of the lens by
1 2
1 1 11n
f R R
Note that this formula differs from Eq (3611) in the text by not having a minus sign
between the two R-terms We like this form better because for a simple converging
lens like a magnifying glass we just use positive values of R for both surfaces If one
of the faces is concave producing divergence use a negative value for R And if a
surface is flat use R = infin If the face is flat the radius is infinite
The relation between the image object and focal distances for a thin lens is the same
as that for a curved mirror
1 1 1
q p f
Use a negative focal length if the lens is diverging
The lateral magnification for a thin lens is the same as for a curved mirror
Image height
Object height
qM
p
Camera
The lens system in a camera projects a real image of an object onto the film (or CCD
array in a digital camera) The position of the image is adjusted to be on the film by
moving the lens into or out of the camera
18
Eye
The eye is like a camera in that a real image is formed on the retina Unlike a camera
the image position is adjusted by changing the focal length of the lens This is done by
the ciliary muscle which squeezes the lens changing its shape
Near Point The near point is the closest distance from the eye for which the lens can
focus an image on the retina It is usually 18-25 cm for young persons
Far Point The far point is the greatest distance from the eye for which the lens can
focus an image on the retina For a person with normal vision the far point is at
infinity
Nearsightedness A person is nearsighted if their far point is at some finite distance
less than infinity This condition can be corrected with a lens that takes an object at
infinity and produces a virtual image at the personrsquos far point
Farsightedness and Presbyopia A person is farsighted if their near point is too far
away for comfortable near work like reading or knitting This can be corrected by a
lens which takes an object at a normal near point distance of 18-25 cm and produces a
virtual image at the personrsquos natural near point Presbyopia involves a similar
problem which nearly all people experience as they age The ciliary muscle becomes
too weak and the lens becomes too stiff to allow the eye to provide for both near and
far vision The solution for this problem is either reading glasses or bifocal lenses
Reading classes are just weak magnifying glasses mounted on eyeglass frames
Bifocal lenses are split into upper and lower halves The lower half is a lens which
gives the proper correction for near work and the upper half is a different lens for
proper focusing at infinity
Angular Size
When an object is brought closer to the eye it appears to be larger because the image
on the retina is larger The size of this image is directly proportional to the objectrsquos
angular size which is the angle subtended by the object measured from the center of
the lens of the eye In optical instruments which are to be used with the eye the
angular size of the final image is whatrsquos important because it determines how large
the image will appear to the viewer
19
Simple Magnifier
A simple magnifier is a single converging lens or magnifying glass It takes an object
closer to the eye than a normal near point and produces a virtual image at or beyond
this near point The angular magnification is defined to be the ratio of the angular size
when viewed through the lens to the angular size of the object when viewed at the
normal near point (without aid of the lens)
Microscope
This instrument has two lenses (1) The objective is near the object being viewed and
produces a greatly magnified real image (2) The eyepiece is a simple magnifier which
the viewer uses to closely examine the image from (1)
Telescope
This instrument also has two lenses (1) The objective at the front of the telescope
takes light from a distant object and produces a real inverted image (which is rather
small) near its focal point (2) This small real image is then examined by the eyepiece
functioning as a simple magnifier to produce a virtual image with a larger angular
size
20
Serway Chapter 37
Two-Slit Interference
If light is incident on two closely spaced narrow slits a pattern of light and dark
stripes is produced beyond the slits The bright stripes or fringes are caused by
constructive interference of the two waves coming from the slits Constructive
interference occurs whenever two waves arrive at a location in phase with each other
This occurs when the distance x1 from slit 1 to a point P on the screen and the
distance x2 from slit 2 to point P differ by in integral number of wavelengths
1 2 where 0 1 2x x m m
where λ is the wavelength of the light When x1 and x2 are much larger than the slit
spacing d this condition reduces to
sind m
where θ is the angle between the direction of the incident light and the direction of the
light arriving at the screen
Thin Films
When light is partially reflected and partially transmitted by a thin film of transparent
material it is possible to have interference between the wave reflected from the front
of the film and light reflected from the back of the film (The colored reflections from
the thin film of oil on the water in a rain-soaked parking lot are an example of this
effect) It is difficult to write down formulas that will work in all cases so we will
just review the important principles here
1 If the two reflected waves are in phase with each other the film has enhanced
reflection (constructive interference) but if the two reflected waves are out of phase
with each other reflection is diminished (destructive interference) Phase shifts occur
due to reflection and due to the extra path length through the film of the wave
reflected from the back of the film
2 The phase change due to reflection is determined by the difference in index of
refraction between the two media involved in the reflection If the wave is incident
21
from a medium with a low index of refraction into a medium with a high index of
refraction a phase change of 180deg occurs and the reflected wave is inverted If
incident from high to low no phase shift occurs and the reflected wave is non-
inverted
3 The extra path length through the film of the wave reflected from the back of the
film is equal to 2t where t is the thickness of the film (The incident light is assumed
to be normal to the surface of the film) The number of wavelengths contained in the
extra path length is equal to 2tλn where λn = λn is the wavelength of the light in the
film
4 Rules for reflection from thin films
If one ray is inverted and the other is not then we have
12 constructive
2 nt m
2 destructivent m
If either both rays are inverted or both are non-inverted we have
2 constructivent m
12 destructive
2 nt m
22
Serway Chapter 38
Diffraction Grating
A diffraction grating is simply a fancy version of two-slit interference with the two
slits replaced by thousands of slits Just as in the two-slit case the bright fringes
occur at angles given by
sin md
but in this case d the distance between neighboring slits is made to be very small
The effect of having many slits instead of two is to make each bright fringe highly
localized with wide dark regions between neighboring maxima
Single Slit Diffraction
When light passes through an opening in an opaque screen an interference pattern is
produced beyond the opening To understand why we may replace the single opening
by many small coherent sources of light These many sources interfere with each
other producing a pattern known as a diffraction pattern (Note that many authors do
not distinguish between interference and diffraction treating them as interchangeable
terms) If the opening is a slit of width a then the diffraction pattern far from the slit
will have a bright central maximum with a succession of minima and weaker maxima
on either side The angle between the incident direction and the minima is given by
sin ma
where m = plusmn1 plusmn2
If the opening is circular with diameter D the angle between the incident direction
and the first minimum is given by
sin 122D
Optical Resolution and Rayleighrsquos Criterion
Two point sources can just be resolved (distinguished from each other) if the peak of
the diffraction image of the first source overlies the first minimum of the diffraction
image of the second source For circular holes of the kind usually encountered in
23
optical devices this condition is approximately satisfied when the angular separation
between the two sources as viewed from the optical instrument is greater than or equal
to the critical angle
min 122D
where λ is the wavelength of the light and where D is the diameter of the aperture in
the instrument
Polarization
We say that an electromagnetic wave is polarized if its electric field vector doesnrsquot
change direction in a random fashion The simplest kind of polarization is linear
polarization in which the electric field vector oscillates back and forth along the
same axis in space Polarized light can be produced from normal unpolarized light by
selective absorption (as in Polaroid sunglasses) by passing light through crystals that
have different indices of refraction for different polarizations (double refraction) by
scattering (the blue sky is polarized) and by reflection (glare) Polarization by
reflection occurs when light reflects from a shiny insulating (non-metallic) surface
The amount of polarization is greatest for reflection at Brewsterrsquos angle
2
1
tan P
n
n
where θP is the incidence angle of light from medium 1 onto medium 2 and where n1
and n2 are the indices of refraction for the media (In our everyday experience n1 = 1
since the light comes in through the air and n2 is the index of refraction of the shiny
insulating material producing the glare eg water glass plastic paint etc)
Malusrsquos Law
The intensity of transmitted polarized light through a perfect polarizer is related to the
incident intensity of polarized light by Malusrsquos law
2cosoI I
where Io is the intensity of the incident light and where θ is the angle between the
electric field vector in the incident wave and the transmission axis the polarizer
24
Serway Chapter 39
Principles of Relativity
All of the weirdness of relativity flows from two simple principles
(1) The laws of physics must be the same in all inertial (non-accelerating) reference
frames
(2) The speed of light in vacuum has the same value in all inertial reference frames
Note that (1) seems reasonable but (2) is very odd It says that if two space ships are
approaching each other at nearly the speed of light and a laser pulse is shot from ship
1 toward ship 2 then when the people on ship 2 measure the speed of the pulse as it
goes by it is moving at 3 times 108 ms the same as if the ships were stationary
Simultaneity
If observer 1 sees two events in her own frame as simultaneous at two different
locations a moving observer 2 will see these two events happening at different times
Relativistic gamma
The factor γ (gamma) appears regularly in the formulas of relativity
2
2
1
1 vc
where v is the relative speed between two inertial frames
Time Dilation
If observer 1 sees two events at the same location in space separated by time Δtp in his
own frame then observer 2 moving at speed v relative to observer 1 will see these
two events separated by a longer time Δt
pt t
Moving clocks run slow
Length Contraction
If observer 1 measures the length of an object along the x-direction at some instant of
time in her frame to be Lp then observer 2 moving in the x-direction at speed v will
measure the length L of the object to be shorter
pLL
25
Moving meter sticks are short
Velocity Addition
If a particle is moving at velocity vac relative to frame c and if frame c is moving at
speed vcb with respect to frame b (with both velocities directed along the same line)
what would be the velocity of the particle in frame b In classical physics the answer is
ab ac cbv v v
but if any of these velocities are an appreciable fraction of the speed of light the
answer changes
21ac cb
abac cb
v Vv
v v c
In using this equation make sure that vac has the same sign in the numerator and in the
denominator and that vbc also has the same sign in both places Note that this equation
is different (and in our opinion easier to understand) than the equations in the text
Momentum and Energy
The momentum and total energy of a particle moving at velocity v (its speed is v v )
are given by
22
2 2
22
1 1 vvc c
m mcm E mc
vp v
Rest Energy
When a particle is at rest (v = 0 and γ = 1) then E = mc2 is called the rest energy It
includes the potential energy of the particle so that if we change the potential energy
of a particle we change its rest mass
Kinetic Energy The kinetic energy is the difference between the total energy and the
rest energy
2 2K mc mc
For v ltlt c this reduces to K = mv22
General Relativity
Einsteinrsquos principle of equivalence in the theory of general relativity is In the vicinity
of a point a gravitational field is equivalent to an accelerated frame of reference in
26
the absence of gravitational effects For example there is no difference whatever in
the physics experienced by an astronaut on earth and one who is in interstellar space
far away from any stars or planets but who has just turned on her spaceshiprsquos engines
and is accelerating at 98 ms2
27
Serway Chapter 40
Photons
We now know that even though light behaves like a wave it is better described as
consisting of small packets of energy called photons The energy of a photon is related
to its frequency by
E hf
where h is Planckrsquos constant
346626 10 J sh
Photons also carry momentum given by
hf hp
c
Since the amount of energy in each photon is fixed the difference between dim light
and intense light (of the same wavelength) is that dim light consists of fewer photons
than intense light
Photoelectric Effect
In the photoelectric effect light shining onto a metal surface gives the electrons in the
metal enough energy to escape and be detected It requires a certain amount of energy
(called the work function typically = 3 ndash 5 eV) just to kick the electron out of the
metal so the light must deliver at least this much energy to an electron to produce the
effect It is observed that red light no matter how intense never produces electrons
But ultraviolet light even if quite dim will eject electrons from the metal Since red
light consists of 2 eV photons and ultraviolet light has photons with energies around
4-6 eV the photon idea explains the behavior of the photoelectric effect The
maximum energy that an ejected electron can have is
maxK hf
where f is the photon frequency The minimum frequency that light can have and cause
electrons to be ejected is called the cutoff energy and is found by setting Kmax = 0
cf h
28
Compton Effect
When high-frequency light interacts with free electrons the scattered light does not
have the same wavelength as the incident light contrary to what classical
electromagnetism would predict But the photon picture in which photons have
momentum and energy allows us to treat scattering as a collision between the photon
and the electron In this collision the electron and the scattered photon both have a
different momentum and energy than they did before And since p = hλ for a photon
if the momentum is different the wavelength will also be different
after before 1 cose
h
m c
where the angle θ is the angle between the incoming photon direction and the
direction of the scattered photon
Particles are Waves
Since photons behave like particles sometimes it is not surprising that elementary
particles can behave like waves sometimes The wavelength of a particle with
momentum p is given by
h
p
Wave-particle Duality
Both photons and elementary particles have a dual nature sometimes they behave like
particles and sometimes they behave like waves It is hard for us to comprehend the
nature of such an object by making mental pictures but experiments definitively show
that this is the case Since we have no direct experience with photons and elementary
particles (because their energies are so much smaller than the energies of the everyday
objects we encounter) it is perhaps not surprising that we have a hard time forming a
mental picture of how they behave
Electron Interference
Since an electron is both a particle and a wave just like a photon it should be able to
produce an interference pattern This is observed If an electron beam is shot at two
closely-spaced slits and if the electrons that pass through the slits are detected
downstream it is found that there are some locations where electrons are never detected
29
and others where lots of electrons are detected The pattern is exactly the same as the
one observed for light waves provided that we use the electron wavelength λ = hp in
place of the wavelength of light This pattern is observed even though each electron is
detected as a single dot on the screen Only after many such dots are collected does the
pattern emerge And if we try to understand how this effect could possibly work by
looking closely at each slit to see which one the electron came through the pattern
disappears the act of measurement destroys the interference This means that each
single electron somehow comes through both slits (as a wave would)
Uncertainty Principle
In classical physics we always imagine that the positions and momenta of moving
particles have definite values It might be hard to measure them but surely at each
instant of time a particle should be precisely located at some point in space and have a
similarly precise momentum This turns out not to be true Instead both position and
momentum are required to be uncertain with their uncertainties Δx and Δp satisfying
the Heisenberg uncertainty relation
2x p
So if the particle were known to be precisely at some particular location (so that Δx =
0) we couldnrsquot know anything about its momentum (Δp = infin) and if its momentum
were exactly known we couldnrsquot know its position
There is a similar relation involving the particlersquos energy E and the time interval Δt
over which this energy is measured
2E t
This means that energy is not actually conserved if we are considering very small time
intervals and this brief non-conservation of energy has been observed
30
Serway Chapter 41
Wave Function ψ and Probability
Quantum mechanics does not predict exactly what an electron or a photon will do Instead
it specifies the wave function or probability amplitude ψ of an electron or a photon
This wave function is a complex-valued function of space and time whose squared
magnitude is the probability density P for finding a particle at a particular place in
space at a certain time
2P
where is the complex conjugate of ψ
For example the wave function of an electron with perfectly specified momentum p
would have a wavelength given by
h
p
and its probability amplitude would be proportional to
2i x ipxe e
The corresponding probability density would then be
21ipx ip ipxP e e e
which means that the electron is equally probable to be anywhere along the x axis
This is in accord with the uncertainty principle since we specified the momentum
precisely we canrsquot have any idea about the position of the electron
As another example you have probably seen ldquofuzzy ballrdquo drawings of electron
orbitals in a chemistry book These fuzzy balls are meant to indicate the distribution
of the probability density 2
P in the orbital
Particle in a Box
A simple example in which we can calculate the wave function is the case of a particle of
mass m constrained to be inside a 1-dimensional box between x = 0 and x = L In this
case the general wave function is a linear superposition of wave functions ψn of the form
sin niE tn
n xx t A e
L
31
where A is a positive constant where n = 1 2 3 and where the energy associated
with each of the quantum states ψn is given by
22
28n
hE n
mL
This wave function is zero at x = 0 and x = L which means that the particle will never
be found at the walls of the box The wave function has maximum values in the
interior and at these places the particle is most likely to be found and it also has
places where it is zero and at these places the particle will also never be found as
expressed by the formula
2sinn x
PL
The particle in the box is interfering with itself producing a probability interference
pattern across the box just like the interference patterns we studied with light and
sound
32
Serway Chapter 42
Atoms
Many people picture an atom as a miniature solar system where electrons orbit around
a massive nucleus at the center This picture is misleading because in atoms the wave
nature of electrons dominates The electrons form 3-dimensional standing waves
(called orbitals) centered at the nucleus And if you ask what is it thatrsquos waving the
answer is ψ the probability amplitude (see Chapter 41)
Standing waves on a string can occur only for certain frequencies (the fundamental
and higher harmonics) Similarly atomic orbitals occur only for certain energies For
the hydrogen atom the energies of the orbitals have the a particularly simple form
2
1136 eVnE
n
where n = 1 2 3 is called the principal quantum number For other atoms the
determination of the orbital energies requires numerical calculation by computers
Atomic Spectra
If an electron is somehow given extra extra energy (we say that it is excited) so that it
occupies a higher orbital it will eventually ldquofallrdquo back down to a lower orbital Each
time an electron falls to a lower orbital it loses the difference in energy between the
two orbitals in the form of a photon Since the orbital energies are discrete so are the
energy differences and so are the wavelengths of the emitted light The entire set of
these discrete wavelengths is called the atomic spectrum and it is unique to each type
of atom For hydrogen the spectrum can be simply written as
H 2 2
1 1 1
f i
Rn n
where RH = 10973732 times 107 m-1 For other atoms the spectrum cannot be expressed as
a formula
Orbital Angular Momentum Electrons in atoms also have quantized values of
angular momentum The orbital quantum number ℓ specifies the value of this
quantized angular momentum through the formula
1L
33
If we want to know the value of the angular momentum along some direction in space
say the z direction the answer is not L but rather
zL m
where mℓ is another quantum number which runs from
1 1m
This quantum number is important when an atom sits in a magnetic field
Spin Angular Momentum It has been found experimentally that electrons and other
charged particles also carry internal angular momentum which we call spin
Electrons have an intrinsic spin angular momentum s along a specified axis that is
extremely quantized it can only take on 2 possible values
1
2zs s
We thus say that electrons have spin s = 12 or that electrons are rdquospin one-half parti-
cles Since s is an angular momentum it obeys the same rule as that for orbital angular
momentum namely that if its value along some axis is s then its total magnitude is
given by
31
2S s s
I know this seems weird but quantum mechanics is weird The only excuse for this
bizarre way of looking at the world is that it predicts what happens in experiments
Exclusion Principle The answer to the question of how many electrons (or any other
spin one-half particle) can be in one particular quantum state was discovered by
Wolfgang Pauli and is called the exclusion principle
ldquoNo two electrons can ever be in the same quantum state therefore no two electrons
in the same atom can have the same set of quantum numbersrdquo
This is the reason that we have atoms with different properties instead of every atom
simple having all of its electrons in the ground state All of the variety we see around
us in the world is the result of chemical differences and these differences would not
exist unless electrons obeyed this important principle The entire structure of the
periodic table (see pages 1377-1379 in Serway) is an expression of this principle
34
Serway Chapter 44
Nuclear Properties
The nucleus is composed of protons with charge +e and mass mp = 1007226 u and of
neutrons with zero charge and mass mn = 1008665 u where u is one atomic mass unit
-271 u=1660540 10 kg
The atomic number Z counts the number of protons in a nucleus while the neutron
number N counts the number of neutrons The mass number A is the sum of the two
A N Z
Protons and nuetrons have about the same mass and some times called baryons (heavy
ones) or nucleons (partcles in the nucleus) A is sometimes called the baryon
number The chemical elements are distinguished by Z ie hydrogen has Z = 1 iron
has Z = 26 uranium has Z = 92 etc But for a given Z there might be several nuclei
with different numbers of neutrons These nuclear siblings are called isotopes
These numbers are used to label nuclei according to the pattern
5626 ie FeA
Z X
denotes the isotope of iron with N = 56 ndash 26 = 30 neutrons
The nucleus is roughly spherical with a radius given approximately by
1 3 150 0where 12 10 mr r A r
Nuclear Stability
Because the positively charged protons electrically repel each other with an enormous
force at distances as small as 10-15 m there must be some really strong force that that
overcomes electrical repulsion to hold protons and neutrons together This force is
called with some lack of imagination the strong force It is a very short-range force
(it only acts over a distance of about 2 x 10-15 m) and attracts protons to protons
neutrons to neutrons and neutrons to protons But in spite of this strong nuclear force
the coulomb repulsion of the protons is still present so anything that might keep the
protons from being right next to each other would help keep the nucleus from
35
exploding This role is played by the neutrons and for nuclei with Z le 20 the stable
nuclei roughly have N = Z
For Z gt 20 the coulomb repulsion force is more powerful and more neutrons are
needed to dilute it up to about N = 15Z around Z = 80 For Z gt 83 no amount of
neutrons can help and these nuclei are unstable (radioactive) There is an ldquoisland of
(relative) stabilityrdquo around Z=90-92 (thorium and uranium respectively) These
elements have one or two isotopes with half-lifes of billions of years so there are
substantial amounts of such elements on Earth This fact makes it possible to have
practical fission devices
Radioactive Decay
There are three types of radioactive decay
Alpha decay the nucleus kicks out a helium nucleus (N = 2 Z = 2 A = 4) So if the
original nucleus is called X and the new nucleus is called Y then the decay would look
like this
4 42 2X Y+ HeA A
Z Z
Beta decay the nucleus either kicks out an electron (endash) or its positively-charged
antimatter twin the positron (e+) plus either an electron neutrino v or an electron
anti-neutrino v
1X Y+e electron decayA AZ Z v
1X Y+e positron decayA AZ Z v
A neutrino is a particle with no charge hardly any mass (much less than the electron
mass) and interacts so weakly with matter that most neutrinos upon encountering the
planet earth just pass right through it as if it werenrsquot there
Gamma decay the nucleons in the nucleus X are in an excited energy state X
(perhaps as a result of having undergone alpha or beta decay) and they drop down to a
lower energy state shedding the energy as a high frequency photon
X XA AZ Z
36
This process is exactly analogous to the way that the electrons in atoms emit photons
Decay Rate and Half Life
There is no way to predict exactly when an unstable or excited nucleus will decay but
there is an average rate at which this decay occurs called the decay constant λ The
meaning of this constant is that if there are a large number N of nuclei in a sample
then the number of decays per second that will be observed (called the decay rate R)
is R = λN In mathematical language
dMR N
dt
This simple differential equation has for its solution
0tN t N e
where N0 is the number of nuclei in the sample at time t = 0
The half-life is the time it takes for half of the nuclei in the sample to decay and is
related to the decay constant by
1 2
ln 2 0693T
Disintegration Energy
When a nucleus decays it is making a transition to an overall state of lower energy
which means according to Einsteinrsquos famous formula E = mc2 that the sum of the
masses after the decay must be less than the mass before with the lost mass appearing
as kinetic energy among the decay products For example in alpha decay this kinetic
energy called the disintegration energy Q is given by
2X YQ M M M c
37
Serway Chapter 45
Nuclear energy
There are two ways to extract energy from the nucleus fission and fusion For nuclei
with Z greater than 26 breaking the nucleus apart into pieces leads to a lower
2mc energy than the original nucleus so energy can be extracted by fission For
nuclei with Z less than 26 a lower energy is achieved by combining nuclei so energy
can be extracted by fusion
Fission
Since neutrons have no charge they are not repelled from nuclei as protons are for
this reason their behavior is the key to understanding how fission works
Fast neutrons mostly bounce off other nuclei with each collision slowing the neutron
down This elastic energy loss is most effective if the other nuclei have low mass (like
hydrogen) and these materials are called moderators because of their ability to slow
down fast neutrons
The reason that slowing neutrons is important is that slow neutrons are much more
likely to be absorbed by a nucleus which then leads to nuclear reactions of various
kinds For a few very large nuclei like uranium-235 and some plutonium isotopes
absorption of a slow neutron causes the nucleus to split into two large fragments plus
2 or 3 fast neutrons (energy is released in the process too) Thus one slow neutron
can produce energy plus 2 or 3 more neutrons which if moderated from fast to slow
can split 2 or 3 more nuclei leading to a runaway chain reaction If the reaction is
allowed to proceed unchecked it produces a large explosion If a neutron absorbing
material is added to the mix (like the cadmium in reactor control rods) it is possible to
keep the reaction under control and to extract the released energy as heat to drive
steam turbines and produce electricity
Fusion
Fusion involves mashing two nuclei together and since they are both charged and repel
each other this reaction is much harder to make go The nuclei must have enough energy to
overcome the coulomb repulsion which is why this reaction requires a high temperature
(hundreds of millions of degrees K) like that in the sun (which is burning hydrogen to
helium via fusion) or like that in the center of the fission explosion that is used to detonate
38
a hydrogen bomb
This reaction is of interest for power production in spite of this difficult temperature
requirement because of the abundance of fusion fuel on the planet There are about
012 g of deuterium ( 21H ) in every gallon of water on earth and it only costs about 4
cents to extract it The fusion energy available from this minuscule amount of
deuterium would run a 1000-Megawatt power plant for 10 seconds Water is so
abundant on earth that if fusion were to work we would have an essentially
inexhaustible source of energy
So why donrsquot we have fusion power plants Well the fuel is cheap but the match is
incredibly expensive The only way we know to control this difficult high-temperature
reaction is with large and expensive pieces of equipment involving either large
magnetic fields and complex high-power electromagnetic antennas or with gigantic
(football-fieldsized) laser facilities involving more than a hundred of the highest-
energy lasers ever built Power plants based on these current methods for controlling
fusion are unattractive to the fiscally-minded people who run the electric power
industry Hopefully better designs will be discovered as experiments continue
39
c actinium 68 Er erbium 101 Md mendelevium 104 Rf rutherfordium l aluminum 63 Eu europium 80 Hg mercury 62 Sm samarium m americium 100 Fm fermium 42 Mo molybdenum 21 Sc scandium b antimony 9 F fluorine 60 Nd neodymium 106 Sg seaborgium r argon 87 Fr francium 10 Ne neon 34 Se selenium s arsenic 64 Gd gadolinium 93 Np neptunium 14 Si silicon t astatine 31 Ga gallium 28 Ni nickel 47 Ag silver a barium 32 Ge germanium 41 Nb niobium 11 Na sodium k berkelium 79 Au gold 7 N nitrogen 38 Sr strontium e beryllium 72 Hf hafnium 102 No nobelium 16 S sulfur i bismuth 108 Hs hassium 76 Os osmium 73 Ta tantalum h bohrium 2 He helium 8 O oxygen 43 Tc technetium
boron 67 Ho holmium 46 Pd palladium 52 Te tellurium r bromine 1 H hydrogen 15 P phosphorus 65 Tb terbium d cadmium 49 In indium 78 Pt platinum 81 Tl thallium a calcium 53 I iodine 94 Pu plutonium 90 Th thorium f californium 77 Ir iridium 84 Po polonium 69 Tm thulium
carbon 26 Fe iron 19 K potassium 50 Sn tin e cerium 36 Kr krypton 59 Pr praseodymium 22 Ti titanium s cesium 57 La lanthanum 61 Pm promethium 74 W tungsten l chlorine 103 Lr lawrencium 91 Pa protactinium 92 U uranium r chromium 82 Pb lead 88 Ra radium 23 V vanadium o cobalt 3 Li lithium 86 Rn radon 54 Xe xenon u copper 71 Lu lutetium 75 Re rhenium 70 Yb ytterbium m curium 12 Mg magnesium 45 Rh rhodium 39 Y yttrium b dubnium 25 Mn manganese 37 Rb rubidium 30 Zn zinc y dysprosium 109 Mt meitnerium 44 Ru ruthenium 40 Zr zirconium s einsteinium
40
UNITS The SI units are given in parentheses Other commonly used units are given in terms of the SI units Symbols conform with the recommendations of the American National Standards Institute (ANSI) and the American Institute of Physics (AIP) Commonly used multiples of SI units
prefix symbol factor tera T 1012 giga G 109
mega M 106 kilo k 103
centi c 10-2 milli m 10-3
micro μ 10-6 nano n 10-9 pico p 10-12
femto f 10-15
length meter (m) angstrom 1 Ǻ = 10-10 m inch 1 in = 254 x 10-2 m foot 1 ft = 03048 m mile 1 mi = 1609 m
mass kilogram (kg) atomic mass unit 1 u = 1661 x 10-27 kgslug 1 slug = 1459 kg
force newton (N = kg bull ms2) dyne 1 dyn = 10-5 N pound 1 lb = 4448 N
pressure Pascal (Pa = kgm bull s2) atmosphere 1 atm = 1013 x 105 Pa poundssquare inch 1 psi = 6895 Pa cm of mercury 1 cm Hg = 1333 Pa bar 1 bar = 1000 x 105 Pa torr 1 torr = 1333 Pa
time second (s) minute 1 min = 60 s hour 1 h =3600 s
frequency hertz (Hz = s-1) radianssecond 1 rads = 12π Hz
energy joule (J = kg bull m2s2) erg 1 erg = 10-7 J electron volt 1 eV = 1602 x 10-19 J calorie 1 cal = 4187 J kilowatt-hour 1 kWh = 36 x 106 J British thermal unit 1 Btu = 1055 J
power watt (W = kg bull m2s3) horsepower 1 hp = 7457 W
charge coulomb (C = A bull s)
electric potential volt (V = kg bull m2 s3 bull A)
current ampere (A)
resistance ohm (Ω = kg bull m2s3 bull A2)
capacitance farad (F = s4 bull A2kg bull m2)
magnetic field tesla (T = kgs2 bull A) gauss 1 G = 10-4 T
magnetic flux weber (Wb = kg m2s2 bull A) maxwell 1 Mx = 10-8 Wb
magnetic inductance henry (H = kg bull m2s2 bullA2)
temperature kelvin (K) degrees Celsius 0degC = 27315 K
angle radian (rad) degree 1 deg = π180 rad revolution 1 rev = 2π rad
41
Some Physical Constantsa Quantity Symbol Valueb Atomic mass unit μ 1660 538 73 (13) x 10-27 kg
931494 013 (37) MeVc2 Avogadros number NA 6022 141 99 (47) x 1023 particlesmol Avogadrorsquos number 6022 x 1023 mol Bohr magneton
2B e
em 9274 008 99 (37) x 10-24 JT
Bohr magneton μB 927 x 10-24 JT Bohr radius
0
2
2e em e k
a 5291 772 083 (19) x 10-11 m
Boltzmanns constant B A
RNk 1380 650 3 (24) x 10-23 JK
Boltzmannrsquos constant kB 1380 x 10-23 JK Compton wavelength
C eh
m c 2426 310 215 (18) x 10-12 m
Deuteron mass md 3343 583 09 (26) x 10-27 kg 2013 553 212 71 (35) u
electron charge e 1602 x 10-19 C Electron mass me 9109 381 88 (72) x 10-31 kg
5485 799 110 (12) x 10-4 u 0510 998 902 (21) MeVc2
electron mass 911 x 10-31 kg Electron volt eV 1602 176 462 (63) x 10-19 J Elementary charge e 1602 176 462 (63) x 10-19 C Gas constant R 8314 472 (15) JKmol Gravitational constant G 6673 (10) x 10-11 Nm2kg2 Neutron mass mn 1674 927 16 (13) x 10-27 kg
1008 664 915 78 (55) u 939565 330 (38) MeVc2
neutron mass 1675 x 10-27 kg Nuclear magneton
2n p
em 5050 783 17 (20) x 10-27 JT
permeability constant μ0 1257 x 10-6 Hm permittivity constant ε0 8854 x 10-12 Fm Plancks constant h
2h
6626 068 76 (52) x 10-34 Js 1054 571 596 (82) x 10-34 Js
Planckrsquos constant h ħ
6626 x 10-34 Js 1055 x 10-34 Js
Proton mass mp 1672621 58 (13) x 10-27 kg 1007 276 466 88 (13) u 938271 998 (38) MeVc2
proton mass 1673 x 10-27 kg Rydberg constant RH 1097 373 156 854 9 (83) x 107 m-1 Speed of light in vacuum c 2997 92458 x 108 ms (exact) speed of light c 300 x 108 ms a
These constants are the values recommended in 1998 by CODATA based on a least-squares adjustment of data from different measurements For a more complete list see P J Mohr and B N Taylor Rev Mod Phys 72351 2000 b The numbers in parentheses for the values above represent the uncertainties of the last two digits
NOTE The ones in red are the ones from Appendix 13 The ones in yellow are the ones that are from the Serway book table but that were already on the Appendix 13 table
42
INDEX
Absolute zero 9
Adiabatic 17
Adiabatic exponent 18
Adiabatic process 17
Alpha decay 46
Amplitude 4
Angle of reflection 25
Angular frequency 3
Angular magnification 30
Angular momentum atomic 43
Angular size 29
Archimedes Principle 1
Atomic spectra 43
Atoms 43
Avogadrorsquos number 10
Beats 8
Bernoullirsquos Equation 2
Beta decay 46
Bifocals 29
Brewsterrsquos angle 34
British Thermal Unit Btu 11
Buoyancy 2
calorie 11
Calorie 11
Camera 28
Carnot cycle 21
Carnot efficiency 22
Celsius scale 9
Ciliary muscle 29
Coefficient of performance 21
Compton effect 39
Constructive interference 7
Continuity equation of 2
Contraction length 35
Convection 14
Curved mirrors 27
Decay constant 46
Decay rate radioactive 46
Decibel scale 5
Degrees of freedom 16
Density 1
Destructive interference 7
Diffraction single slit 33
Diffraction grating 33
Dilation time 35
Disintegration energy 47
Dispersion of light 26
Doppler effect 6
Double slit interference 31
Efficiency 20
Electron interference 39
Emissivity 15
Energy nuclear decay 47
Energy relativistic 36
43
Engineering work 20
Entropy 22
Entropy ideal gas 24
Equipartition of energy 18
Equivalence principle 36
Exclusion principle 44
Expansion thermal 9
Expansion coefficient linear 9
Eye 29
Fahrenheit scale 9
Far point 29
Farsightedness 29
First Law of Thermodynamics 12
Fission 48
Flux volume 2
Free expansion 23
Fringes 31
Fusion 48
Fusion heat of 11
Gamma relativistic 35
Gamma decay 46
General relativity 36
Half-life 46
Heat 11
Heat capacity 11
Heat conduction 13
Heat engine 20
Heat of fusion 11
Heat of vaporization 11
Heat pump 20
Heats of transformation 11
Hydrostatics 1
Ideal Gas Law 10
Images real and virtual 27
Intensity sound 5
Interference 7
Interference two-slit 31
Internal energy 12
Internal energy degrees of freedom 16
Irreversible process 19 23
Isotopes 45
Joule 11
Kelvin scale 9
Kinetic energy relativity 36
Kinetic theory 16
Latent heat 11
Length contraction 35
Linear expansion coefficient 9
Linear polarization 34
Linear superposition 7
Longitudinal wave 3
Loudness 5
Magnification lateral 27
Magnifying glass 30
Malusrsquos law 34
Microscope 30
Momentum relativistic 36
Muscle ciliary 29
44
Musical instruments 8
Musical scale 8
Near point 29
Nearsightedness 29
Nonlinear 7
Nuclear energy 48
Nuclear properties 45
Nuclear stability 45
Octave 8
Optical resolution 33
Orbital quantum number ℓ 43
Orbitals 43
Particle in a Box 41
Particles are waves 39
Pascalrsquos Principle 1
Period 3
Photoelectric effect 38
Photons 38
Pitch 8
Polarization 34
Power sound 5
Presbyopia 29
Pressure 1
Principal quantum number n 43
Principle of equivalence 36
Principle of linear superposition 7
Probability amplitude ψ 41
Processes thermodynamic 13
R-value 14
Radiation thermal 14
Radioactive decay 46
Radioactive decay rate 46
Ray tracing 27
Rayleighrsquos criterion 33
Reading glasses 29
Real image 27
Refraction 25
Refrigerator 20
Relativistic gamma 35
Relativity principles 35
Resolved for light sources 33
Rest energy 36
Reversible process 19 22
Rope wave speed 4
Second Law of Thermodynamics 19
Shock waves 6
Simple magnifier 30
Simultaneity 35
Single slit diffraction 33
Snellrsquos law 25
Sound speed 5
Specific heat 11
Spin Angular momentum 44
Standing waves 7
Stefanrsquos law 14
Strong force 45
Telescope 30
Temperature 9
45
Temperature Scales 9
Thermal conductivity 14
Thermal energy 12
Thermal expansion 9
Thin film interference 31
Thin lenses 28
Time dilation 35
Tone musical 8
Total internal reflection 25
Transverse wave 3
Traveling Waves 3
Two-slit interference 31
Uncertainty principle 40
Vaporization heat of 11
Velocity addition relativity 36
Virtual image 27
Volume flux 2
Wave function 41
Wave Function ψ and Probability 41
Wave speed 3
Wave-particle duality 39
Wavelength 3
Wavenumber 3
Wien Displacement Law 16
Work 11
4
siny x t A kx t
where A is the amplitude of the wave As usual ω and k are related by the relation ω = kv where
v is the wave speed the speed at which the crests travel the minus sign means that the wave is
traveling to the right towards higher values of x φ is the phase and constrols where the maxima
and minima occur
Speed of Transverse Waves on a Rope
The velocity of transverse waves on a perfectly flexible rope is given by
Tv
where T is the tension in the rope and where μ is its linear mass density (mass per unit length)
Common Test Questions
A common question is to be given an equation for a wave and to be asked what is amplitude
velocity etc Another common type of question is to be given facts about a wave like its
direction velocity and magnitude at a given place and time and to be asked which of several
equations correctly describe the wave
5
Serway Chapter 17
Sound Speed in Solids Liquids and Gases
The speed of sound in a liquid or solid of bulk modulus B and volume mass density ρ
Bv
The speed of sound in air is
331 1 273v m s T C
The speed of sound in air at room temperature is about 343 ms Notice it doesnrsquot depend on
pressure or frequency
Power Intensity and Loudness
The intensity of a sound wave is defined to be the power per unit area ie wattsm2 in the wave
For spherical waves traveling away from a small source emitting waves with average power Рav
the intensity falls off with distance from the source r according to the inverse-square law
av24
Ir
Hence the intensity I2 at distance r2 is related to the intensity I1 at a different distance r1 by
21 2
22 1
I r
I r
Experiments on human hearing have shown that we hear intensity differences logarithmically so
the decibel loudness scale for sound intensity was invented The loudness β of a sound in
decibels is related to its intensity I in Wm2 by the formula
1010logo
I
I
Where Io is a sound intensity near the threshold of hearing defined to be Io = 10-12 Wm2 Note
that on this scale a sound is made 10 times more intense by adding 10 decibels Remember that
intensity is proportional to velocity squared Amplitude is the A is the previous chapterrsquos
equation for a traveling wave If you make the A 10 times as much the intensity will increase by
100 times (20 dB)
Doppler Effect
If sound waves are traveling through a medium and if either the receiver of the waves or the
source of waves is moving then the frequency received is related to the frequency emitted by
6
0
s
v vf f
v v
where fprime is the frequency detected by the observer f is the frequency emitted by the source v is
the speed of the waves vo is the speed of the observer and vs is the speed of the source This
formula assumes that the source and receiver are either moving directly toward each other or
directly away from each other To know which signs to use remember that when observer and
source approach each other the observed frequency is higher while if they move away from each
other it is lower Just examine the signs in the formula and make the answer come out right
For electromagnetic waves (light radio waves X-rays) traveling in vacuum Einsteinrsquos theory of
relativity (and careful experiments) show that the Doppler shift is given by
r
r
c vf f
c v
where vr is the relative speed between the source and the observer
For all kinds of waves (sound light etc) if the relative speed of the source and the observer is
small compared to the speed of the waves then there is a simple approximation to the Doppler
effect For example if the relative speed is 1 of the wave speed then the frequency shifts by
1 Remember that this is only an approximation
Shock Waves
When an object moves through a medium at a speed greater than the speed of waves a V-shaped
shock wave is produced The V-shaped wake behind a speeding boat is a good example of this
effect and the cone of sonic-boom behind a supersonic aircraft is another The angle that V-line
makes with the direction of travel of the source is given by
sins
v
v
where v is the wave speed and vs is the source speed
7
Serway Chapter 18
Principle of Linear Superposition
We say that a system obeys the principle of linear superposition if two or more different motions
of the system can simply be added together to find the net motion of the system Light waves
obey this principle as they propagate through the air as can be seen by shining two flashlights so
that their beams cross The beams propagate along without affecting each other (Light sabers are
a spectacular but unfortunately fictional example of systems that do not obey the principle of
superposition) Wave pulses on an ideal rope also obey this principle two different pulses pass
through each other without change Standing waves are an example of this effect being simply
the linear superposition of two traveling waves of the same frequency but moving in opposite
directions Light waves in matter do not always obey this principle For instance two powerful
laser beams could be made to cross in a piece of glass in such a way that their combined heating
effect in the crossing region could melt the glass and scatter the beams in complicated ways This
is an example of a nonlinear effect
Interference
When two or more waves are present in the same medium at the same time their net effect may
often be obtained simply by adding them at each point in the medium according to the principle
of linear superposition (Note this wonrsquot work if the medium is nonlinear) When this addition
makes the total amplitude be greater than the individual amplitudes of the various waves we say
that the interference is constructive When the addition produces cancellation and an amplitude
less than the amplitudes of the separate waves we have destructive interference
Standing Waves
A standing wave is the superposition of two identical traveling waves moving in opposite
directions Nodes are places where the two waves perfectly destructively interfere to produce
zero amplitude at all times Anti-nodes are places where the two waves perfectly constructively
interfere to produce an amplitude maximum The distance between nodes is λ2 Standing waves
on a string fixed at both ends have nodes at each end of the string Standing waves in an air
column enclosed in a tube have displacement anti-nodes at open ends of the tube and
displacement nodes at closed ends The frequency of the standing wave with the lowest possible
frequency is called the fundamental frequency Standing waves on strings or in air columns all
have frequencies which are integer multiples of the fundamental frequency and are called
8
harmonics (The fundamental is called the ldquofirst harmonicrdquo)
Beats
Beats are heard when two waves with slightly different frequencies f1 and f2 are combined The
waves constructively interfere for a number of cycles then destructively interfere for a number
of cycles We hear a periodic ldquowah-wahrdquo frequency equal to the difference of the two wave
frequencies
1 2bf f f
Musical Instruments
Musical instruments produce tones by exciting standing waves on strings (violins piano) and in
tubes (trumpet organ) The fundamental frequency of the standing wave is called the pitch of the
tone The pitch of concert A is 440 Hz by definition Two tones are an octave apart if one pitch
has twice the frequency of the other In written music there are 12 intervals in each octave with
the ratio between successive intervals equal to 2112 = 105946 The ratios for each tone in an
octave starting at A and ending at the next higher A are
A A B C C D D E F F G G A
1 10595 11225 11892 12599 13348 14142 14983 15874 16818 17818 18877 2
A musical tone is actually a superposition of the fundamental frequency and the higher
harmonics The tone quality of a musical instrument is determined by the amplitudes of the
various harmonics that it produces A violin and a trumpet can play the same pitch but they
donrsquot sound at all alike to our ears The difference between them is in the various amplitudes of
their harmonics
9
Serway Chapter 19
Temperature
Formally temperature is what is measured by a thermometer Roughly high temperature is what
we call hot and low temperature is what we call cold On the atomic level temperature refers to
the kinetic energy of the molecules A collection of molecules is called ldquohotrdquo if the molecules
have rapid random motion while a collection of molecules is called ldquocoldrdquo if the random motion
is slow When two bodies are placed in close contact with each other they exchange molecular
kinetic energy until they come to the same temperature This is the microscopic picture of the
Zeroth Law of Thermodynamics
Absolute Zero
Absolute zero is the lowest possible temperature that any object can have This is the temperature
at which all of the energy than can be removed an object has been removed (This removable
energy we call thermal energy) There is still motion at absolute zero Electrons continue to orbit
around atomic nuclei and even atoms continue to move about with a small amount of kinetic
energy but this small energy cannot be removed from the object For example at absolute zero
helium is a liquid whose atoms still move and slide past each other
Temperature Scales
Kelvin Scale Absolute zero is at T = 0 K water freezes at T = 27315 K room temperature is
around T = 295 K and water boils at T = 373 K Note we donrsquot use a deg symbol Kelvin is
prefered SI Unit
Celsius Scale Absolute zero is at T = -273degC water freezes at T = 0degC room temperature is
around T = 22degC and water boils at T = 100degC
Fahrenheit Scale Absolute zero is at T = -459degF water freezes at TF = 32degF room temperature
is around T = 72degF and water boils at TF = 212degF TF =18TC + 32 Notice that temperature
differences are the same for the Kelvin and Celsius scales
Thermal Expansion
When materials are heated they usually expand and when they are cooled they usually contract
(Water near freezing is a spectacular counterexample it works the other way around) The
coefficient of linear expansion is defined by the relation
1
i
L
L T
10
where Li is the initial length of a rod of the material and ΔL is the change in its length due to a
small temperature change ΔT The coefficient of volume expansion is defined similarly
1
i
V
V T
where Vi is the initial volume of a piece of material and where ΔV is the change in its volume due
to a small temperature change ΔT
Avogadrorsquos Number (N or NA)
One mole of any substance corresponds to 6022 times 1023 molecules The atomic mass given on the
periodic table (p A32-33 in the text) is the mass in grams of one mole of atoms For example
the atomic mass of oxygen is 15999 gmol The mass of a single atom is given by the same
number in units of atomic mass units (u) For example the mass of an oxygen atom is 15999 u
(on the average)
Ideal Gas Law (an example of an equation of state)
When the molecules of a gas are sufficiently inert and widely separated that interactions between
them are negligible we say that it is an ideal gas The pressure P volume V and temperature T
(in kelvins) of such a gas are State Variables and are related by the ideal gas law
Bor PV=NkPV nRT T
where n is the number of moles of the gas where R is the gas constant
8314 Jmol KR
where N is the number of molecules and where kB is Boltzmannrsquos constant
231380 10 J KBk
It works well for air at atmosphere pressure and even better for partial vaccuums The relative
ease of measuring pressure and the linear relationship between pressure and temperature (if V
and n are held fixed) makes an ideal gas an ideal thermometer Thermometers can be based on
properties of solids or liquids but the behavior of these materials with temperature is more
complicated
11
Serway Chapter 20
Heat
Heat is energy that flows between a system and its environment because of a tempera-
ture difference between them The units of heat are Joules as expected for an energy
Unfortunately there are several competing units of energy They are related by
1 cal 4186 J 1 Cal 4186 J 1 Btu 1054 J
Heat Capacity
There is often a simple linear relation between the heat that flows in or out of part of a
system and the temperature change that results from this energy transfer When this
linear relation holds it is convenient to define the heat capacity C and the specific
heat c as follows
For the entire object the heat Q it absorbs is given by Q = C(Tf - Ti)
For a particular material the specific heat is defined by c = Cm which is the heat
capacity per unit mass so that
Q = mc(Tf ndash Ti)
Note C has units of energy (J or Cal)(Kelvin kg)
Heats of Transformation or Latent Heat Q = plusmn mL
When a substance changes phase from solid to liquid or from liquid to gas it absorbs
heat without a change in temperature The latent heat or heat of transformation is
usually given per unit mass of the substance For example for water the heat of fusion
(melting) is L = 333 kJkg while the heat of vaporization (boiling) is L = 2260 kJkg
Note that heat for boiling is considerably bigger than melting for water You have to
be careful with signs heat is given off (negative) if you go down in temperature and
condense steam
Work
In general the small amount of work done on a system as a force Fon is exerted on it
through a vector displacement dx is given by
on xdW d F
12
But if the displacement is done very slowly (as we always assume in thermodynamics)
then the force exerted on the system and the force exerted by the system are in
balance so the force exerted by the system is ndash Fon In thermodynamics it is more
convenient to talk about the force exerted by the system so we change the above
formula for the work done on the system to
xdW d F
where F is the force exerted by the system This has confused students for more than a
century now but this is the way your book and many other books do it so you are
stuck You will need to memorize the minus sign in this definition of the work to be
able to use your textbook
There are many chances to get signs wrong in this and the next two chapters (Mosiah
2 )
When an external agent changes the volume of a gas at pressure P by a small amount
dV the (small amount) of work done on the system is given by
dW PdV
Notice that this minus sign is just what we need to make dW be positive if the external
agent compresses the gas for then dV is negative If on the other hand the external
agent gives way allowing the gas to expand against it then dV is positive and we say
that the work done on the gas is negative
The work done on the system (eg by the gas in a cylinder) in a thermodynamic
process is the area under the curve in a PV diagram It is positive for compressions
and negative for expansions If the volume of gas remains constant in a process then
no work is done by the gas
Cyclic processes are important For cyclic processes represented by PV diagrams the
magnitude of the net work during one cycle is simply the area enclosed by the cycle
on the diagram Be careful to keep track of signs when you are calculating that
enclosed area In cyclic process Q = ndashW for a cycle Put a PV diagram in your notes
1
Path A-B B-C C-D D-A A to A net
Q
W
ΔU
ΔS
Internal Energy
The energy stored in a substance is called its internal energy Eint This energy may be
stored as random kinetic energy or as potential energy in each molecule (stretched
chemical bonds electrons in excited states etc) For ideal gases all states with the
same temperature will have the same Eint
First Law of Thermodynamics
The change ΔEint in the internal energy of a system is given by
intE Q W
where Q is the heat absorbed by the system and where W is the work done on the
system Hence if a system absorbs heat (and if Wge0) the internal energy increases
Likewise if the system does work (W on the system is negative) and if Qge0 the
internal energy decreases Potential Pitfall Many times people talk about work done
by the system It is the minus of W on the system Donrsquot get tripped up
Processes
Adiabatic process No heat is exchanged Q = 0 requiring that the system be insulated
from the environment A process may be approximately adiabatic if it happens so
rapidly that heat does not have time to enter or leave the system Work + or ndash is done
and ΔEint = W
Adiabatic free expansion A gas is allowed to expand into a vacuum without pushing
on anythingndashit just rushes into the vacuum In this process Q = W = 0 so the internal
energy and temperature does not change (Note the difference between an adiabatic
process and a free expansion is that NO work is done in the adiabatic free expansion)
Isobaric process The pressure is held fixed ΔP = 0 For example usually the
pressure increases when a gas is heated but if it were allowed to expand during the
2
heating process in just the right way its pressure could remain fixed In isobaric
processes both Q and W are non-zero (Work is eacy = ndashP(Vf ndash Vi)
Isovolumetric process The volume is held fixed ΔV = 0 The work done by the gas is
then zero and so we have ΔEint = Q
Isothermal process The temperature is held fixed ΔT = 0 There is no change in Eint
so Q = ndashW for isothermal processes For an ideal gas PV = nRT so PV = constant in
an isothermal process The work done on the gas is then given by
lnf
i
Vi
Vf
VW PdV nRT
V
Heat Conduction
The quantity P is defined to be the rate at which heat flows through an object and is a
power having units of watts It is analogous to electric current which is the rate at
which charge flows through an object If the flow of heat through a slab of length L
and cross-sectional area A is steady in time then P is given by the equation
h cT TdQkA
dt L
P =
where Th and Tc are the (hot and cold) temperatures of the two ends of the slab The
heat flows of course because of this temperature difference The quantity k is called
the thermal conductivity and is a constant that is characteristic of the material It is
analogous to the electrical conductivity h cT TL is sometimes called the temperature
gradient and is written dTd dTdx
R-Values
It is common to have the heat-conducting properties of materials described by their R-
values especially for insulating materials like fiberglass batting The connection
between k and R is R = Lk where L is the material thickness In this country R-values
always have units of 2ft F hourBtu
Polyurethane foam 6 (1 in thick) Air space 1 (35 in thick) Fiber glass batting 11
(35 in thick) Wood about 1 (1 in thick) Glass 09 (0125 in thick)
The heat flow rate through a slab of area A is given by
3
h cT TA
R
P
in units of Btuhour Note that A must be in square feet and the temperatures must be
in degrees Fahrenheit
Convection
Convection is the transfer of thermal energy by flow of material For instance a home
furnace doesnrsquot heat a house by waiting for the heat from the burner to slowly conduct
throughout the house instead it quickly pumps warm air to all of the rooms
Generally convection is a much faster way to transfer heat than conduction
Radiation
Electromagnetic radiation can also transfer heat When you warm yourself near a
campfire which has burned itself down into a bed of glowing embers you are
receiving radiant heat from the infrared portion of the electromagnetic spectrum The
rate at which an object emits radiant heat is given by Stefanrsquos law
4AeTP
where P is the radiated power in watts σ is a constant
8 2 45696 10 W m K
A is the surface area of the object in m2 and T is the temperature in kelvins The
constant e is called the emissivity and it varies from substance to substance A perfect
absorber (think black velvet) has e = 1 while a perfect reflector (think mirror) has e =
0 Hence black objects radiate very well while shiny ones do not Also an object that
is hotter than its surroundings radiates more energy than it absorbs whereas an object
that is cooler than its surroundings absorbs more energy than it radiates
Terminology
Transfer variables vs state variables
Energy transfer by heat as well as work done depends on the initial final and
intermediate states of the system They are transfer variables But their sum (Q + W =
4
Eint) is a state variable
Figure 205
5
Serway Chapter 21
Kinetic Theory
The ideal gas law works for all atoms and molecules at low pressure It is rather
amazing that it does Kinetic theory explains why The properties of an ideal gas can
be understood by thinking of it as N rapidly moving particles of mass m As these
particles collide with the container walls momentum is imparted to the walls which
we call the force of gas pressure In this picture the pressure is related to the average
of the square of the particle velocity 2v by
22 1( )
3 2
NP mv
V
Using the ideal gas law we obtain the average translational kinetic energy per
molecule
21 3
2 2 Bmv k T
The rms speed is then given by
2rms
3 3Bk T RTv v
m M
where M is the molecular mass in kgmol
Degrees of Freedom
Roughly speaking a degree of freedom is a way in which a molecule can store energy
For instance since there are three different directions in space along which a molecule
can move there are three degrees of freedom for the translational kinetic energy
There are also three different axes of rotation about which a polyatomic molecule can
spin so we say there are three degrees of freedom for the rotational kinetic energy
There are even degrees of freedom associated with the various ways in which a
molecule can vibrate and with the different energy levels in which the electrons of
the molecule can exist
Internal Energy and Degrees of Freedom The internal energy of an ideal gas made
up of molecules with J degrees of freedom is given by
int 2 2 B
J JE nRT Nk T
6
Heat Capacities of an Ideal Gas The heat capacity of a gas is described by means of
molar heat capacities CV and CP These are the heat capacities per mole and the
subscript V on CV means that the volume is being held constant while for CP the
pressure is held constant For example to raise the temperature of n moles of a gas
whose pressure is held constant by 10 K we would have to supply an amount of heat Q
= nCP (10) K
Molar Specific Heat of an Ideal Gas at Constant Volume
VQ nC T
3monatomic
2VC R
5diatomic
2VC R
5polyatomic
2VC
Real gases deviate from these formulas because in addition to the translational and ro-
tational degrees of freedom they also have vibrational and electronic degrees of
freedom These are unimportant at low temperatures due to quantum mechanical
effects but become increasingly important at higher temperatures The rough rule is
No of degrees of freedom
2VC R
Molar Specific Heat of an Ideal Gas at Constant Pressure
PQ nC T
P VC C R
The internal energy of an ideal gas depends only on the temperature
int VE nC T
Adiabatic Processes in an Ideal Gas
7
An adiabatic process is one in which no heat is exchanged between the system and the
environment When an ideal gas expands or contracts adiabatically not only does its
pressure change as expected from the ideal gas law but its temperature changes as
well Under these conditions the final pressure Pf can be computed from the initial
pressure Pi and from the final and initial volumes Vf and Vi by
or constantf f i iP V PV PV
where γ = CPCV The quantity γ is called the adiabatic exponent Note that this
doesnrsquot mean that the ideal gas law no longer holds it does and in fact it can be
combined with the adiabatic law for pressure given above to obtain the adiabatic law
for temperatures
1 constantTV
Compressions in sound waves are adiabatic because they happen too rapidly for any
appreciable amount of heat to flow This is why the adiabatic exponent appears in the
formula for the speed of sound in an ideal gas
RTv
M
Note that v depends only on T and not on P Because it depends only on the
temperature the speed of sound is the same in Provo as at sea level in spite of the
lower pressure here due to the difference in elevation
Equipartition of Energy
Every kind of molecule has a certain number of degrees of freedom which are
independent ways in which it can store energy Each such degree of freedom has
associated with it ndash on average ndash an energy of 12 Bk T per molecule (or 1
2 RT per mole)
(Note since a molecule has so many possible degrees of freedom it would seem that
there should be a lot of 12 sBk T to spread around But because energy is quantized
some of these degrees of freedom are not ldquoactiverdquo until the temperature becomes high
enough that 12 Bk T is as big as the lowest quantum of energy
8
Serway Chapter 22
Second Law of Thermodynamics
There are several equivalent forms of this important law
Kelvin It is not possible to change heat completely into work with no other change
taking place Or in other words there are no perfect heat engines
Clausius It is not possible for heat to flow from one body to another body at a higher
temperature with no other change taking place Or in other words there are no
perfect refrigerators
Entropy In any thermodynamic process that proceeds from one equilibrium state to
another the entropy of the system + environment either remains unchanged or
increases The total entropy never decreases This law is a bit of an oddity among the
laws of physics because it is not absolute Things are forbidden by the second law not
because it is impossible for them to happen but because it is extremely unlikely for
them to happen (See below for more information about entropy)
Reversible and Irreversible Processes
A reversible process is one which occurs so slowly that it is in thermal equilibrium (or
very nearly so) at all times A hallmark of such processes is that a motion picture of
them looks perfectly normal whether run forward or backward Imagine for instance
the slow expansion of a gas at constant temperature in a cylinder whose volume is being
increased by a slowly moving piston Run the movie backwards and what do you see
You see the slow compression of a gas at constant temperature which looks perfectly
normal
An irreversible process is one which occurs in such a way that thermal equilibrium is
not maintained throughout the process The mark of this kind of process is that a motion
picture of it looks very odd when run backward Imagine the sudden expansion of a gas
into a previously evacuated chamber because a hole was punched in the wall between a
pressurized chamber and the evacuated one Run the movie backward and what do you
see You see the gas in the soon-to-be-evacuated chamber gather itself together and
stream through a tiny hole into a chamber in which there is already plenty of gas If you
have ever seen this happen get in touch with the support group for those who have
witnessed the spontaneous combustion of large mammals by calling 1-800-PYROCOW
9
Heat Engines
Because of the vexing difference in sign between work done on and system and work done
by a system we will invent a new work variable Weng Heat engines do work and so the net
W for these engines is negative But in engineering applications hidden minus signs are
regarded as evil so for heat engines we donrsquot talk about W instead we talk about its
magnitude engW W So for heat engines the first law is
int engE Q W
But for heat pumps and refrigerators work is done on the system so we use the usual work
W when we talk about these systems
A heat engine is a machine that absorbs heat converts part of it to work and exhausts the
rest The heat must be absorbed at high temperature and exhausted at low temperature If the
absorbed heat is Qh the exhausted heat is Qc and the work done by the engine is Weng then
eng h cW Q Q
and the efficiency of the engine is defined to be
eng
h
We
Q
A perfect engine would convert the heat hQ completely into work Weng giving an effi-
ciency of e = 1 Energy conservation alone allows a perfect engine but the second law
requires e lt 1
Refrigerators and Heat Pumps
A refrigerator is a machine that absorbs heat at low temperature and exhausts it at high
temperature the ldquobackwardsrdquo heat transfer being driven by the work done on the machine
by some source of power A heat pump is a machine that either works like a refrigerator
keeping a place cold by transferring heat from this cold place to a higher temperature
environment (cooling mode like an air conditioner) or it functions as a heater
transferring heat into a warm place from a cooler one (heating mode like a window unit
that heats a house by extracting thermal energy from the cold outdoors) The coefficient
of performance of a refrigerator or of a heat pump in cooling mode is defined to be
COP cooling modecQ
W
10
For a heat pump in heating mode the coefficient of performance is
COP heating modehQ
W
Note that we donrsquot have to use the engineering work here because in these systems
work is done on the system and W is naturally positive
A perfect refrigerator would take in heat Qc and exhaust the same amount of heat without
doing any work giving an infinite coefficient of performance Energy conservation alone
allows a perfect refrigerator but the second law requires COP lt infin A perfect heat pump
in heating mode would transfer Qh into the house without doing any work and so would
also have an infinite coefficient of performance The second law forbids this too A good
coefficient of performance for a real device would be around 5 or 6
Carnot Cycle
The most efficient of all possible engines is one that uses the Carnot cycle This cycle
employs an ideal gas has no friction and operates very slowly so that the gas can be in
thermal equilibrium at all parts of the cycle This means of course that it canrsquot
possibly be built and even if it could be built it would not run fast enough to be useful
Nevertheless this cycle is very important because it gives an upper bound on the
efficiency of real engines There cannot possibly be an engine that is more efficient
than one based on the Carnot cycle This cycle consists of the following four steps
1 The ideal gas absorbs heat Qh at constant temperature Th while the gas increases its
volume The reason that heat is absorbed is that expansion tends to cool the gas but
thermal contact with the environment at Th keeps the temperature high by heat
conduction into the ideal gas
2 The ideal gas further increases its volume by an adiabatic expansion This expansion
causes the gas to cool so at the end of this part of the cycle the gas is at temperature Tc
3 The gas exhausts heat Qc at constant temperature Tc while the gas decreases in
volume The reason that heat is exhausted is that compression tends to heat the gas
but thermal contact with the environment at Tc keeps the temperature low by heat
conduction out of the ideal gas
4 The gas is adiabatically compressed back to its original volume (the volume it started
with in step 1) This compression heats the gas from Tc up to Th
11
The efficiency of a Carnot engine is given by the very simple formula
1 cC
h
Te
T
where the temperatures must in be Kelvin No real engine can be more efficient than
this
The coefficient of performance of a Carnot refrigerator or heat pump in cooling mode
is given by
COP cooling mode cC
h c
T
T T
and no real refrigerator can have a coefficient of performance greater than this
The coefficient of performance of a Carnot heat pump in heating mode is
COP heating mode hC
h c
T
T T
So why donrsquot we just use these wonderful Carnot engines and have perfect efficiency
To make the reversible steps in the cycle really reversible they would have to occur
infinitely slowly So the price you pay for making a perfect engine is that it takes
forever to get it to do any work
Entropy
The entropy of a system is defined in terms of its molecular makeup and measures
roughly the disorder of the system If the system is packed into a very small volume
then it is quite ordered and the entropy will be low If it occupies a large volume the
entropy is high (To see what this has to do with disorder note that socks in a drawer
occupy a small volume while socks on the bed in the corner by the door and
hanging from the chandelier occupy a large volume) If the system is very cold then
the molecules hardly move and may even reach out to each other and form a crystal
This is a highly ordered state and therefore has low entropy If the system is very hot
with rapidly speeding molecules crashing into the container walls and bouncing off
each other things are disordered and the entropy is high
It is possible to calculate the entropy of a system in terms of its macroscopic thermody-
namic properties ie pressure volume temperature number of moles etc The key to this
calculation is the concept of a reversible process A reversible process is one that is
12
carried out without friction and so slowly that the process can be reversed at any stage by
making an infinitesimal change in the environment of the system The slow expansion of the
gas at Th in the Carnot cycle with tiny amounts of heat being transferred to the environment
is reversible If we turned around and began slowly to compress the gas would just slowly
exhaust heat to the environment in the exact reverse way that it absorbed it during expansion
Most processes however are irreversible For example if a gas-filled box were suddenly
increased in size so that the particles were free to wander into the void created by the sudden
expansion then the gas would eventually fill the new volume uniformly at the same
temperature as before the expansion (The temperature is unchanged in this imaginary
process because the kinetic energy of the molecules would be unaffected by such an
instantaneous expansion of the container walls) This imaginary but highly thought-
stimulating process is called a free expansion and it is impossible to reverse it During the
expansion we didnrsquot push on any of the molecules so reversing this process would mean
making them go back into their original volume without pushing on them they simply will
not cooperate to this extent Another way to see that just pushing them back where they came
from does not reverse the free expansion is to think about what would happen if we just
compressed either adiabatically or isothermally An adiabatic compression back to the
original volume would heat the gas above its original temperature and an isothermal
compression would require that heat be exhausted to the environment But the free expansion
involved neither temperature changes nor heat exchanges so neither of these two processes
is the reverse of the free expansion It is simply impossible to reverse this rapid expansion in
a way that takes us back to the initial state
It is possible to calculate the change of entropy for both reversible and irreversible processes
Letrsquos consider a reversible process first In a reversible process the entropy change is given
by the formula
dQS
T
where dQ is the amount of heat added to the system during a small step of the process
The total energy change during the process may then simply be calculated by integration
f f
f i i i
dQS S S dS
T
13
(Just as in the case of energy we are mainly interested in differences rather than in
absolute magnitudes)
This integration method does not work for irreversible processes For instance in the
rapid free expansion discussed above no heat is added to the system but its disorder
obviously goes up We need to find some other way to calculate the entropy The key
is the fact that the entropy of a system depends only on its current state and not at all
on how it arrived there So to calculate the entropy change in an irreversible process
first find out what the initial and final conditions of the process are Then invent a
reversible process that takes the system from the initial state to the final state Since
the entropy depends only on the state of system and not on the process the entropy
change for the reversible process is the same as that for the irreversible process
Entropy is thus like pressure or temperature if the pressure changes from Pl to to P2
then the pressure difference is P2 - Pl regardless of how the pressure change was
made This seems obvious for pressure but not for entropy only because you arent
comfortable with entropy yet
Entropy of an Ideal Gas
For n moles of an ideal gas the difference in entropy between a state with temperature
T and volume V and some standard state with temperature To and volume Vo is given
by
ln lno o Vo o
T VS T V S T V nC nR
T V
Entropy in a Phase Change
Calculating entropy change in a phase change like melting or boiling is easy It is
QT Where Q is in the latent heat for example the flows in to cause the melting and
T is the temperature of the phase change There is NO integral to do
14
Serway Chapter 35
Angle of Reflection
If plane waves are incident on a reflecting surface with the propagation direction of
the waves making angle θ1 with the normal direction to the surface then the reflected
angle θ1prime relative to the surface normal is simply
1 1
ie the incident angles and reflected angles are the same
Refraction
If plane waves traveling through medium 1 are incident on a plane interface between
medium 1 and medium 2 then the angle of incidence of the incoming wave θ1 and
the angle of refraction of the transmitted wave θ2 are related by Snellrsquos law
2 2
1 1
sin
sin
v
v
where v1 and v2 are the wave speeds in medium 1 and medium 2 The angle of
incidence and the angle of refraction are both measured between the wave propagation
direction and the normal to the interface In terms of indices of refraction in the case
of light waves Snellrsquos law takes the more familiar form
1 1 2 2sin sinn n
where ni = cvi
Total Internal Reflection
If a wave is incident from a medium of low wave speed into a medium of high wave
speed the law of refraction requires that the angle of refraction be greater than the
angle of incidence If the angle of refraction is required to be greater than 90deg then no
refracted wave can exist and total internal reflection occurs The critical incident
angle θc beyond which total internal reflection occurs is given by
2
1
sin c
n
n
15
Dispersion of Light
In addition to the speed of light varying from material to material it also varies with
wavelength within each material This means that the index of refraction is generally a
function of wavelength
c
nv
Since the wave speed is not constant such a medium is dispersive meaning in this
context that refraction actually disperses white light into its various colors because
Snellrsquos law gives a different angle for each wavelength In most materials the
variation with wavelength is quite small but this small effect is responsible for some
of the most spectacular color effects we ever see including rainbows a flashing
crystal chandelier and the colored fire of a diamond solitaire by candlelight
16
Serway Chapter 36
Real and Virtual Images
When light rays are focused at a certain plane producing an image if a sheet of white
paper is placed there we call the image a real image The images produced by film
projectors and overhead projectors are examples of real images
When light rays appear to come from a certain location but no image is produced
when a screen is placed there we say that there is a virtual image at that location For
instance when you look in a mirror it appears that someone is behind the mirror but
a screen placed back there in the dark would show nothing Your image in the mirror
is a virtual image
Ray Tracing
There are lots of rules about how to find the images in optical systems but the best
way to keep things straight is to learn how to draw the principal rays for curved
mirrors and lenses The rays for convex and concave mirrors are shown in Fig 3615
The rays for converging and diverging thin lenses are shown in Fig 3627 You should
memorize the principal rays and know how to use them to locate images
Curved Mirrors
The focal length of a curved mirror with radius of curvature R is given by
2
Rf
The relation between object distance p image distance q and focal length f is
1 1 1
p q f
If the mirror is a diverging mirror f should be negative and if q should turn out to be
negative the image is virtual
Lateral Magnification
The lateral magnification in an optical system is defined by the ratio of the image size
to the object size
17
Image height
Object height
qM
p
for curved mirrors
As usual there are sign conventions here too but this formula just gives the
magnitude It is better to keep keep track of upright images versus inverted images by
means of ray diagrams rather than by memorizing sign conventions
Thin Lenses
The focal length of a thin lens is related to the radii of curvature of the two faces R1
and R2 of the lens by
1 2
1 1 11n
f R R
Note that this formula differs from Eq (3611) in the text by not having a minus sign
between the two R-terms We like this form better because for a simple converging
lens like a magnifying glass we just use positive values of R for both surfaces If one
of the faces is concave producing divergence use a negative value for R And if a
surface is flat use R = infin If the face is flat the radius is infinite
The relation between the image object and focal distances for a thin lens is the same
as that for a curved mirror
1 1 1
q p f
Use a negative focal length if the lens is diverging
The lateral magnification for a thin lens is the same as for a curved mirror
Image height
Object height
qM
p
Camera
The lens system in a camera projects a real image of an object onto the film (or CCD
array in a digital camera) The position of the image is adjusted to be on the film by
moving the lens into or out of the camera
18
Eye
The eye is like a camera in that a real image is formed on the retina Unlike a camera
the image position is adjusted by changing the focal length of the lens This is done by
the ciliary muscle which squeezes the lens changing its shape
Near Point The near point is the closest distance from the eye for which the lens can
focus an image on the retina It is usually 18-25 cm for young persons
Far Point The far point is the greatest distance from the eye for which the lens can
focus an image on the retina For a person with normal vision the far point is at
infinity
Nearsightedness A person is nearsighted if their far point is at some finite distance
less than infinity This condition can be corrected with a lens that takes an object at
infinity and produces a virtual image at the personrsquos far point
Farsightedness and Presbyopia A person is farsighted if their near point is too far
away for comfortable near work like reading or knitting This can be corrected by a
lens which takes an object at a normal near point distance of 18-25 cm and produces a
virtual image at the personrsquos natural near point Presbyopia involves a similar
problem which nearly all people experience as they age The ciliary muscle becomes
too weak and the lens becomes too stiff to allow the eye to provide for both near and
far vision The solution for this problem is either reading glasses or bifocal lenses
Reading classes are just weak magnifying glasses mounted on eyeglass frames
Bifocal lenses are split into upper and lower halves The lower half is a lens which
gives the proper correction for near work and the upper half is a different lens for
proper focusing at infinity
Angular Size
When an object is brought closer to the eye it appears to be larger because the image
on the retina is larger The size of this image is directly proportional to the objectrsquos
angular size which is the angle subtended by the object measured from the center of
the lens of the eye In optical instruments which are to be used with the eye the
angular size of the final image is whatrsquos important because it determines how large
the image will appear to the viewer
19
Simple Magnifier
A simple magnifier is a single converging lens or magnifying glass It takes an object
closer to the eye than a normal near point and produces a virtual image at or beyond
this near point The angular magnification is defined to be the ratio of the angular size
when viewed through the lens to the angular size of the object when viewed at the
normal near point (without aid of the lens)
Microscope
This instrument has two lenses (1) The objective is near the object being viewed and
produces a greatly magnified real image (2) The eyepiece is a simple magnifier which
the viewer uses to closely examine the image from (1)
Telescope
This instrument also has two lenses (1) The objective at the front of the telescope
takes light from a distant object and produces a real inverted image (which is rather
small) near its focal point (2) This small real image is then examined by the eyepiece
functioning as a simple magnifier to produce a virtual image with a larger angular
size
20
Serway Chapter 37
Two-Slit Interference
If light is incident on two closely spaced narrow slits a pattern of light and dark
stripes is produced beyond the slits The bright stripes or fringes are caused by
constructive interference of the two waves coming from the slits Constructive
interference occurs whenever two waves arrive at a location in phase with each other
This occurs when the distance x1 from slit 1 to a point P on the screen and the
distance x2 from slit 2 to point P differ by in integral number of wavelengths
1 2 where 0 1 2x x m m
where λ is the wavelength of the light When x1 and x2 are much larger than the slit
spacing d this condition reduces to
sind m
where θ is the angle between the direction of the incident light and the direction of the
light arriving at the screen
Thin Films
When light is partially reflected and partially transmitted by a thin film of transparent
material it is possible to have interference between the wave reflected from the front
of the film and light reflected from the back of the film (The colored reflections from
the thin film of oil on the water in a rain-soaked parking lot are an example of this
effect) It is difficult to write down formulas that will work in all cases so we will
just review the important principles here
1 If the two reflected waves are in phase with each other the film has enhanced
reflection (constructive interference) but if the two reflected waves are out of phase
with each other reflection is diminished (destructive interference) Phase shifts occur
due to reflection and due to the extra path length through the film of the wave
reflected from the back of the film
2 The phase change due to reflection is determined by the difference in index of
refraction between the two media involved in the reflection If the wave is incident
21
from a medium with a low index of refraction into a medium with a high index of
refraction a phase change of 180deg occurs and the reflected wave is inverted If
incident from high to low no phase shift occurs and the reflected wave is non-
inverted
3 The extra path length through the film of the wave reflected from the back of the
film is equal to 2t where t is the thickness of the film (The incident light is assumed
to be normal to the surface of the film) The number of wavelengths contained in the
extra path length is equal to 2tλn where λn = λn is the wavelength of the light in the
film
4 Rules for reflection from thin films
If one ray is inverted and the other is not then we have
12 constructive
2 nt m
2 destructivent m
If either both rays are inverted or both are non-inverted we have
2 constructivent m
12 destructive
2 nt m
22
Serway Chapter 38
Diffraction Grating
A diffraction grating is simply a fancy version of two-slit interference with the two
slits replaced by thousands of slits Just as in the two-slit case the bright fringes
occur at angles given by
sin md
but in this case d the distance between neighboring slits is made to be very small
The effect of having many slits instead of two is to make each bright fringe highly
localized with wide dark regions between neighboring maxima
Single Slit Diffraction
When light passes through an opening in an opaque screen an interference pattern is
produced beyond the opening To understand why we may replace the single opening
by many small coherent sources of light These many sources interfere with each
other producing a pattern known as a diffraction pattern (Note that many authors do
not distinguish between interference and diffraction treating them as interchangeable
terms) If the opening is a slit of width a then the diffraction pattern far from the slit
will have a bright central maximum with a succession of minima and weaker maxima
on either side The angle between the incident direction and the minima is given by
sin ma
where m = plusmn1 plusmn2
If the opening is circular with diameter D the angle between the incident direction
and the first minimum is given by
sin 122D
Optical Resolution and Rayleighrsquos Criterion
Two point sources can just be resolved (distinguished from each other) if the peak of
the diffraction image of the first source overlies the first minimum of the diffraction
image of the second source For circular holes of the kind usually encountered in
23
optical devices this condition is approximately satisfied when the angular separation
between the two sources as viewed from the optical instrument is greater than or equal
to the critical angle
min 122D
where λ is the wavelength of the light and where D is the diameter of the aperture in
the instrument
Polarization
We say that an electromagnetic wave is polarized if its electric field vector doesnrsquot
change direction in a random fashion The simplest kind of polarization is linear
polarization in which the electric field vector oscillates back and forth along the
same axis in space Polarized light can be produced from normal unpolarized light by
selective absorption (as in Polaroid sunglasses) by passing light through crystals that
have different indices of refraction for different polarizations (double refraction) by
scattering (the blue sky is polarized) and by reflection (glare) Polarization by
reflection occurs when light reflects from a shiny insulating (non-metallic) surface
The amount of polarization is greatest for reflection at Brewsterrsquos angle
2
1
tan P
n
n
where θP is the incidence angle of light from medium 1 onto medium 2 and where n1
and n2 are the indices of refraction for the media (In our everyday experience n1 = 1
since the light comes in through the air and n2 is the index of refraction of the shiny
insulating material producing the glare eg water glass plastic paint etc)
Malusrsquos Law
The intensity of transmitted polarized light through a perfect polarizer is related to the
incident intensity of polarized light by Malusrsquos law
2cosoI I
where Io is the intensity of the incident light and where θ is the angle between the
electric field vector in the incident wave and the transmission axis the polarizer
24
Serway Chapter 39
Principles of Relativity
All of the weirdness of relativity flows from two simple principles
(1) The laws of physics must be the same in all inertial (non-accelerating) reference
frames
(2) The speed of light in vacuum has the same value in all inertial reference frames
Note that (1) seems reasonable but (2) is very odd It says that if two space ships are
approaching each other at nearly the speed of light and a laser pulse is shot from ship
1 toward ship 2 then when the people on ship 2 measure the speed of the pulse as it
goes by it is moving at 3 times 108 ms the same as if the ships were stationary
Simultaneity
If observer 1 sees two events in her own frame as simultaneous at two different
locations a moving observer 2 will see these two events happening at different times
Relativistic gamma
The factor γ (gamma) appears regularly in the formulas of relativity
2
2
1
1 vc
where v is the relative speed between two inertial frames
Time Dilation
If observer 1 sees two events at the same location in space separated by time Δtp in his
own frame then observer 2 moving at speed v relative to observer 1 will see these
two events separated by a longer time Δt
pt t
Moving clocks run slow
Length Contraction
If observer 1 measures the length of an object along the x-direction at some instant of
time in her frame to be Lp then observer 2 moving in the x-direction at speed v will
measure the length L of the object to be shorter
pLL
25
Moving meter sticks are short
Velocity Addition
If a particle is moving at velocity vac relative to frame c and if frame c is moving at
speed vcb with respect to frame b (with both velocities directed along the same line)
what would be the velocity of the particle in frame b In classical physics the answer is
ab ac cbv v v
but if any of these velocities are an appreciable fraction of the speed of light the
answer changes
21ac cb
abac cb
v Vv
v v c
In using this equation make sure that vac has the same sign in the numerator and in the
denominator and that vbc also has the same sign in both places Note that this equation
is different (and in our opinion easier to understand) than the equations in the text
Momentum and Energy
The momentum and total energy of a particle moving at velocity v (its speed is v v )
are given by
22
2 2
22
1 1 vvc c
m mcm E mc
vp v
Rest Energy
When a particle is at rest (v = 0 and γ = 1) then E = mc2 is called the rest energy It
includes the potential energy of the particle so that if we change the potential energy
of a particle we change its rest mass
Kinetic Energy The kinetic energy is the difference between the total energy and the
rest energy
2 2K mc mc
For v ltlt c this reduces to K = mv22
General Relativity
Einsteinrsquos principle of equivalence in the theory of general relativity is In the vicinity
of a point a gravitational field is equivalent to an accelerated frame of reference in
26
the absence of gravitational effects For example there is no difference whatever in
the physics experienced by an astronaut on earth and one who is in interstellar space
far away from any stars or planets but who has just turned on her spaceshiprsquos engines
and is accelerating at 98 ms2
27
Serway Chapter 40
Photons
We now know that even though light behaves like a wave it is better described as
consisting of small packets of energy called photons The energy of a photon is related
to its frequency by
E hf
where h is Planckrsquos constant
346626 10 J sh
Photons also carry momentum given by
hf hp
c
Since the amount of energy in each photon is fixed the difference between dim light
and intense light (of the same wavelength) is that dim light consists of fewer photons
than intense light
Photoelectric Effect
In the photoelectric effect light shining onto a metal surface gives the electrons in the
metal enough energy to escape and be detected It requires a certain amount of energy
(called the work function typically = 3 ndash 5 eV) just to kick the electron out of the
metal so the light must deliver at least this much energy to an electron to produce the
effect It is observed that red light no matter how intense never produces electrons
But ultraviolet light even if quite dim will eject electrons from the metal Since red
light consists of 2 eV photons and ultraviolet light has photons with energies around
4-6 eV the photon idea explains the behavior of the photoelectric effect The
maximum energy that an ejected electron can have is
maxK hf
where f is the photon frequency The minimum frequency that light can have and cause
electrons to be ejected is called the cutoff energy and is found by setting Kmax = 0
cf h
28
Compton Effect
When high-frequency light interacts with free electrons the scattered light does not
have the same wavelength as the incident light contrary to what classical
electromagnetism would predict But the photon picture in which photons have
momentum and energy allows us to treat scattering as a collision between the photon
and the electron In this collision the electron and the scattered photon both have a
different momentum and energy than they did before And since p = hλ for a photon
if the momentum is different the wavelength will also be different
after before 1 cose
h
m c
where the angle θ is the angle between the incoming photon direction and the
direction of the scattered photon
Particles are Waves
Since photons behave like particles sometimes it is not surprising that elementary
particles can behave like waves sometimes The wavelength of a particle with
momentum p is given by
h
p
Wave-particle Duality
Both photons and elementary particles have a dual nature sometimes they behave like
particles and sometimes they behave like waves It is hard for us to comprehend the
nature of such an object by making mental pictures but experiments definitively show
that this is the case Since we have no direct experience with photons and elementary
particles (because their energies are so much smaller than the energies of the everyday
objects we encounter) it is perhaps not surprising that we have a hard time forming a
mental picture of how they behave
Electron Interference
Since an electron is both a particle and a wave just like a photon it should be able to
produce an interference pattern This is observed If an electron beam is shot at two
closely-spaced slits and if the electrons that pass through the slits are detected
downstream it is found that there are some locations where electrons are never detected
29
and others where lots of electrons are detected The pattern is exactly the same as the
one observed for light waves provided that we use the electron wavelength λ = hp in
place of the wavelength of light This pattern is observed even though each electron is
detected as a single dot on the screen Only after many such dots are collected does the
pattern emerge And if we try to understand how this effect could possibly work by
looking closely at each slit to see which one the electron came through the pattern
disappears the act of measurement destroys the interference This means that each
single electron somehow comes through both slits (as a wave would)
Uncertainty Principle
In classical physics we always imagine that the positions and momenta of moving
particles have definite values It might be hard to measure them but surely at each
instant of time a particle should be precisely located at some point in space and have a
similarly precise momentum This turns out not to be true Instead both position and
momentum are required to be uncertain with their uncertainties Δx and Δp satisfying
the Heisenberg uncertainty relation
2x p
So if the particle were known to be precisely at some particular location (so that Δx =
0) we couldnrsquot know anything about its momentum (Δp = infin) and if its momentum
were exactly known we couldnrsquot know its position
There is a similar relation involving the particlersquos energy E and the time interval Δt
over which this energy is measured
2E t
This means that energy is not actually conserved if we are considering very small time
intervals and this brief non-conservation of energy has been observed
30
Serway Chapter 41
Wave Function ψ and Probability
Quantum mechanics does not predict exactly what an electron or a photon will do Instead
it specifies the wave function or probability amplitude ψ of an electron or a photon
This wave function is a complex-valued function of space and time whose squared
magnitude is the probability density P for finding a particle at a particular place in
space at a certain time
2P
where is the complex conjugate of ψ
For example the wave function of an electron with perfectly specified momentum p
would have a wavelength given by
h
p
and its probability amplitude would be proportional to
2i x ipxe e
The corresponding probability density would then be
21ipx ip ipxP e e e
which means that the electron is equally probable to be anywhere along the x axis
This is in accord with the uncertainty principle since we specified the momentum
precisely we canrsquot have any idea about the position of the electron
As another example you have probably seen ldquofuzzy ballrdquo drawings of electron
orbitals in a chemistry book These fuzzy balls are meant to indicate the distribution
of the probability density 2
P in the orbital
Particle in a Box
A simple example in which we can calculate the wave function is the case of a particle of
mass m constrained to be inside a 1-dimensional box between x = 0 and x = L In this
case the general wave function is a linear superposition of wave functions ψn of the form
sin niE tn
n xx t A e
L
31
where A is a positive constant where n = 1 2 3 and where the energy associated
with each of the quantum states ψn is given by
22
28n
hE n
mL
This wave function is zero at x = 0 and x = L which means that the particle will never
be found at the walls of the box The wave function has maximum values in the
interior and at these places the particle is most likely to be found and it also has
places where it is zero and at these places the particle will also never be found as
expressed by the formula
2sinn x
PL
The particle in the box is interfering with itself producing a probability interference
pattern across the box just like the interference patterns we studied with light and
sound
32
Serway Chapter 42
Atoms
Many people picture an atom as a miniature solar system where electrons orbit around
a massive nucleus at the center This picture is misleading because in atoms the wave
nature of electrons dominates The electrons form 3-dimensional standing waves
(called orbitals) centered at the nucleus And if you ask what is it thatrsquos waving the
answer is ψ the probability amplitude (see Chapter 41)
Standing waves on a string can occur only for certain frequencies (the fundamental
and higher harmonics) Similarly atomic orbitals occur only for certain energies For
the hydrogen atom the energies of the orbitals have the a particularly simple form
2
1136 eVnE
n
where n = 1 2 3 is called the principal quantum number For other atoms the
determination of the orbital energies requires numerical calculation by computers
Atomic Spectra
If an electron is somehow given extra extra energy (we say that it is excited) so that it
occupies a higher orbital it will eventually ldquofallrdquo back down to a lower orbital Each
time an electron falls to a lower orbital it loses the difference in energy between the
two orbitals in the form of a photon Since the orbital energies are discrete so are the
energy differences and so are the wavelengths of the emitted light The entire set of
these discrete wavelengths is called the atomic spectrum and it is unique to each type
of atom For hydrogen the spectrum can be simply written as
H 2 2
1 1 1
f i
Rn n
where RH = 10973732 times 107 m-1 For other atoms the spectrum cannot be expressed as
a formula
Orbital Angular Momentum Electrons in atoms also have quantized values of
angular momentum The orbital quantum number ℓ specifies the value of this
quantized angular momentum through the formula
1L
33
If we want to know the value of the angular momentum along some direction in space
say the z direction the answer is not L but rather
zL m
where mℓ is another quantum number which runs from
1 1m
This quantum number is important when an atom sits in a magnetic field
Spin Angular Momentum It has been found experimentally that electrons and other
charged particles also carry internal angular momentum which we call spin
Electrons have an intrinsic spin angular momentum s along a specified axis that is
extremely quantized it can only take on 2 possible values
1
2zs s
We thus say that electrons have spin s = 12 or that electrons are rdquospin one-half parti-
cles Since s is an angular momentum it obeys the same rule as that for orbital angular
momentum namely that if its value along some axis is s then its total magnitude is
given by
31
2S s s
I know this seems weird but quantum mechanics is weird The only excuse for this
bizarre way of looking at the world is that it predicts what happens in experiments
Exclusion Principle The answer to the question of how many electrons (or any other
spin one-half particle) can be in one particular quantum state was discovered by
Wolfgang Pauli and is called the exclusion principle
ldquoNo two electrons can ever be in the same quantum state therefore no two electrons
in the same atom can have the same set of quantum numbersrdquo
This is the reason that we have atoms with different properties instead of every atom
simple having all of its electrons in the ground state All of the variety we see around
us in the world is the result of chemical differences and these differences would not
exist unless electrons obeyed this important principle The entire structure of the
periodic table (see pages 1377-1379 in Serway) is an expression of this principle
34
Serway Chapter 44
Nuclear Properties
The nucleus is composed of protons with charge +e and mass mp = 1007226 u and of
neutrons with zero charge and mass mn = 1008665 u where u is one atomic mass unit
-271 u=1660540 10 kg
The atomic number Z counts the number of protons in a nucleus while the neutron
number N counts the number of neutrons The mass number A is the sum of the two
A N Z
Protons and nuetrons have about the same mass and some times called baryons (heavy
ones) or nucleons (partcles in the nucleus) A is sometimes called the baryon
number The chemical elements are distinguished by Z ie hydrogen has Z = 1 iron
has Z = 26 uranium has Z = 92 etc But for a given Z there might be several nuclei
with different numbers of neutrons These nuclear siblings are called isotopes
These numbers are used to label nuclei according to the pattern
5626 ie FeA
Z X
denotes the isotope of iron with N = 56 ndash 26 = 30 neutrons
The nucleus is roughly spherical with a radius given approximately by
1 3 150 0where 12 10 mr r A r
Nuclear Stability
Because the positively charged protons electrically repel each other with an enormous
force at distances as small as 10-15 m there must be some really strong force that that
overcomes electrical repulsion to hold protons and neutrons together This force is
called with some lack of imagination the strong force It is a very short-range force
(it only acts over a distance of about 2 x 10-15 m) and attracts protons to protons
neutrons to neutrons and neutrons to protons But in spite of this strong nuclear force
the coulomb repulsion of the protons is still present so anything that might keep the
protons from being right next to each other would help keep the nucleus from
35
exploding This role is played by the neutrons and for nuclei with Z le 20 the stable
nuclei roughly have N = Z
For Z gt 20 the coulomb repulsion force is more powerful and more neutrons are
needed to dilute it up to about N = 15Z around Z = 80 For Z gt 83 no amount of
neutrons can help and these nuclei are unstable (radioactive) There is an ldquoisland of
(relative) stabilityrdquo around Z=90-92 (thorium and uranium respectively) These
elements have one or two isotopes with half-lifes of billions of years so there are
substantial amounts of such elements on Earth This fact makes it possible to have
practical fission devices
Radioactive Decay
There are three types of radioactive decay
Alpha decay the nucleus kicks out a helium nucleus (N = 2 Z = 2 A = 4) So if the
original nucleus is called X and the new nucleus is called Y then the decay would look
like this
4 42 2X Y+ HeA A
Z Z
Beta decay the nucleus either kicks out an electron (endash) or its positively-charged
antimatter twin the positron (e+) plus either an electron neutrino v or an electron
anti-neutrino v
1X Y+e electron decayA AZ Z v
1X Y+e positron decayA AZ Z v
A neutrino is a particle with no charge hardly any mass (much less than the electron
mass) and interacts so weakly with matter that most neutrinos upon encountering the
planet earth just pass right through it as if it werenrsquot there
Gamma decay the nucleons in the nucleus X are in an excited energy state X
(perhaps as a result of having undergone alpha or beta decay) and they drop down to a
lower energy state shedding the energy as a high frequency photon
X XA AZ Z
36
This process is exactly analogous to the way that the electrons in atoms emit photons
Decay Rate and Half Life
There is no way to predict exactly when an unstable or excited nucleus will decay but
there is an average rate at which this decay occurs called the decay constant λ The
meaning of this constant is that if there are a large number N of nuclei in a sample
then the number of decays per second that will be observed (called the decay rate R)
is R = λN In mathematical language
dMR N
dt
This simple differential equation has for its solution
0tN t N e
where N0 is the number of nuclei in the sample at time t = 0
The half-life is the time it takes for half of the nuclei in the sample to decay and is
related to the decay constant by
1 2
ln 2 0693T
Disintegration Energy
When a nucleus decays it is making a transition to an overall state of lower energy
which means according to Einsteinrsquos famous formula E = mc2 that the sum of the
masses after the decay must be less than the mass before with the lost mass appearing
as kinetic energy among the decay products For example in alpha decay this kinetic
energy called the disintegration energy Q is given by
2X YQ M M M c
37
Serway Chapter 45
Nuclear energy
There are two ways to extract energy from the nucleus fission and fusion For nuclei
with Z greater than 26 breaking the nucleus apart into pieces leads to a lower
2mc energy than the original nucleus so energy can be extracted by fission For
nuclei with Z less than 26 a lower energy is achieved by combining nuclei so energy
can be extracted by fusion
Fission
Since neutrons have no charge they are not repelled from nuclei as protons are for
this reason their behavior is the key to understanding how fission works
Fast neutrons mostly bounce off other nuclei with each collision slowing the neutron
down This elastic energy loss is most effective if the other nuclei have low mass (like
hydrogen) and these materials are called moderators because of their ability to slow
down fast neutrons
The reason that slowing neutrons is important is that slow neutrons are much more
likely to be absorbed by a nucleus which then leads to nuclear reactions of various
kinds For a few very large nuclei like uranium-235 and some plutonium isotopes
absorption of a slow neutron causes the nucleus to split into two large fragments plus
2 or 3 fast neutrons (energy is released in the process too) Thus one slow neutron
can produce energy plus 2 or 3 more neutrons which if moderated from fast to slow
can split 2 or 3 more nuclei leading to a runaway chain reaction If the reaction is
allowed to proceed unchecked it produces a large explosion If a neutron absorbing
material is added to the mix (like the cadmium in reactor control rods) it is possible to
keep the reaction under control and to extract the released energy as heat to drive
steam turbines and produce electricity
Fusion
Fusion involves mashing two nuclei together and since they are both charged and repel
each other this reaction is much harder to make go The nuclei must have enough energy to
overcome the coulomb repulsion which is why this reaction requires a high temperature
(hundreds of millions of degrees K) like that in the sun (which is burning hydrogen to
helium via fusion) or like that in the center of the fission explosion that is used to detonate
38
a hydrogen bomb
This reaction is of interest for power production in spite of this difficult temperature
requirement because of the abundance of fusion fuel on the planet There are about
012 g of deuterium ( 21H ) in every gallon of water on earth and it only costs about 4
cents to extract it The fusion energy available from this minuscule amount of
deuterium would run a 1000-Megawatt power plant for 10 seconds Water is so
abundant on earth that if fusion were to work we would have an essentially
inexhaustible source of energy
So why donrsquot we have fusion power plants Well the fuel is cheap but the match is
incredibly expensive The only way we know to control this difficult high-temperature
reaction is with large and expensive pieces of equipment involving either large
magnetic fields and complex high-power electromagnetic antennas or with gigantic
(football-fieldsized) laser facilities involving more than a hundred of the highest-
energy lasers ever built Power plants based on these current methods for controlling
fusion are unattractive to the fiscally-minded people who run the electric power
industry Hopefully better designs will be discovered as experiments continue
39
c actinium 68 Er erbium 101 Md mendelevium 104 Rf rutherfordium l aluminum 63 Eu europium 80 Hg mercury 62 Sm samarium m americium 100 Fm fermium 42 Mo molybdenum 21 Sc scandium b antimony 9 F fluorine 60 Nd neodymium 106 Sg seaborgium r argon 87 Fr francium 10 Ne neon 34 Se selenium s arsenic 64 Gd gadolinium 93 Np neptunium 14 Si silicon t astatine 31 Ga gallium 28 Ni nickel 47 Ag silver a barium 32 Ge germanium 41 Nb niobium 11 Na sodium k berkelium 79 Au gold 7 N nitrogen 38 Sr strontium e beryllium 72 Hf hafnium 102 No nobelium 16 S sulfur i bismuth 108 Hs hassium 76 Os osmium 73 Ta tantalum h bohrium 2 He helium 8 O oxygen 43 Tc technetium
boron 67 Ho holmium 46 Pd palladium 52 Te tellurium r bromine 1 H hydrogen 15 P phosphorus 65 Tb terbium d cadmium 49 In indium 78 Pt platinum 81 Tl thallium a calcium 53 I iodine 94 Pu plutonium 90 Th thorium f californium 77 Ir iridium 84 Po polonium 69 Tm thulium
carbon 26 Fe iron 19 K potassium 50 Sn tin e cerium 36 Kr krypton 59 Pr praseodymium 22 Ti titanium s cesium 57 La lanthanum 61 Pm promethium 74 W tungsten l chlorine 103 Lr lawrencium 91 Pa protactinium 92 U uranium r chromium 82 Pb lead 88 Ra radium 23 V vanadium o cobalt 3 Li lithium 86 Rn radon 54 Xe xenon u copper 71 Lu lutetium 75 Re rhenium 70 Yb ytterbium m curium 12 Mg magnesium 45 Rh rhodium 39 Y yttrium b dubnium 25 Mn manganese 37 Rb rubidium 30 Zn zinc y dysprosium 109 Mt meitnerium 44 Ru ruthenium 40 Zr zirconium s einsteinium
40
UNITS The SI units are given in parentheses Other commonly used units are given in terms of the SI units Symbols conform with the recommendations of the American National Standards Institute (ANSI) and the American Institute of Physics (AIP) Commonly used multiples of SI units
prefix symbol factor tera T 1012 giga G 109
mega M 106 kilo k 103
centi c 10-2 milli m 10-3
micro μ 10-6 nano n 10-9 pico p 10-12
femto f 10-15
length meter (m) angstrom 1 Ǻ = 10-10 m inch 1 in = 254 x 10-2 m foot 1 ft = 03048 m mile 1 mi = 1609 m
mass kilogram (kg) atomic mass unit 1 u = 1661 x 10-27 kgslug 1 slug = 1459 kg
force newton (N = kg bull ms2) dyne 1 dyn = 10-5 N pound 1 lb = 4448 N
pressure Pascal (Pa = kgm bull s2) atmosphere 1 atm = 1013 x 105 Pa poundssquare inch 1 psi = 6895 Pa cm of mercury 1 cm Hg = 1333 Pa bar 1 bar = 1000 x 105 Pa torr 1 torr = 1333 Pa
time second (s) minute 1 min = 60 s hour 1 h =3600 s
frequency hertz (Hz = s-1) radianssecond 1 rads = 12π Hz
energy joule (J = kg bull m2s2) erg 1 erg = 10-7 J electron volt 1 eV = 1602 x 10-19 J calorie 1 cal = 4187 J kilowatt-hour 1 kWh = 36 x 106 J British thermal unit 1 Btu = 1055 J
power watt (W = kg bull m2s3) horsepower 1 hp = 7457 W
charge coulomb (C = A bull s)
electric potential volt (V = kg bull m2 s3 bull A)
current ampere (A)
resistance ohm (Ω = kg bull m2s3 bull A2)
capacitance farad (F = s4 bull A2kg bull m2)
magnetic field tesla (T = kgs2 bull A) gauss 1 G = 10-4 T
magnetic flux weber (Wb = kg m2s2 bull A) maxwell 1 Mx = 10-8 Wb
magnetic inductance henry (H = kg bull m2s2 bullA2)
temperature kelvin (K) degrees Celsius 0degC = 27315 K
angle radian (rad) degree 1 deg = π180 rad revolution 1 rev = 2π rad
41
Some Physical Constantsa Quantity Symbol Valueb Atomic mass unit μ 1660 538 73 (13) x 10-27 kg
931494 013 (37) MeVc2 Avogadros number NA 6022 141 99 (47) x 1023 particlesmol Avogadrorsquos number 6022 x 1023 mol Bohr magneton
2B e
em 9274 008 99 (37) x 10-24 JT
Bohr magneton μB 927 x 10-24 JT Bohr radius
0
2
2e em e k
a 5291 772 083 (19) x 10-11 m
Boltzmanns constant B A
RNk 1380 650 3 (24) x 10-23 JK
Boltzmannrsquos constant kB 1380 x 10-23 JK Compton wavelength
C eh
m c 2426 310 215 (18) x 10-12 m
Deuteron mass md 3343 583 09 (26) x 10-27 kg 2013 553 212 71 (35) u
electron charge e 1602 x 10-19 C Electron mass me 9109 381 88 (72) x 10-31 kg
5485 799 110 (12) x 10-4 u 0510 998 902 (21) MeVc2
electron mass 911 x 10-31 kg Electron volt eV 1602 176 462 (63) x 10-19 J Elementary charge e 1602 176 462 (63) x 10-19 C Gas constant R 8314 472 (15) JKmol Gravitational constant G 6673 (10) x 10-11 Nm2kg2 Neutron mass mn 1674 927 16 (13) x 10-27 kg
1008 664 915 78 (55) u 939565 330 (38) MeVc2
neutron mass 1675 x 10-27 kg Nuclear magneton
2n p
em 5050 783 17 (20) x 10-27 JT
permeability constant μ0 1257 x 10-6 Hm permittivity constant ε0 8854 x 10-12 Fm Plancks constant h
2h
6626 068 76 (52) x 10-34 Js 1054 571 596 (82) x 10-34 Js
Planckrsquos constant h ħ
6626 x 10-34 Js 1055 x 10-34 Js
Proton mass mp 1672621 58 (13) x 10-27 kg 1007 276 466 88 (13) u 938271 998 (38) MeVc2
proton mass 1673 x 10-27 kg Rydberg constant RH 1097 373 156 854 9 (83) x 107 m-1 Speed of light in vacuum c 2997 92458 x 108 ms (exact) speed of light c 300 x 108 ms a
These constants are the values recommended in 1998 by CODATA based on a least-squares adjustment of data from different measurements For a more complete list see P J Mohr and B N Taylor Rev Mod Phys 72351 2000 b The numbers in parentheses for the values above represent the uncertainties of the last two digits
NOTE The ones in red are the ones from Appendix 13 The ones in yellow are the ones that are from the Serway book table but that were already on the Appendix 13 table
42
INDEX
Absolute zero 9
Adiabatic 17
Adiabatic exponent 18
Adiabatic process 17
Alpha decay 46
Amplitude 4
Angle of reflection 25
Angular frequency 3
Angular magnification 30
Angular momentum atomic 43
Angular size 29
Archimedes Principle 1
Atomic spectra 43
Atoms 43
Avogadrorsquos number 10
Beats 8
Bernoullirsquos Equation 2
Beta decay 46
Bifocals 29
Brewsterrsquos angle 34
British Thermal Unit Btu 11
Buoyancy 2
calorie 11
Calorie 11
Camera 28
Carnot cycle 21
Carnot efficiency 22
Celsius scale 9
Ciliary muscle 29
Coefficient of performance 21
Compton effect 39
Constructive interference 7
Continuity equation of 2
Contraction length 35
Convection 14
Curved mirrors 27
Decay constant 46
Decay rate radioactive 46
Decibel scale 5
Degrees of freedom 16
Density 1
Destructive interference 7
Diffraction single slit 33
Diffraction grating 33
Dilation time 35
Disintegration energy 47
Dispersion of light 26
Doppler effect 6
Double slit interference 31
Efficiency 20
Electron interference 39
Emissivity 15
Energy nuclear decay 47
Energy relativistic 36
43
Engineering work 20
Entropy 22
Entropy ideal gas 24
Equipartition of energy 18
Equivalence principle 36
Exclusion principle 44
Expansion thermal 9
Expansion coefficient linear 9
Eye 29
Fahrenheit scale 9
Far point 29
Farsightedness 29
First Law of Thermodynamics 12
Fission 48
Flux volume 2
Free expansion 23
Fringes 31
Fusion 48
Fusion heat of 11
Gamma relativistic 35
Gamma decay 46
General relativity 36
Half-life 46
Heat 11
Heat capacity 11
Heat conduction 13
Heat engine 20
Heat of fusion 11
Heat of vaporization 11
Heat pump 20
Heats of transformation 11
Hydrostatics 1
Ideal Gas Law 10
Images real and virtual 27
Intensity sound 5
Interference 7
Interference two-slit 31
Internal energy 12
Internal energy degrees of freedom 16
Irreversible process 19 23
Isotopes 45
Joule 11
Kelvin scale 9
Kinetic energy relativity 36
Kinetic theory 16
Latent heat 11
Length contraction 35
Linear expansion coefficient 9
Linear polarization 34
Linear superposition 7
Longitudinal wave 3
Loudness 5
Magnification lateral 27
Magnifying glass 30
Malusrsquos law 34
Microscope 30
Momentum relativistic 36
Muscle ciliary 29
44
Musical instruments 8
Musical scale 8
Near point 29
Nearsightedness 29
Nonlinear 7
Nuclear energy 48
Nuclear properties 45
Nuclear stability 45
Octave 8
Optical resolution 33
Orbital quantum number ℓ 43
Orbitals 43
Particle in a Box 41
Particles are waves 39
Pascalrsquos Principle 1
Period 3
Photoelectric effect 38
Photons 38
Pitch 8
Polarization 34
Power sound 5
Presbyopia 29
Pressure 1
Principal quantum number n 43
Principle of equivalence 36
Principle of linear superposition 7
Probability amplitude ψ 41
Processes thermodynamic 13
R-value 14
Radiation thermal 14
Radioactive decay 46
Radioactive decay rate 46
Ray tracing 27
Rayleighrsquos criterion 33
Reading glasses 29
Real image 27
Refraction 25
Refrigerator 20
Relativistic gamma 35
Relativity principles 35
Resolved for light sources 33
Rest energy 36
Reversible process 19 22
Rope wave speed 4
Second Law of Thermodynamics 19
Shock waves 6
Simple magnifier 30
Simultaneity 35
Single slit diffraction 33
Snellrsquos law 25
Sound speed 5
Specific heat 11
Spin Angular momentum 44
Standing waves 7
Stefanrsquos law 14
Strong force 45
Telescope 30
Temperature 9
45
Temperature Scales 9
Thermal conductivity 14
Thermal energy 12
Thermal expansion 9
Thin film interference 31
Thin lenses 28
Time dilation 35
Tone musical 8
Total internal reflection 25
Transverse wave 3
Traveling Waves 3
Two-slit interference 31
Uncertainty principle 40
Vaporization heat of 11
Velocity addition relativity 36
Virtual image 27
Volume flux 2
Wave function 41
Wave Function ψ and Probability 41
Wave speed 3
Wave-particle duality 39
Wavelength 3
Wavenumber 3
Wien Displacement Law 16
Work 11
5
Serway Chapter 17
Sound Speed in Solids Liquids and Gases
The speed of sound in a liquid or solid of bulk modulus B and volume mass density ρ
Bv
The speed of sound in air is
331 1 273v m s T C
The speed of sound in air at room temperature is about 343 ms Notice it doesnrsquot depend on
pressure or frequency
Power Intensity and Loudness
The intensity of a sound wave is defined to be the power per unit area ie wattsm2 in the wave
For spherical waves traveling away from a small source emitting waves with average power Рav
the intensity falls off with distance from the source r according to the inverse-square law
av24
Ir
Hence the intensity I2 at distance r2 is related to the intensity I1 at a different distance r1 by
21 2
22 1
I r
I r
Experiments on human hearing have shown that we hear intensity differences logarithmically so
the decibel loudness scale for sound intensity was invented The loudness β of a sound in
decibels is related to its intensity I in Wm2 by the formula
1010logo
I
I
Where Io is a sound intensity near the threshold of hearing defined to be Io = 10-12 Wm2 Note
that on this scale a sound is made 10 times more intense by adding 10 decibels Remember that
intensity is proportional to velocity squared Amplitude is the A is the previous chapterrsquos
equation for a traveling wave If you make the A 10 times as much the intensity will increase by
100 times (20 dB)
Doppler Effect
If sound waves are traveling through a medium and if either the receiver of the waves or the
source of waves is moving then the frequency received is related to the frequency emitted by
6
0
s
v vf f
v v
where fprime is the frequency detected by the observer f is the frequency emitted by the source v is
the speed of the waves vo is the speed of the observer and vs is the speed of the source This
formula assumes that the source and receiver are either moving directly toward each other or
directly away from each other To know which signs to use remember that when observer and
source approach each other the observed frequency is higher while if they move away from each
other it is lower Just examine the signs in the formula and make the answer come out right
For electromagnetic waves (light radio waves X-rays) traveling in vacuum Einsteinrsquos theory of
relativity (and careful experiments) show that the Doppler shift is given by
r
r
c vf f
c v
where vr is the relative speed between the source and the observer
For all kinds of waves (sound light etc) if the relative speed of the source and the observer is
small compared to the speed of the waves then there is a simple approximation to the Doppler
effect For example if the relative speed is 1 of the wave speed then the frequency shifts by
1 Remember that this is only an approximation
Shock Waves
When an object moves through a medium at a speed greater than the speed of waves a V-shaped
shock wave is produced The V-shaped wake behind a speeding boat is a good example of this
effect and the cone of sonic-boom behind a supersonic aircraft is another The angle that V-line
makes with the direction of travel of the source is given by
sins
v
v
where v is the wave speed and vs is the source speed
7
Serway Chapter 18
Principle of Linear Superposition
We say that a system obeys the principle of linear superposition if two or more different motions
of the system can simply be added together to find the net motion of the system Light waves
obey this principle as they propagate through the air as can be seen by shining two flashlights so
that their beams cross The beams propagate along without affecting each other (Light sabers are
a spectacular but unfortunately fictional example of systems that do not obey the principle of
superposition) Wave pulses on an ideal rope also obey this principle two different pulses pass
through each other without change Standing waves are an example of this effect being simply
the linear superposition of two traveling waves of the same frequency but moving in opposite
directions Light waves in matter do not always obey this principle For instance two powerful
laser beams could be made to cross in a piece of glass in such a way that their combined heating
effect in the crossing region could melt the glass and scatter the beams in complicated ways This
is an example of a nonlinear effect
Interference
When two or more waves are present in the same medium at the same time their net effect may
often be obtained simply by adding them at each point in the medium according to the principle
of linear superposition (Note this wonrsquot work if the medium is nonlinear) When this addition
makes the total amplitude be greater than the individual amplitudes of the various waves we say
that the interference is constructive When the addition produces cancellation and an amplitude
less than the amplitudes of the separate waves we have destructive interference
Standing Waves
A standing wave is the superposition of two identical traveling waves moving in opposite
directions Nodes are places where the two waves perfectly destructively interfere to produce
zero amplitude at all times Anti-nodes are places where the two waves perfectly constructively
interfere to produce an amplitude maximum The distance between nodes is λ2 Standing waves
on a string fixed at both ends have nodes at each end of the string Standing waves in an air
column enclosed in a tube have displacement anti-nodes at open ends of the tube and
displacement nodes at closed ends The frequency of the standing wave with the lowest possible
frequency is called the fundamental frequency Standing waves on strings or in air columns all
have frequencies which are integer multiples of the fundamental frequency and are called
8
harmonics (The fundamental is called the ldquofirst harmonicrdquo)
Beats
Beats are heard when two waves with slightly different frequencies f1 and f2 are combined The
waves constructively interfere for a number of cycles then destructively interfere for a number
of cycles We hear a periodic ldquowah-wahrdquo frequency equal to the difference of the two wave
frequencies
1 2bf f f
Musical Instruments
Musical instruments produce tones by exciting standing waves on strings (violins piano) and in
tubes (trumpet organ) The fundamental frequency of the standing wave is called the pitch of the
tone The pitch of concert A is 440 Hz by definition Two tones are an octave apart if one pitch
has twice the frequency of the other In written music there are 12 intervals in each octave with
the ratio between successive intervals equal to 2112 = 105946 The ratios for each tone in an
octave starting at A and ending at the next higher A are
A A B C C D D E F F G G A
1 10595 11225 11892 12599 13348 14142 14983 15874 16818 17818 18877 2
A musical tone is actually a superposition of the fundamental frequency and the higher
harmonics The tone quality of a musical instrument is determined by the amplitudes of the
various harmonics that it produces A violin and a trumpet can play the same pitch but they
donrsquot sound at all alike to our ears The difference between them is in the various amplitudes of
their harmonics
9
Serway Chapter 19
Temperature
Formally temperature is what is measured by a thermometer Roughly high temperature is what
we call hot and low temperature is what we call cold On the atomic level temperature refers to
the kinetic energy of the molecules A collection of molecules is called ldquohotrdquo if the molecules
have rapid random motion while a collection of molecules is called ldquocoldrdquo if the random motion
is slow When two bodies are placed in close contact with each other they exchange molecular
kinetic energy until they come to the same temperature This is the microscopic picture of the
Zeroth Law of Thermodynamics
Absolute Zero
Absolute zero is the lowest possible temperature that any object can have This is the temperature
at which all of the energy than can be removed an object has been removed (This removable
energy we call thermal energy) There is still motion at absolute zero Electrons continue to orbit
around atomic nuclei and even atoms continue to move about with a small amount of kinetic
energy but this small energy cannot be removed from the object For example at absolute zero
helium is a liquid whose atoms still move and slide past each other
Temperature Scales
Kelvin Scale Absolute zero is at T = 0 K water freezes at T = 27315 K room temperature is
around T = 295 K and water boils at T = 373 K Note we donrsquot use a deg symbol Kelvin is
prefered SI Unit
Celsius Scale Absolute zero is at T = -273degC water freezes at T = 0degC room temperature is
around T = 22degC and water boils at T = 100degC
Fahrenheit Scale Absolute zero is at T = -459degF water freezes at TF = 32degF room temperature
is around T = 72degF and water boils at TF = 212degF TF =18TC + 32 Notice that temperature
differences are the same for the Kelvin and Celsius scales
Thermal Expansion
When materials are heated they usually expand and when they are cooled they usually contract
(Water near freezing is a spectacular counterexample it works the other way around) The
coefficient of linear expansion is defined by the relation
1
i
L
L T
10
where Li is the initial length of a rod of the material and ΔL is the change in its length due to a
small temperature change ΔT The coefficient of volume expansion is defined similarly
1
i
V
V T
where Vi is the initial volume of a piece of material and where ΔV is the change in its volume due
to a small temperature change ΔT
Avogadrorsquos Number (N or NA)
One mole of any substance corresponds to 6022 times 1023 molecules The atomic mass given on the
periodic table (p A32-33 in the text) is the mass in grams of one mole of atoms For example
the atomic mass of oxygen is 15999 gmol The mass of a single atom is given by the same
number in units of atomic mass units (u) For example the mass of an oxygen atom is 15999 u
(on the average)
Ideal Gas Law (an example of an equation of state)
When the molecules of a gas are sufficiently inert and widely separated that interactions between
them are negligible we say that it is an ideal gas The pressure P volume V and temperature T
(in kelvins) of such a gas are State Variables and are related by the ideal gas law
Bor PV=NkPV nRT T
where n is the number of moles of the gas where R is the gas constant
8314 Jmol KR
where N is the number of molecules and where kB is Boltzmannrsquos constant
231380 10 J KBk
It works well for air at atmosphere pressure and even better for partial vaccuums The relative
ease of measuring pressure and the linear relationship between pressure and temperature (if V
and n are held fixed) makes an ideal gas an ideal thermometer Thermometers can be based on
properties of solids or liquids but the behavior of these materials with temperature is more
complicated
11
Serway Chapter 20
Heat
Heat is energy that flows between a system and its environment because of a tempera-
ture difference between them The units of heat are Joules as expected for an energy
Unfortunately there are several competing units of energy They are related by
1 cal 4186 J 1 Cal 4186 J 1 Btu 1054 J
Heat Capacity
There is often a simple linear relation between the heat that flows in or out of part of a
system and the temperature change that results from this energy transfer When this
linear relation holds it is convenient to define the heat capacity C and the specific
heat c as follows
For the entire object the heat Q it absorbs is given by Q = C(Tf - Ti)
For a particular material the specific heat is defined by c = Cm which is the heat
capacity per unit mass so that
Q = mc(Tf ndash Ti)
Note C has units of energy (J or Cal)(Kelvin kg)
Heats of Transformation or Latent Heat Q = plusmn mL
When a substance changes phase from solid to liquid or from liquid to gas it absorbs
heat without a change in temperature The latent heat or heat of transformation is
usually given per unit mass of the substance For example for water the heat of fusion
(melting) is L = 333 kJkg while the heat of vaporization (boiling) is L = 2260 kJkg
Note that heat for boiling is considerably bigger than melting for water You have to
be careful with signs heat is given off (negative) if you go down in temperature and
condense steam
Work
In general the small amount of work done on a system as a force Fon is exerted on it
through a vector displacement dx is given by
on xdW d F
12
But if the displacement is done very slowly (as we always assume in thermodynamics)
then the force exerted on the system and the force exerted by the system are in
balance so the force exerted by the system is ndash Fon In thermodynamics it is more
convenient to talk about the force exerted by the system so we change the above
formula for the work done on the system to
xdW d F
where F is the force exerted by the system This has confused students for more than a
century now but this is the way your book and many other books do it so you are
stuck You will need to memorize the minus sign in this definition of the work to be
able to use your textbook
There are many chances to get signs wrong in this and the next two chapters (Mosiah
2 )
When an external agent changes the volume of a gas at pressure P by a small amount
dV the (small amount) of work done on the system is given by
dW PdV
Notice that this minus sign is just what we need to make dW be positive if the external
agent compresses the gas for then dV is negative If on the other hand the external
agent gives way allowing the gas to expand against it then dV is positive and we say
that the work done on the gas is negative
The work done on the system (eg by the gas in a cylinder) in a thermodynamic
process is the area under the curve in a PV diagram It is positive for compressions
and negative for expansions If the volume of gas remains constant in a process then
no work is done by the gas
Cyclic processes are important For cyclic processes represented by PV diagrams the
magnitude of the net work during one cycle is simply the area enclosed by the cycle
on the diagram Be careful to keep track of signs when you are calculating that
enclosed area In cyclic process Q = ndashW for a cycle Put a PV diagram in your notes
1
Path A-B B-C C-D D-A A to A net
Q
W
ΔU
ΔS
Internal Energy
The energy stored in a substance is called its internal energy Eint This energy may be
stored as random kinetic energy or as potential energy in each molecule (stretched
chemical bonds electrons in excited states etc) For ideal gases all states with the
same temperature will have the same Eint
First Law of Thermodynamics
The change ΔEint in the internal energy of a system is given by
intE Q W
where Q is the heat absorbed by the system and where W is the work done on the
system Hence if a system absorbs heat (and if Wge0) the internal energy increases
Likewise if the system does work (W on the system is negative) and if Qge0 the
internal energy decreases Potential Pitfall Many times people talk about work done
by the system It is the minus of W on the system Donrsquot get tripped up
Processes
Adiabatic process No heat is exchanged Q = 0 requiring that the system be insulated
from the environment A process may be approximately adiabatic if it happens so
rapidly that heat does not have time to enter or leave the system Work + or ndash is done
and ΔEint = W
Adiabatic free expansion A gas is allowed to expand into a vacuum without pushing
on anythingndashit just rushes into the vacuum In this process Q = W = 0 so the internal
energy and temperature does not change (Note the difference between an adiabatic
process and a free expansion is that NO work is done in the adiabatic free expansion)
Isobaric process The pressure is held fixed ΔP = 0 For example usually the
pressure increases when a gas is heated but if it were allowed to expand during the
2
heating process in just the right way its pressure could remain fixed In isobaric
processes both Q and W are non-zero (Work is eacy = ndashP(Vf ndash Vi)
Isovolumetric process The volume is held fixed ΔV = 0 The work done by the gas is
then zero and so we have ΔEint = Q
Isothermal process The temperature is held fixed ΔT = 0 There is no change in Eint
so Q = ndashW for isothermal processes For an ideal gas PV = nRT so PV = constant in
an isothermal process The work done on the gas is then given by
lnf
i
Vi
Vf
VW PdV nRT
V
Heat Conduction
The quantity P is defined to be the rate at which heat flows through an object and is a
power having units of watts It is analogous to electric current which is the rate at
which charge flows through an object If the flow of heat through a slab of length L
and cross-sectional area A is steady in time then P is given by the equation
h cT TdQkA
dt L
P =
where Th and Tc are the (hot and cold) temperatures of the two ends of the slab The
heat flows of course because of this temperature difference The quantity k is called
the thermal conductivity and is a constant that is characteristic of the material It is
analogous to the electrical conductivity h cT TL is sometimes called the temperature
gradient and is written dTd dTdx
R-Values
It is common to have the heat-conducting properties of materials described by their R-
values especially for insulating materials like fiberglass batting The connection
between k and R is R = Lk where L is the material thickness In this country R-values
always have units of 2ft F hourBtu
Polyurethane foam 6 (1 in thick) Air space 1 (35 in thick) Fiber glass batting 11
(35 in thick) Wood about 1 (1 in thick) Glass 09 (0125 in thick)
The heat flow rate through a slab of area A is given by
3
h cT TA
R
P
in units of Btuhour Note that A must be in square feet and the temperatures must be
in degrees Fahrenheit
Convection
Convection is the transfer of thermal energy by flow of material For instance a home
furnace doesnrsquot heat a house by waiting for the heat from the burner to slowly conduct
throughout the house instead it quickly pumps warm air to all of the rooms
Generally convection is a much faster way to transfer heat than conduction
Radiation
Electromagnetic radiation can also transfer heat When you warm yourself near a
campfire which has burned itself down into a bed of glowing embers you are
receiving radiant heat from the infrared portion of the electromagnetic spectrum The
rate at which an object emits radiant heat is given by Stefanrsquos law
4AeTP
where P is the radiated power in watts σ is a constant
8 2 45696 10 W m K
A is the surface area of the object in m2 and T is the temperature in kelvins The
constant e is called the emissivity and it varies from substance to substance A perfect
absorber (think black velvet) has e = 1 while a perfect reflector (think mirror) has e =
0 Hence black objects radiate very well while shiny ones do not Also an object that
is hotter than its surroundings radiates more energy than it absorbs whereas an object
that is cooler than its surroundings absorbs more energy than it radiates
Terminology
Transfer variables vs state variables
Energy transfer by heat as well as work done depends on the initial final and
intermediate states of the system They are transfer variables But their sum (Q + W =
4
Eint) is a state variable
Figure 205
5
Serway Chapter 21
Kinetic Theory
The ideal gas law works for all atoms and molecules at low pressure It is rather
amazing that it does Kinetic theory explains why The properties of an ideal gas can
be understood by thinking of it as N rapidly moving particles of mass m As these
particles collide with the container walls momentum is imparted to the walls which
we call the force of gas pressure In this picture the pressure is related to the average
of the square of the particle velocity 2v by
22 1( )
3 2
NP mv
V
Using the ideal gas law we obtain the average translational kinetic energy per
molecule
21 3
2 2 Bmv k T
The rms speed is then given by
2rms
3 3Bk T RTv v
m M
where M is the molecular mass in kgmol
Degrees of Freedom
Roughly speaking a degree of freedom is a way in which a molecule can store energy
For instance since there are three different directions in space along which a molecule
can move there are three degrees of freedom for the translational kinetic energy
There are also three different axes of rotation about which a polyatomic molecule can
spin so we say there are three degrees of freedom for the rotational kinetic energy
There are even degrees of freedom associated with the various ways in which a
molecule can vibrate and with the different energy levels in which the electrons of
the molecule can exist
Internal Energy and Degrees of Freedom The internal energy of an ideal gas made
up of molecules with J degrees of freedom is given by
int 2 2 B
J JE nRT Nk T
6
Heat Capacities of an Ideal Gas The heat capacity of a gas is described by means of
molar heat capacities CV and CP These are the heat capacities per mole and the
subscript V on CV means that the volume is being held constant while for CP the
pressure is held constant For example to raise the temperature of n moles of a gas
whose pressure is held constant by 10 K we would have to supply an amount of heat Q
= nCP (10) K
Molar Specific Heat of an Ideal Gas at Constant Volume
VQ nC T
3monatomic
2VC R
5diatomic
2VC R
5polyatomic
2VC
Real gases deviate from these formulas because in addition to the translational and ro-
tational degrees of freedom they also have vibrational and electronic degrees of
freedom These are unimportant at low temperatures due to quantum mechanical
effects but become increasingly important at higher temperatures The rough rule is
No of degrees of freedom
2VC R
Molar Specific Heat of an Ideal Gas at Constant Pressure
PQ nC T
P VC C R
The internal energy of an ideal gas depends only on the temperature
int VE nC T
Adiabatic Processes in an Ideal Gas
7
An adiabatic process is one in which no heat is exchanged between the system and the
environment When an ideal gas expands or contracts adiabatically not only does its
pressure change as expected from the ideal gas law but its temperature changes as
well Under these conditions the final pressure Pf can be computed from the initial
pressure Pi and from the final and initial volumes Vf and Vi by
or constantf f i iP V PV PV
where γ = CPCV The quantity γ is called the adiabatic exponent Note that this
doesnrsquot mean that the ideal gas law no longer holds it does and in fact it can be
combined with the adiabatic law for pressure given above to obtain the adiabatic law
for temperatures
1 constantTV
Compressions in sound waves are adiabatic because they happen too rapidly for any
appreciable amount of heat to flow This is why the adiabatic exponent appears in the
formula for the speed of sound in an ideal gas
RTv
M
Note that v depends only on T and not on P Because it depends only on the
temperature the speed of sound is the same in Provo as at sea level in spite of the
lower pressure here due to the difference in elevation
Equipartition of Energy
Every kind of molecule has a certain number of degrees of freedom which are
independent ways in which it can store energy Each such degree of freedom has
associated with it ndash on average ndash an energy of 12 Bk T per molecule (or 1
2 RT per mole)
(Note since a molecule has so many possible degrees of freedom it would seem that
there should be a lot of 12 sBk T to spread around But because energy is quantized
some of these degrees of freedom are not ldquoactiverdquo until the temperature becomes high
enough that 12 Bk T is as big as the lowest quantum of energy
8
Serway Chapter 22
Second Law of Thermodynamics
There are several equivalent forms of this important law
Kelvin It is not possible to change heat completely into work with no other change
taking place Or in other words there are no perfect heat engines
Clausius It is not possible for heat to flow from one body to another body at a higher
temperature with no other change taking place Or in other words there are no
perfect refrigerators
Entropy In any thermodynamic process that proceeds from one equilibrium state to
another the entropy of the system + environment either remains unchanged or
increases The total entropy never decreases This law is a bit of an oddity among the
laws of physics because it is not absolute Things are forbidden by the second law not
because it is impossible for them to happen but because it is extremely unlikely for
them to happen (See below for more information about entropy)
Reversible and Irreversible Processes
A reversible process is one which occurs so slowly that it is in thermal equilibrium (or
very nearly so) at all times A hallmark of such processes is that a motion picture of
them looks perfectly normal whether run forward or backward Imagine for instance
the slow expansion of a gas at constant temperature in a cylinder whose volume is being
increased by a slowly moving piston Run the movie backwards and what do you see
You see the slow compression of a gas at constant temperature which looks perfectly
normal
An irreversible process is one which occurs in such a way that thermal equilibrium is
not maintained throughout the process The mark of this kind of process is that a motion
picture of it looks very odd when run backward Imagine the sudden expansion of a gas
into a previously evacuated chamber because a hole was punched in the wall between a
pressurized chamber and the evacuated one Run the movie backward and what do you
see You see the gas in the soon-to-be-evacuated chamber gather itself together and
stream through a tiny hole into a chamber in which there is already plenty of gas If you
have ever seen this happen get in touch with the support group for those who have
witnessed the spontaneous combustion of large mammals by calling 1-800-PYROCOW
9
Heat Engines
Because of the vexing difference in sign between work done on and system and work done
by a system we will invent a new work variable Weng Heat engines do work and so the net
W for these engines is negative But in engineering applications hidden minus signs are
regarded as evil so for heat engines we donrsquot talk about W instead we talk about its
magnitude engW W So for heat engines the first law is
int engE Q W
But for heat pumps and refrigerators work is done on the system so we use the usual work
W when we talk about these systems
A heat engine is a machine that absorbs heat converts part of it to work and exhausts the
rest The heat must be absorbed at high temperature and exhausted at low temperature If the
absorbed heat is Qh the exhausted heat is Qc and the work done by the engine is Weng then
eng h cW Q Q
and the efficiency of the engine is defined to be
eng
h
We
Q
A perfect engine would convert the heat hQ completely into work Weng giving an effi-
ciency of e = 1 Energy conservation alone allows a perfect engine but the second law
requires e lt 1
Refrigerators and Heat Pumps
A refrigerator is a machine that absorbs heat at low temperature and exhausts it at high
temperature the ldquobackwardsrdquo heat transfer being driven by the work done on the machine
by some source of power A heat pump is a machine that either works like a refrigerator
keeping a place cold by transferring heat from this cold place to a higher temperature
environment (cooling mode like an air conditioner) or it functions as a heater
transferring heat into a warm place from a cooler one (heating mode like a window unit
that heats a house by extracting thermal energy from the cold outdoors) The coefficient
of performance of a refrigerator or of a heat pump in cooling mode is defined to be
COP cooling modecQ
W
10
For a heat pump in heating mode the coefficient of performance is
COP heating modehQ
W
Note that we donrsquot have to use the engineering work here because in these systems
work is done on the system and W is naturally positive
A perfect refrigerator would take in heat Qc and exhaust the same amount of heat without
doing any work giving an infinite coefficient of performance Energy conservation alone
allows a perfect refrigerator but the second law requires COP lt infin A perfect heat pump
in heating mode would transfer Qh into the house without doing any work and so would
also have an infinite coefficient of performance The second law forbids this too A good
coefficient of performance for a real device would be around 5 or 6
Carnot Cycle
The most efficient of all possible engines is one that uses the Carnot cycle This cycle
employs an ideal gas has no friction and operates very slowly so that the gas can be in
thermal equilibrium at all parts of the cycle This means of course that it canrsquot
possibly be built and even if it could be built it would not run fast enough to be useful
Nevertheless this cycle is very important because it gives an upper bound on the
efficiency of real engines There cannot possibly be an engine that is more efficient
than one based on the Carnot cycle This cycle consists of the following four steps
1 The ideal gas absorbs heat Qh at constant temperature Th while the gas increases its
volume The reason that heat is absorbed is that expansion tends to cool the gas but
thermal contact with the environment at Th keeps the temperature high by heat
conduction into the ideal gas
2 The ideal gas further increases its volume by an adiabatic expansion This expansion
causes the gas to cool so at the end of this part of the cycle the gas is at temperature Tc
3 The gas exhausts heat Qc at constant temperature Tc while the gas decreases in
volume The reason that heat is exhausted is that compression tends to heat the gas
but thermal contact with the environment at Tc keeps the temperature low by heat
conduction out of the ideal gas
4 The gas is adiabatically compressed back to its original volume (the volume it started
with in step 1) This compression heats the gas from Tc up to Th
11
The efficiency of a Carnot engine is given by the very simple formula
1 cC
h
Te
T
where the temperatures must in be Kelvin No real engine can be more efficient than
this
The coefficient of performance of a Carnot refrigerator or heat pump in cooling mode
is given by
COP cooling mode cC
h c
T
T T
and no real refrigerator can have a coefficient of performance greater than this
The coefficient of performance of a Carnot heat pump in heating mode is
COP heating mode hC
h c
T
T T
So why donrsquot we just use these wonderful Carnot engines and have perfect efficiency
To make the reversible steps in the cycle really reversible they would have to occur
infinitely slowly So the price you pay for making a perfect engine is that it takes
forever to get it to do any work
Entropy
The entropy of a system is defined in terms of its molecular makeup and measures
roughly the disorder of the system If the system is packed into a very small volume
then it is quite ordered and the entropy will be low If it occupies a large volume the
entropy is high (To see what this has to do with disorder note that socks in a drawer
occupy a small volume while socks on the bed in the corner by the door and
hanging from the chandelier occupy a large volume) If the system is very cold then
the molecules hardly move and may even reach out to each other and form a crystal
This is a highly ordered state and therefore has low entropy If the system is very hot
with rapidly speeding molecules crashing into the container walls and bouncing off
each other things are disordered and the entropy is high
It is possible to calculate the entropy of a system in terms of its macroscopic thermody-
namic properties ie pressure volume temperature number of moles etc The key to this
calculation is the concept of a reversible process A reversible process is one that is
12
carried out without friction and so slowly that the process can be reversed at any stage by
making an infinitesimal change in the environment of the system The slow expansion of the
gas at Th in the Carnot cycle with tiny amounts of heat being transferred to the environment
is reversible If we turned around and began slowly to compress the gas would just slowly
exhaust heat to the environment in the exact reverse way that it absorbed it during expansion
Most processes however are irreversible For example if a gas-filled box were suddenly
increased in size so that the particles were free to wander into the void created by the sudden
expansion then the gas would eventually fill the new volume uniformly at the same
temperature as before the expansion (The temperature is unchanged in this imaginary
process because the kinetic energy of the molecules would be unaffected by such an
instantaneous expansion of the container walls) This imaginary but highly thought-
stimulating process is called a free expansion and it is impossible to reverse it During the
expansion we didnrsquot push on any of the molecules so reversing this process would mean
making them go back into their original volume without pushing on them they simply will
not cooperate to this extent Another way to see that just pushing them back where they came
from does not reverse the free expansion is to think about what would happen if we just
compressed either adiabatically or isothermally An adiabatic compression back to the
original volume would heat the gas above its original temperature and an isothermal
compression would require that heat be exhausted to the environment But the free expansion
involved neither temperature changes nor heat exchanges so neither of these two processes
is the reverse of the free expansion It is simply impossible to reverse this rapid expansion in
a way that takes us back to the initial state
It is possible to calculate the change of entropy for both reversible and irreversible processes
Letrsquos consider a reversible process first In a reversible process the entropy change is given
by the formula
dQS
T
where dQ is the amount of heat added to the system during a small step of the process
The total energy change during the process may then simply be calculated by integration
f f
f i i i
dQS S S dS
T
13
(Just as in the case of energy we are mainly interested in differences rather than in
absolute magnitudes)
This integration method does not work for irreversible processes For instance in the
rapid free expansion discussed above no heat is added to the system but its disorder
obviously goes up We need to find some other way to calculate the entropy The key
is the fact that the entropy of a system depends only on its current state and not at all
on how it arrived there So to calculate the entropy change in an irreversible process
first find out what the initial and final conditions of the process are Then invent a
reversible process that takes the system from the initial state to the final state Since
the entropy depends only on the state of system and not on the process the entropy
change for the reversible process is the same as that for the irreversible process
Entropy is thus like pressure or temperature if the pressure changes from Pl to to P2
then the pressure difference is P2 - Pl regardless of how the pressure change was
made This seems obvious for pressure but not for entropy only because you arent
comfortable with entropy yet
Entropy of an Ideal Gas
For n moles of an ideal gas the difference in entropy between a state with temperature
T and volume V and some standard state with temperature To and volume Vo is given
by
ln lno o Vo o
T VS T V S T V nC nR
T V
Entropy in a Phase Change
Calculating entropy change in a phase change like melting or boiling is easy It is
QT Where Q is in the latent heat for example the flows in to cause the melting and
T is the temperature of the phase change There is NO integral to do
14
Serway Chapter 35
Angle of Reflection
If plane waves are incident on a reflecting surface with the propagation direction of
the waves making angle θ1 with the normal direction to the surface then the reflected
angle θ1prime relative to the surface normal is simply
1 1
ie the incident angles and reflected angles are the same
Refraction
If plane waves traveling through medium 1 are incident on a plane interface between
medium 1 and medium 2 then the angle of incidence of the incoming wave θ1 and
the angle of refraction of the transmitted wave θ2 are related by Snellrsquos law
2 2
1 1
sin
sin
v
v
where v1 and v2 are the wave speeds in medium 1 and medium 2 The angle of
incidence and the angle of refraction are both measured between the wave propagation
direction and the normal to the interface In terms of indices of refraction in the case
of light waves Snellrsquos law takes the more familiar form
1 1 2 2sin sinn n
where ni = cvi
Total Internal Reflection
If a wave is incident from a medium of low wave speed into a medium of high wave
speed the law of refraction requires that the angle of refraction be greater than the
angle of incidence If the angle of refraction is required to be greater than 90deg then no
refracted wave can exist and total internal reflection occurs The critical incident
angle θc beyond which total internal reflection occurs is given by
2
1
sin c
n
n
15
Dispersion of Light
In addition to the speed of light varying from material to material it also varies with
wavelength within each material This means that the index of refraction is generally a
function of wavelength
c
nv
Since the wave speed is not constant such a medium is dispersive meaning in this
context that refraction actually disperses white light into its various colors because
Snellrsquos law gives a different angle for each wavelength In most materials the
variation with wavelength is quite small but this small effect is responsible for some
of the most spectacular color effects we ever see including rainbows a flashing
crystal chandelier and the colored fire of a diamond solitaire by candlelight
16
Serway Chapter 36
Real and Virtual Images
When light rays are focused at a certain plane producing an image if a sheet of white
paper is placed there we call the image a real image The images produced by film
projectors and overhead projectors are examples of real images
When light rays appear to come from a certain location but no image is produced
when a screen is placed there we say that there is a virtual image at that location For
instance when you look in a mirror it appears that someone is behind the mirror but
a screen placed back there in the dark would show nothing Your image in the mirror
is a virtual image
Ray Tracing
There are lots of rules about how to find the images in optical systems but the best
way to keep things straight is to learn how to draw the principal rays for curved
mirrors and lenses The rays for convex and concave mirrors are shown in Fig 3615
The rays for converging and diverging thin lenses are shown in Fig 3627 You should
memorize the principal rays and know how to use them to locate images
Curved Mirrors
The focal length of a curved mirror with radius of curvature R is given by
2
Rf
The relation between object distance p image distance q and focal length f is
1 1 1
p q f
If the mirror is a diverging mirror f should be negative and if q should turn out to be
negative the image is virtual
Lateral Magnification
The lateral magnification in an optical system is defined by the ratio of the image size
to the object size
17
Image height
Object height
qM
p
for curved mirrors
As usual there are sign conventions here too but this formula just gives the
magnitude It is better to keep keep track of upright images versus inverted images by
means of ray diagrams rather than by memorizing sign conventions
Thin Lenses
The focal length of a thin lens is related to the radii of curvature of the two faces R1
and R2 of the lens by
1 2
1 1 11n
f R R
Note that this formula differs from Eq (3611) in the text by not having a minus sign
between the two R-terms We like this form better because for a simple converging
lens like a magnifying glass we just use positive values of R for both surfaces If one
of the faces is concave producing divergence use a negative value for R And if a
surface is flat use R = infin If the face is flat the radius is infinite
The relation between the image object and focal distances for a thin lens is the same
as that for a curved mirror
1 1 1
q p f
Use a negative focal length if the lens is diverging
The lateral magnification for a thin lens is the same as for a curved mirror
Image height
Object height
qM
p
Camera
The lens system in a camera projects a real image of an object onto the film (or CCD
array in a digital camera) The position of the image is adjusted to be on the film by
moving the lens into or out of the camera
18
Eye
The eye is like a camera in that a real image is formed on the retina Unlike a camera
the image position is adjusted by changing the focal length of the lens This is done by
the ciliary muscle which squeezes the lens changing its shape
Near Point The near point is the closest distance from the eye for which the lens can
focus an image on the retina It is usually 18-25 cm for young persons
Far Point The far point is the greatest distance from the eye for which the lens can
focus an image on the retina For a person with normal vision the far point is at
infinity
Nearsightedness A person is nearsighted if their far point is at some finite distance
less than infinity This condition can be corrected with a lens that takes an object at
infinity and produces a virtual image at the personrsquos far point
Farsightedness and Presbyopia A person is farsighted if their near point is too far
away for comfortable near work like reading or knitting This can be corrected by a
lens which takes an object at a normal near point distance of 18-25 cm and produces a
virtual image at the personrsquos natural near point Presbyopia involves a similar
problem which nearly all people experience as they age The ciliary muscle becomes
too weak and the lens becomes too stiff to allow the eye to provide for both near and
far vision The solution for this problem is either reading glasses or bifocal lenses
Reading classes are just weak magnifying glasses mounted on eyeglass frames
Bifocal lenses are split into upper and lower halves The lower half is a lens which
gives the proper correction for near work and the upper half is a different lens for
proper focusing at infinity
Angular Size
When an object is brought closer to the eye it appears to be larger because the image
on the retina is larger The size of this image is directly proportional to the objectrsquos
angular size which is the angle subtended by the object measured from the center of
the lens of the eye In optical instruments which are to be used with the eye the
angular size of the final image is whatrsquos important because it determines how large
the image will appear to the viewer
19
Simple Magnifier
A simple magnifier is a single converging lens or magnifying glass It takes an object
closer to the eye than a normal near point and produces a virtual image at or beyond
this near point The angular magnification is defined to be the ratio of the angular size
when viewed through the lens to the angular size of the object when viewed at the
normal near point (without aid of the lens)
Microscope
This instrument has two lenses (1) The objective is near the object being viewed and
produces a greatly magnified real image (2) The eyepiece is a simple magnifier which
the viewer uses to closely examine the image from (1)
Telescope
This instrument also has two lenses (1) The objective at the front of the telescope
takes light from a distant object and produces a real inverted image (which is rather
small) near its focal point (2) This small real image is then examined by the eyepiece
functioning as a simple magnifier to produce a virtual image with a larger angular
size
20
Serway Chapter 37
Two-Slit Interference
If light is incident on two closely spaced narrow slits a pattern of light and dark
stripes is produced beyond the slits The bright stripes or fringes are caused by
constructive interference of the two waves coming from the slits Constructive
interference occurs whenever two waves arrive at a location in phase with each other
This occurs when the distance x1 from slit 1 to a point P on the screen and the
distance x2 from slit 2 to point P differ by in integral number of wavelengths
1 2 where 0 1 2x x m m
where λ is the wavelength of the light When x1 and x2 are much larger than the slit
spacing d this condition reduces to
sind m
where θ is the angle between the direction of the incident light and the direction of the
light arriving at the screen
Thin Films
When light is partially reflected and partially transmitted by a thin film of transparent
material it is possible to have interference between the wave reflected from the front
of the film and light reflected from the back of the film (The colored reflections from
the thin film of oil on the water in a rain-soaked parking lot are an example of this
effect) It is difficult to write down formulas that will work in all cases so we will
just review the important principles here
1 If the two reflected waves are in phase with each other the film has enhanced
reflection (constructive interference) but if the two reflected waves are out of phase
with each other reflection is diminished (destructive interference) Phase shifts occur
due to reflection and due to the extra path length through the film of the wave
reflected from the back of the film
2 The phase change due to reflection is determined by the difference in index of
refraction between the two media involved in the reflection If the wave is incident
21
from a medium with a low index of refraction into a medium with a high index of
refraction a phase change of 180deg occurs and the reflected wave is inverted If
incident from high to low no phase shift occurs and the reflected wave is non-
inverted
3 The extra path length through the film of the wave reflected from the back of the
film is equal to 2t where t is the thickness of the film (The incident light is assumed
to be normal to the surface of the film) The number of wavelengths contained in the
extra path length is equal to 2tλn where λn = λn is the wavelength of the light in the
film
4 Rules for reflection from thin films
If one ray is inverted and the other is not then we have
12 constructive
2 nt m
2 destructivent m
If either both rays are inverted or both are non-inverted we have
2 constructivent m
12 destructive
2 nt m
22
Serway Chapter 38
Diffraction Grating
A diffraction grating is simply a fancy version of two-slit interference with the two
slits replaced by thousands of slits Just as in the two-slit case the bright fringes
occur at angles given by
sin md
but in this case d the distance between neighboring slits is made to be very small
The effect of having many slits instead of two is to make each bright fringe highly
localized with wide dark regions between neighboring maxima
Single Slit Diffraction
When light passes through an opening in an opaque screen an interference pattern is
produced beyond the opening To understand why we may replace the single opening
by many small coherent sources of light These many sources interfere with each
other producing a pattern known as a diffraction pattern (Note that many authors do
not distinguish between interference and diffraction treating them as interchangeable
terms) If the opening is a slit of width a then the diffraction pattern far from the slit
will have a bright central maximum with a succession of minima and weaker maxima
on either side The angle between the incident direction and the minima is given by
sin ma
where m = plusmn1 plusmn2
If the opening is circular with diameter D the angle between the incident direction
and the first minimum is given by
sin 122D
Optical Resolution and Rayleighrsquos Criterion
Two point sources can just be resolved (distinguished from each other) if the peak of
the diffraction image of the first source overlies the first minimum of the diffraction
image of the second source For circular holes of the kind usually encountered in
23
optical devices this condition is approximately satisfied when the angular separation
between the two sources as viewed from the optical instrument is greater than or equal
to the critical angle
min 122D
where λ is the wavelength of the light and where D is the diameter of the aperture in
the instrument
Polarization
We say that an electromagnetic wave is polarized if its electric field vector doesnrsquot
change direction in a random fashion The simplest kind of polarization is linear
polarization in which the electric field vector oscillates back and forth along the
same axis in space Polarized light can be produced from normal unpolarized light by
selective absorption (as in Polaroid sunglasses) by passing light through crystals that
have different indices of refraction for different polarizations (double refraction) by
scattering (the blue sky is polarized) and by reflection (glare) Polarization by
reflection occurs when light reflects from a shiny insulating (non-metallic) surface
The amount of polarization is greatest for reflection at Brewsterrsquos angle
2
1
tan P
n
n
where θP is the incidence angle of light from medium 1 onto medium 2 and where n1
and n2 are the indices of refraction for the media (In our everyday experience n1 = 1
since the light comes in through the air and n2 is the index of refraction of the shiny
insulating material producing the glare eg water glass plastic paint etc)
Malusrsquos Law
The intensity of transmitted polarized light through a perfect polarizer is related to the
incident intensity of polarized light by Malusrsquos law
2cosoI I
where Io is the intensity of the incident light and where θ is the angle between the
electric field vector in the incident wave and the transmission axis the polarizer
24
Serway Chapter 39
Principles of Relativity
All of the weirdness of relativity flows from two simple principles
(1) The laws of physics must be the same in all inertial (non-accelerating) reference
frames
(2) The speed of light in vacuum has the same value in all inertial reference frames
Note that (1) seems reasonable but (2) is very odd It says that if two space ships are
approaching each other at nearly the speed of light and a laser pulse is shot from ship
1 toward ship 2 then when the people on ship 2 measure the speed of the pulse as it
goes by it is moving at 3 times 108 ms the same as if the ships were stationary
Simultaneity
If observer 1 sees two events in her own frame as simultaneous at two different
locations a moving observer 2 will see these two events happening at different times
Relativistic gamma
The factor γ (gamma) appears regularly in the formulas of relativity
2
2
1
1 vc
where v is the relative speed between two inertial frames
Time Dilation
If observer 1 sees two events at the same location in space separated by time Δtp in his
own frame then observer 2 moving at speed v relative to observer 1 will see these
two events separated by a longer time Δt
pt t
Moving clocks run slow
Length Contraction
If observer 1 measures the length of an object along the x-direction at some instant of
time in her frame to be Lp then observer 2 moving in the x-direction at speed v will
measure the length L of the object to be shorter
pLL
25
Moving meter sticks are short
Velocity Addition
If a particle is moving at velocity vac relative to frame c and if frame c is moving at
speed vcb with respect to frame b (with both velocities directed along the same line)
what would be the velocity of the particle in frame b In classical physics the answer is
ab ac cbv v v
but if any of these velocities are an appreciable fraction of the speed of light the
answer changes
21ac cb
abac cb
v Vv
v v c
In using this equation make sure that vac has the same sign in the numerator and in the
denominator and that vbc also has the same sign in both places Note that this equation
is different (and in our opinion easier to understand) than the equations in the text
Momentum and Energy
The momentum and total energy of a particle moving at velocity v (its speed is v v )
are given by
22
2 2
22
1 1 vvc c
m mcm E mc
vp v
Rest Energy
When a particle is at rest (v = 0 and γ = 1) then E = mc2 is called the rest energy It
includes the potential energy of the particle so that if we change the potential energy
of a particle we change its rest mass
Kinetic Energy The kinetic energy is the difference between the total energy and the
rest energy
2 2K mc mc
For v ltlt c this reduces to K = mv22
General Relativity
Einsteinrsquos principle of equivalence in the theory of general relativity is In the vicinity
of a point a gravitational field is equivalent to an accelerated frame of reference in
26
the absence of gravitational effects For example there is no difference whatever in
the physics experienced by an astronaut on earth and one who is in interstellar space
far away from any stars or planets but who has just turned on her spaceshiprsquos engines
and is accelerating at 98 ms2
27
Serway Chapter 40
Photons
We now know that even though light behaves like a wave it is better described as
consisting of small packets of energy called photons The energy of a photon is related
to its frequency by
E hf
where h is Planckrsquos constant
346626 10 J sh
Photons also carry momentum given by
hf hp
c
Since the amount of energy in each photon is fixed the difference between dim light
and intense light (of the same wavelength) is that dim light consists of fewer photons
than intense light
Photoelectric Effect
In the photoelectric effect light shining onto a metal surface gives the electrons in the
metal enough energy to escape and be detected It requires a certain amount of energy
(called the work function typically = 3 ndash 5 eV) just to kick the electron out of the
metal so the light must deliver at least this much energy to an electron to produce the
effect It is observed that red light no matter how intense never produces electrons
But ultraviolet light even if quite dim will eject electrons from the metal Since red
light consists of 2 eV photons and ultraviolet light has photons with energies around
4-6 eV the photon idea explains the behavior of the photoelectric effect The
maximum energy that an ejected electron can have is
maxK hf
where f is the photon frequency The minimum frequency that light can have and cause
electrons to be ejected is called the cutoff energy and is found by setting Kmax = 0
cf h
28
Compton Effect
When high-frequency light interacts with free electrons the scattered light does not
have the same wavelength as the incident light contrary to what classical
electromagnetism would predict But the photon picture in which photons have
momentum and energy allows us to treat scattering as a collision between the photon
and the electron In this collision the electron and the scattered photon both have a
different momentum and energy than they did before And since p = hλ for a photon
if the momentum is different the wavelength will also be different
after before 1 cose
h
m c
where the angle θ is the angle between the incoming photon direction and the
direction of the scattered photon
Particles are Waves
Since photons behave like particles sometimes it is not surprising that elementary
particles can behave like waves sometimes The wavelength of a particle with
momentum p is given by
h
p
Wave-particle Duality
Both photons and elementary particles have a dual nature sometimes they behave like
particles and sometimes they behave like waves It is hard for us to comprehend the
nature of such an object by making mental pictures but experiments definitively show
that this is the case Since we have no direct experience with photons and elementary
particles (because their energies are so much smaller than the energies of the everyday
objects we encounter) it is perhaps not surprising that we have a hard time forming a
mental picture of how they behave
Electron Interference
Since an electron is both a particle and a wave just like a photon it should be able to
produce an interference pattern This is observed If an electron beam is shot at two
closely-spaced slits and if the electrons that pass through the slits are detected
downstream it is found that there are some locations where electrons are never detected
29
and others where lots of electrons are detected The pattern is exactly the same as the
one observed for light waves provided that we use the electron wavelength λ = hp in
place of the wavelength of light This pattern is observed even though each electron is
detected as a single dot on the screen Only after many such dots are collected does the
pattern emerge And if we try to understand how this effect could possibly work by
looking closely at each slit to see which one the electron came through the pattern
disappears the act of measurement destroys the interference This means that each
single electron somehow comes through both slits (as a wave would)
Uncertainty Principle
In classical physics we always imagine that the positions and momenta of moving
particles have definite values It might be hard to measure them but surely at each
instant of time a particle should be precisely located at some point in space and have a
similarly precise momentum This turns out not to be true Instead both position and
momentum are required to be uncertain with their uncertainties Δx and Δp satisfying
the Heisenberg uncertainty relation
2x p
So if the particle were known to be precisely at some particular location (so that Δx =
0) we couldnrsquot know anything about its momentum (Δp = infin) and if its momentum
were exactly known we couldnrsquot know its position
There is a similar relation involving the particlersquos energy E and the time interval Δt
over which this energy is measured
2E t
This means that energy is not actually conserved if we are considering very small time
intervals and this brief non-conservation of energy has been observed
30
Serway Chapter 41
Wave Function ψ and Probability
Quantum mechanics does not predict exactly what an electron or a photon will do Instead
it specifies the wave function or probability amplitude ψ of an electron or a photon
This wave function is a complex-valued function of space and time whose squared
magnitude is the probability density P for finding a particle at a particular place in
space at a certain time
2P
where is the complex conjugate of ψ
For example the wave function of an electron with perfectly specified momentum p
would have a wavelength given by
h
p
and its probability amplitude would be proportional to
2i x ipxe e
The corresponding probability density would then be
21ipx ip ipxP e e e
which means that the electron is equally probable to be anywhere along the x axis
This is in accord with the uncertainty principle since we specified the momentum
precisely we canrsquot have any idea about the position of the electron
As another example you have probably seen ldquofuzzy ballrdquo drawings of electron
orbitals in a chemistry book These fuzzy balls are meant to indicate the distribution
of the probability density 2
P in the orbital
Particle in a Box
A simple example in which we can calculate the wave function is the case of a particle of
mass m constrained to be inside a 1-dimensional box between x = 0 and x = L In this
case the general wave function is a linear superposition of wave functions ψn of the form
sin niE tn
n xx t A e
L
31
where A is a positive constant where n = 1 2 3 and where the energy associated
with each of the quantum states ψn is given by
22
28n
hE n
mL
This wave function is zero at x = 0 and x = L which means that the particle will never
be found at the walls of the box The wave function has maximum values in the
interior and at these places the particle is most likely to be found and it also has
places where it is zero and at these places the particle will also never be found as
expressed by the formula
2sinn x
PL
The particle in the box is interfering with itself producing a probability interference
pattern across the box just like the interference patterns we studied with light and
sound
32
Serway Chapter 42
Atoms
Many people picture an atom as a miniature solar system where electrons orbit around
a massive nucleus at the center This picture is misleading because in atoms the wave
nature of electrons dominates The electrons form 3-dimensional standing waves
(called orbitals) centered at the nucleus And if you ask what is it thatrsquos waving the
answer is ψ the probability amplitude (see Chapter 41)
Standing waves on a string can occur only for certain frequencies (the fundamental
and higher harmonics) Similarly atomic orbitals occur only for certain energies For
the hydrogen atom the energies of the orbitals have the a particularly simple form
2
1136 eVnE
n
where n = 1 2 3 is called the principal quantum number For other atoms the
determination of the orbital energies requires numerical calculation by computers
Atomic Spectra
If an electron is somehow given extra extra energy (we say that it is excited) so that it
occupies a higher orbital it will eventually ldquofallrdquo back down to a lower orbital Each
time an electron falls to a lower orbital it loses the difference in energy between the
two orbitals in the form of a photon Since the orbital energies are discrete so are the
energy differences and so are the wavelengths of the emitted light The entire set of
these discrete wavelengths is called the atomic spectrum and it is unique to each type
of atom For hydrogen the spectrum can be simply written as
H 2 2
1 1 1
f i
Rn n
where RH = 10973732 times 107 m-1 For other atoms the spectrum cannot be expressed as
a formula
Orbital Angular Momentum Electrons in atoms also have quantized values of
angular momentum The orbital quantum number ℓ specifies the value of this
quantized angular momentum through the formula
1L
33
If we want to know the value of the angular momentum along some direction in space
say the z direction the answer is not L but rather
zL m
where mℓ is another quantum number which runs from
1 1m
This quantum number is important when an atom sits in a magnetic field
Spin Angular Momentum It has been found experimentally that electrons and other
charged particles also carry internal angular momentum which we call spin
Electrons have an intrinsic spin angular momentum s along a specified axis that is
extremely quantized it can only take on 2 possible values
1
2zs s
We thus say that electrons have spin s = 12 or that electrons are rdquospin one-half parti-
cles Since s is an angular momentum it obeys the same rule as that for orbital angular
momentum namely that if its value along some axis is s then its total magnitude is
given by
31
2S s s
I know this seems weird but quantum mechanics is weird The only excuse for this
bizarre way of looking at the world is that it predicts what happens in experiments
Exclusion Principle The answer to the question of how many electrons (or any other
spin one-half particle) can be in one particular quantum state was discovered by
Wolfgang Pauli and is called the exclusion principle
ldquoNo two electrons can ever be in the same quantum state therefore no two electrons
in the same atom can have the same set of quantum numbersrdquo
This is the reason that we have atoms with different properties instead of every atom
simple having all of its electrons in the ground state All of the variety we see around
us in the world is the result of chemical differences and these differences would not
exist unless electrons obeyed this important principle The entire structure of the
periodic table (see pages 1377-1379 in Serway) is an expression of this principle
34
Serway Chapter 44
Nuclear Properties
The nucleus is composed of protons with charge +e and mass mp = 1007226 u and of
neutrons with zero charge and mass mn = 1008665 u where u is one atomic mass unit
-271 u=1660540 10 kg
The atomic number Z counts the number of protons in a nucleus while the neutron
number N counts the number of neutrons The mass number A is the sum of the two
A N Z
Protons and nuetrons have about the same mass and some times called baryons (heavy
ones) or nucleons (partcles in the nucleus) A is sometimes called the baryon
number The chemical elements are distinguished by Z ie hydrogen has Z = 1 iron
has Z = 26 uranium has Z = 92 etc But for a given Z there might be several nuclei
with different numbers of neutrons These nuclear siblings are called isotopes
These numbers are used to label nuclei according to the pattern
5626 ie FeA
Z X
denotes the isotope of iron with N = 56 ndash 26 = 30 neutrons
The nucleus is roughly spherical with a radius given approximately by
1 3 150 0where 12 10 mr r A r
Nuclear Stability
Because the positively charged protons electrically repel each other with an enormous
force at distances as small as 10-15 m there must be some really strong force that that
overcomes electrical repulsion to hold protons and neutrons together This force is
called with some lack of imagination the strong force It is a very short-range force
(it only acts over a distance of about 2 x 10-15 m) and attracts protons to protons
neutrons to neutrons and neutrons to protons But in spite of this strong nuclear force
the coulomb repulsion of the protons is still present so anything that might keep the
protons from being right next to each other would help keep the nucleus from
35
exploding This role is played by the neutrons and for nuclei with Z le 20 the stable
nuclei roughly have N = Z
For Z gt 20 the coulomb repulsion force is more powerful and more neutrons are
needed to dilute it up to about N = 15Z around Z = 80 For Z gt 83 no amount of
neutrons can help and these nuclei are unstable (radioactive) There is an ldquoisland of
(relative) stabilityrdquo around Z=90-92 (thorium and uranium respectively) These
elements have one or two isotopes with half-lifes of billions of years so there are
substantial amounts of such elements on Earth This fact makes it possible to have
practical fission devices
Radioactive Decay
There are three types of radioactive decay
Alpha decay the nucleus kicks out a helium nucleus (N = 2 Z = 2 A = 4) So if the
original nucleus is called X and the new nucleus is called Y then the decay would look
like this
4 42 2X Y+ HeA A
Z Z
Beta decay the nucleus either kicks out an electron (endash) or its positively-charged
antimatter twin the positron (e+) plus either an electron neutrino v or an electron
anti-neutrino v
1X Y+e electron decayA AZ Z v
1X Y+e positron decayA AZ Z v
A neutrino is a particle with no charge hardly any mass (much less than the electron
mass) and interacts so weakly with matter that most neutrinos upon encountering the
planet earth just pass right through it as if it werenrsquot there
Gamma decay the nucleons in the nucleus X are in an excited energy state X
(perhaps as a result of having undergone alpha or beta decay) and they drop down to a
lower energy state shedding the energy as a high frequency photon
X XA AZ Z
36
This process is exactly analogous to the way that the electrons in atoms emit photons
Decay Rate and Half Life
There is no way to predict exactly when an unstable or excited nucleus will decay but
there is an average rate at which this decay occurs called the decay constant λ The
meaning of this constant is that if there are a large number N of nuclei in a sample
then the number of decays per second that will be observed (called the decay rate R)
is R = λN In mathematical language
dMR N
dt
This simple differential equation has for its solution
0tN t N e
where N0 is the number of nuclei in the sample at time t = 0
The half-life is the time it takes for half of the nuclei in the sample to decay and is
related to the decay constant by
1 2
ln 2 0693T
Disintegration Energy
When a nucleus decays it is making a transition to an overall state of lower energy
which means according to Einsteinrsquos famous formula E = mc2 that the sum of the
masses after the decay must be less than the mass before with the lost mass appearing
as kinetic energy among the decay products For example in alpha decay this kinetic
energy called the disintegration energy Q is given by
2X YQ M M M c
37
Serway Chapter 45
Nuclear energy
There are two ways to extract energy from the nucleus fission and fusion For nuclei
with Z greater than 26 breaking the nucleus apart into pieces leads to a lower
2mc energy than the original nucleus so energy can be extracted by fission For
nuclei with Z less than 26 a lower energy is achieved by combining nuclei so energy
can be extracted by fusion
Fission
Since neutrons have no charge they are not repelled from nuclei as protons are for
this reason their behavior is the key to understanding how fission works
Fast neutrons mostly bounce off other nuclei with each collision slowing the neutron
down This elastic energy loss is most effective if the other nuclei have low mass (like
hydrogen) and these materials are called moderators because of their ability to slow
down fast neutrons
The reason that slowing neutrons is important is that slow neutrons are much more
likely to be absorbed by a nucleus which then leads to nuclear reactions of various
kinds For a few very large nuclei like uranium-235 and some plutonium isotopes
absorption of a slow neutron causes the nucleus to split into two large fragments plus
2 or 3 fast neutrons (energy is released in the process too) Thus one slow neutron
can produce energy plus 2 or 3 more neutrons which if moderated from fast to slow
can split 2 or 3 more nuclei leading to a runaway chain reaction If the reaction is
allowed to proceed unchecked it produces a large explosion If a neutron absorbing
material is added to the mix (like the cadmium in reactor control rods) it is possible to
keep the reaction under control and to extract the released energy as heat to drive
steam turbines and produce electricity
Fusion
Fusion involves mashing two nuclei together and since they are both charged and repel
each other this reaction is much harder to make go The nuclei must have enough energy to
overcome the coulomb repulsion which is why this reaction requires a high temperature
(hundreds of millions of degrees K) like that in the sun (which is burning hydrogen to
helium via fusion) or like that in the center of the fission explosion that is used to detonate
38
a hydrogen bomb
This reaction is of interest for power production in spite of this difficult temperature
requirement because of the abundance of fusion fuel on the planet There are about
012 g of deuterium ( 21H ) in every gallon of water on earth and it only costs about 4
cents to extract it The fusion energy available from this minuscule amount of
deuterium would run a 1000-Megawatt power plant for 10 seconds Water is so
abundant on earth that if fusion were to work we would have an essentially
inexhaustible source of energy
So why donrsquot we have fusion power plants Well the fuel is cheap but the match is
incredibly expensive The only way we know to control this difficult high-temperature
reaction is with large and expensive pieces of equipment involving either large
magnetic fields and complex high-power electromagnetic antennas or with gigantic
(football-fieldsized) laser facilities involving more than a hundred of the highest-
energy lasers ever built Power plants based on these current methods for controlling
fusion are unattractive to the fiscally-minded people who run the electric power
industry Hopefully better designs will be discovered as experiments continue
39
c actinium 68 Er erbium 101 Md mendelevium 104 Rf rutherfordium l aluminum 63 Eu europium 80 Hg mercury 62 Sm samarium m americium 100 Fm fermium 42 Mo molybdenum 21 Sc scandium b antimony 9 F fluorine 60 Nd neodymium 106 Sg seaborgium r argon 87 Fr francium 10 Ne neon 34 Se selenium s arsenic 64 Gd gadolinium 93 Np neptunium 14 Si silicon t astatine 31 Ga gallium 28 Ni nickel 47 Ag silver a barium 32 Ge germanium 41 Nb niobium 11 Na sodium k berkelium 79 Au gold 7 N nitrogen 38 Sr strontium e beryllium 72 Hf hafnium 102 No nobelium 16 S sulfur i bismuth 108 Hs hassium 76 Os osmium 73 Ta tantalum h bohrium 2 He helium 8 O oxygen 43 Tc technetium
boron 67 Ho holmium 46 Pd palladium 52 Te tellurium r bromine 1 H hydrogen 15 P phosphorus 65 Tb terbium d cadmium 49 In indium 78 Pt platinum 81 Tl thallium a calcium 53 I iodine 94 Pu plutonium 90 Th thorium f californium 77 Ir iridium 84 Po polonium 69 Tm thulium
carbon 26 Fe iron 19 K potassium 50 Sn tin e cerium 36 Kr krypton 59 Pr praseodymium 22 Ti titanium s cesium 57 La lanthanum 61 Pm promethium 74 W tungsten l chlorine 103 Lr lawrencium 91 Pa protactinium 92 U uranium r chromium 82 Pb lead 88 Ra radium 23 V vanadium o cobalt 3 Li lithium 86 Rn radon 54 Xe xenon u copper 71 Lu lutetium 75 Re rhenium 70 Yb ytterbium m curium 12 Mg magnesium 45 Rh rhodium 39 Y yttrium b dubnium 25 Mn manganese 37 Rb rubidium 30 Zn zinc y dysprosium 109 Mt meitnerium 44 Ru ruthenium 40 Zr zirconium s einsteinium
40
UNITS The SI units are given in parentheses Other commonly used units are given in terms of the SI units Symbols conform with the recommendations of the American National Standards Institute (ANSI) and the American Institute of Physics (AIP) Commonly used multiples of SI units
prefix symbol factor tera T 1012 giga G 109
mega M 106 kilo k 103
centi c 10-2 milli m 10-3
micro μ 10-6 nano n 10-9 pico p 10-12
femto f 10-15
length meter (m) angstrom 1 Ǻ = 10-10 m inch 1 in = 254 x 10-2 m foot 1 ft = 03048 m mile 1 mi = 1609 m
mass kilogram (kg) atomic mass unit 1 u = 1661 x 10-27 kgslug 1 slug = 1459 kg
force newton (N = kg bull ms2) dyne 1 dyn = 10-5 N pound 1 lb = 4448 N
pressure Pascal (Pa = kgm bull s2) atmosphere 1 atm = 1013 x 105 Pa poundssquare inch 1 psi = 6895 Pa cm of mercury 1 cm Hg = 1333 Pa bar 1 bar = 1000 x 105 Pa torr 1 torr = 1333 Pa
time second (s) minute 1 min = 60 s hour 1 h =3600 s
frequency hertz (Hz = s-1) radianssecond 1 rads = 12π Hz
energy joule (J = kg bull m2s2) erg 1 erg = 10-7 J electron volt 1 eV = 1602 x 10-19 J calorie 1 cal = 4187 J kilowatt-hour 1 kWh = 36 x 106 J British thermal unit 1 Btu = 1055 J
power watt (W = kg bull m2s3) horsepower 1 hp = 7457 W
charge coulomb (C = A bull s)
electric potential volt (V = kg bull m2 s3 bull A)
current ampere (A)
resistance ohm (Ω = kg bull m2s3 bull A2)
capacitance farad (F = s4 bull A2kg bull m2)
magnetic field tesla (T = kgs2 bull A) gauss 1 G = 10-4 T
magnetic flux weber (Wb = kg m2s2 bull A) maxwell 1 Mx = 10-8 Wb
magnetic inductance henry (H = kg bull m2s2 bullA2)
temperature kelvin (K) degrees Celsius 0degC = 27315 K
angle radian (rad) degree 1 deg = π180 rad revolution 1 rev = 2π rad
41
Some Physical Constantsa Quantity Symbol Valueb Atomic mass unit μ 1660 538 73 (13) x 10-27 kg
931494 013 (37) MeVc2 Avogadros number NA 6022 141 99 (47) x 1023 particlesmol Avogadrorsquos number 6022 x 1023 mol Bohr magneton
2B e
em 9274 008 99 (37) x 10-24 JT
Bohr magneton μB 927 x 10-24 JT Bohr radius
0
2
2e em e k
a 5291 772 083 (19) x 10-11 m
Boltzmanns constant B A
RNk 1380 650 3 (24) x 10-23 JK
Boltzmannrsquos constant kB 1380 x 10-23 JK Compton wavelength
C eh
m c 2426 310 215 (18) x 10-12 m
Deuteron mass md 3343 583 09 (26) x 10-27 kg 2013 553 212 71 (35) u
electron charge e 1602 x 10-19 C Electron mass me 9109 381 88 (72) x 10-31 kg
5485 799 110 (12) x 10-4 u 0510 998 902 (21) MeVc2
electron mass 911 x 10-31 kg Electron volt eV 1602 176 462 (63) x 10-19 J Elementary charge e 1602 176 462 (63) x 10-19 C Gas constant R 8314 472 (15) JKmol Gravitational constant G 6673 (10) x 10-11 Nm2kg2 Neutron mass mn 1674 927 16 (13) x 10-27 kg
1008 664 915 78 (55) u 939565 330 (38) MeVc2
neutron mass 1675 x 10-27 kg Nuclear magneton
2n p
em 5050 783 17 (20) x 10-27 JT
permeability constant μ0 1257 x 10-6 Hm permittivity constant ε0 8854 x 10-12 Fm Plancks constant h
2h
6626 068 76 (52) x 10-34 Js 1054 571 596 (82) x 10-34 Js
Planckrsquos constant h ħ
6626 x 10-34 Js 1055 x 10-34 Js
Proton mass mp 1672621 58 (13) x 10-27 kg 1007 276 466 88 (13) u 938271 998 (38) MeVc2
proton mass 1673 x 10-27 kg Rydberg constant RH 1097 373 156 854 9 (83) x 107 m-1 Speed of light in vacuum c 2997 92458 x 108 ms (exact) speed of light c 300 x 108 ms a
These constants are the values recommended in 1998 by CODATA based on a least-squares adjustment of data from different measurements For a more complete list see P J Mohr and B N Taylor Rev Mod Phys 72351 2000 b The numbers in parentheses for the values above represent the uncertainties of the last two digits
NOTE The ones in red are the ones from Appendix 13 The ones in yellow are the ones that are from the Serway book table but that were already on the Appendix 13 table
42
INDEX
Absolute zero 9
Adiabatic 17
Adiabatic exponent 18
Adiabatic process 17
Alpha decay 46
Amplitude 4
Angle of reflection 25
Angular frequency 3
Angular magnification 30
Angular momentum atomic 43
Angular size 29
Archimedes Principle 1
Atomic spectra 43
Atoms 43
Avogadrorsquos number 10
Beats 8
Bernoullirsquos Equation 2
Beta decay 46
Bifocals 29
Brewsterrsquos angle 34
British Thermal Unit Btu 11
Buoyancy 2
calorie 11
Calorie 11
Camera 28
Carnot cycle 21
Carnot efficiency 22
Celsius scale 9
Ciliary muscle 29
Coefficient of performance 21
Compton effect 39
Constructive interference 7
Continuity equation of 2
Contraction length 35
Convection 14
Curved mirrors 27
Decay constant 46
Decay rate radioactive 46
Decibel scale 5
Degrees of freedom 16
Density 1
Destructive interference 7
Diffraction single slit 33
Diffraction grating 33
Dilation time 35
Disintegration energy 47
Dispersion of light 26
Doppler effect 6
Double slit interference 31
Efficiency 20
Electron interference 39
Emissivity 15
Energy nuclear decay 47
Energy relativistic 36
43
Engineering work 20
Entropy 22
Entropy ideal gas 24
Equipartition of energy 18
Equivalence principle 36
Exclusion principle 44
Expansion thermal 9
Expansion coefficient linear 9
Eye 29
Fahrenheit scale 9
Far point 29
Farsightedness 29
First Law of Thermodynamics 12
Fission 48
Flux volume 2
Free expansion 23
Fringes 31
Fusion 48
Fusion heat of 11
Gamma relativistic 35
Gamma decay 46
General relativity 36
Half-life 46
Heat 11
Heat capacity 11
Heat conduction 13
Heat engine 20
Heat of fusion 11
Heat of vaporization 11
Heat pump 20
Heats of transformation 11
Hydrostatics 1
Ideal Gas Law 10
Images real and virtual 27
Intensity sound 5
Interference 7
Interference two-slit 31
Internal energy 12
Internal energy degrees of freedom 16
Irreversible process 19 23
Isotopes 45
Joule 11
Kelvin scale 9
Kinetic energy relativity 36
Kinetic theory 16
Latent heat 11
Length contraction 35
Linear expansion coefficient 9
Linear polarization 34
Linear superposition 7
Longitudinal wave 3
Loudness 5
Magnification lateral 27
Magnifying glass 30
Malusrsquos law 34
Microscope 30
Momentum relativistic 36
Muscle ciliary 29
44
Musical instruments 8
Musical scale 8
Near point 29
Nearsightedness 29
Nonlinear 7
Nuclear energy 48
Nuclear properties 45
Nuclear stability 45
Octave 8
Optical resolution 33
Orbital quantum number ℓ 43
Orbitals 43
Particle in a Box 41
Particles are waves 39
Pascalrsquos Principle 1
Period 3
Photoelectric effect 38
Photons 38
Pitch 8
Polarization 34
Power sound 5
Presbyopia 29
Pressure 1
Principal quantum number n 43
Principle of equivalence 36
Principle of linear superposition 7
Probability amplitude ψ 41
Processes thermodynamic 13
R-value 14
Radiation thermal 14
Radioactive decay 46
Radioactive decay rate 46
Ray tracing 27
Rayleighrsquos criterion 33
Reading glasses 29
Real image 27
Refraction 25
Refrigerator 20
Relativistic gamma 35
Relativity principles 35
Resolved for light sources 33
Rest energy 36
Reversible process 19 22
Rope wave speed 4
Second Law of Thermodynamics 19
Shock waves 6
Simple magnifier 30
Simultaneity 35
Single slit diffraction 33
Snellrsquos law 25
Sound speed 5
Specific heat 11
Spin Angular momentum 44
Standing waves 7
Stefanrsquos law 14
Strong force 45
Telescope 30
Temperature 9
45
Temperature Scales 9
Thermal conductivity 14
Thermal energy 12
Thermal expansion 9
Thin film interference 31
Thin lenses 28
Time dilation 35
Tone musical 8
Total internal reflection 25
Transverse wave 3
Traveling Waves 3
Two-slit interference 31
Uncertainty principle 40
Vaporization heat of 11
Velocity addition relativity 36
Virtual image 27
Volume flux 2
Wave function 41
Wave Function ψ and Probability 41
Wave speed 3
Wave-particle duality 39
Wavelength 3
Wavenumber 3
Wien Displacement Law 16
Work 11
6
0
s
v vf f
v v
where fprime is the frequency detected by the observer f is the frequency emitted by the source v is
the speed of the waves vo is the speed of the observer and vs is the speed of the source This
formula assumes that the source and receiver are either moving directly toward each other or
directly away from each other To know which signs to use remember that when observer and
source approach each other the observed frequency is higher while if they move away from each
other it is lower Just examine the signs in the formula and make the answer come out right
For electromagnetic waves (light radio waves X-rays) traveling in vacuum Einsteinrsquos theory of
relativity (and careful experiments) show that the Doppler shift is given by
r
r
c vf f
c v
where vr is the relative speed between the source and the observer
For all kinds of waves (sound light etc) if the relative speed of the source and the observer is
small compared to the speed of the waves then there is a simple approximation to the Doppler
effect For example if the relative speed is 1 of the wave speed then the frequency shifts by
1 Remember that this is only an approximation
Shock Waves
When an object moves through a medium at a speed greater than the speed of waves a V-shaped
shock wave is produced The V-shaped wake behind a speeding boat is a good example of this
effect and the cone of sonic-boom behind a supersonic aircraft is another The angle that V-line
makes with the direction of travel of the source is given by
sins
v
v
where v is the wave speed and vs is the source speed
7
Serway Chapter 18
Principle of Linear Superposition
We say that a system obeys the principle of linear superposition if two or more different motions
of the system can simply be added together to find the net motion of the system Light waves
obey this principle as they propagate through the air as can be seen by shining two flashlights so
that their beams cross The beams propagate along without affecting each other (Light sabers are
a spectacular but unfortunately fictional example of systems that do not obey the principle of
superposition) Wave pulses on an ideal rope also obey this principle two different pulses pass
through each other without change Standing waves are an example of this effect being simply
the linear superposition of two traveling waves of the same frequency but moving in opposite
directions Light waves in matter do not always obey this principle For instance two powerful
laser beams could be made to cross in a piece of glass in such a way that their combined heating
effect in the crossing region could melt the glass and scatter the beams in complicated ways This
is an example of a nonlinear effect
Interference
When two or more waves are present in the same medium at the same time their net effect may
often be obtained simply by adding them at each point in the medium according to the principle
of linear superposition (Note this wonrsquot work if the medium is nonlinear) When this addition
makes the total amplitude be greater than the individual amplitudes of the various waves we say
that the interference is constructive When the addition produces cancellation and an amplitude
less than the amplitudes of the separate waves we have destructive interference
Standing Waves
A standing wave is the superposition of two identical traveling waves moving in opposite
directions Nodes are places where the two waves perfectly destructively interfere to produce
zero amplitude at all times Anti-nodes are places where the two waves perfectly constructively
interfere to produce an amplitude maximum The distance between nodes is λ2 Standing waves
on a string fixed at both ends have nodes at each end of the string Standing waves in an air
column enclosed in a tube have displacement anti-nodes at open ends of the tube and
displacement nodes at closed ends The frequency of the standing wave with the lowest possible
frequency is called the fundamental frequency Standing waves on strings or in air columns all
have frequencies which are integer multiples of the fundamental frequency and are called
8
harmonics (The fundamental is called the ldquofirst harmonicrdquo)
Beats
Beats are heard when two waves with slightly different frequencies f1 and f2 are combined The
waves constructively interfere for a number of cycles then destructively interfere for a number
of cycles We hear a periodic ldquowah-wahrdquo frequency equal to the difference of the two wave
frequencies
1 2bf f f
Musical Instruments
Musical instruments produce tones by exciting standing waves on strings (violins piano) and in
tubes (trumpet organ) The fundamental frequency of the standing wave is called the pitch of the
tone The pitch of concert A is 440 Hz by definition Two tones are an octave apart if one pitch
has twice the frequency of the other In written music there are 12 intervals in each octave with
the ratio between successive intervals equal to 2112 = 105946 The ratios for each tone in an
octave starting at A and ending at the next higher A are
A A B C C D D E F F G G A
1 10595 11225 11892 12599 13348 14142 14983 15874 16818 17818 18877 2
A musical tone is actually a superposition of the fundamental frequency and the higher
harmonics The tone quality of a musical instrument is determined by the amplitudes of the
various harmonics that it produces A violin and a trumpet can play the same pitch but they
donrsquot sound at all alike to our ears The difference between them is in the various amplitudes of
their harmonics
9
Serway Chapter 19
Temperature
Formally temperature is what is measured by a thermometer Roughly high temperature is what
we call hot and low temperature is what we call cold On the atomic level temperature refers to
the kinetic energy of the molecules A collection of molecules is called ldquohotrdquo if the molecules
have rapid random motion while a collection of molecules is called ldquocoldrdquo if the random motion
is slow When two bodies are placed in close contact with each other they exchange molecular
kinetic energy until they come to the same temperature This is the microscopic picture of the
Zeroth Law of Thermodynamics
Absolute Zero
Absolute zero is the lowest possible temperature that any object can have This is the temperature
at which all of the energy than can be removed an object has been removed (This removable
energy we call thermal energy) There is still motion at absolute zero Electrons continue to orbit
around atomic nuclei and even atoms continue to move about with a small amount of kinetic
energy but this small energy cannot be removed from the object For example at absolute zero
helium is a liquid whose atoms still move and slide past each other
Temperature Scales
Kelvin Scale Absolute zero is at T = 0 K water freezes at T = 27315 K room temperature is
around T = 295 K and water boils at T = 373 K Note we donrsquot use a deg symbol Kelvin is
prefered SI Unit
Celsius Scale Absolute zero is at T = -273degC water freezes at T = 0degC room temperature is
around T = 22degC and water boils at T = 100degC
Fahrenheit Scale Absolute zero is at T = -459degF water freezes at TF = 32degF room temperature
is around T = 72degF and water boils at TF = 212degF TF =18TC + 32 Notice that temperature
differences are the same for the Kelvin and Celsius scales
Thermal Expansion
When materials are heated they usually expand and when they are cooled they usually contract
(Water near freezing is a spectacular counterexample it works the other way around) The
coefficient of linear expansion is defined by the relation
1
i
L
L T
10
where Li is the initial length of a rod of the material and ΔL is the change in its length due to a
small temperature change ΔT The coefficient of volume expansion is defined similarly
1
i
V
V T
where Vi is the initial volume of a piece of material and where ΔV is the change in its volume due
to a small temperature change ΔT
Avogadrorsquos Number (N or NA)
One mole of any substance corresponds to 6022 times 1023 molecules The atomic mass given on the
periodic table (p A32-33 in the text) is the mass in grams of one mole of atoms For example
the atomic mass of oxygen is 15999 gmol The mass of a single atom is given by the same
number in units of atomic mass units (u) For example the mass of an oxygen atom is 15999 u
(on the average)
Ideal Gas Law (an example of an equation of state)
When the molecules of a gas are sufficiently inert and widely separated that interactions between
them are negligible we say that it is an ideal gas The pressure P volume V and temperature T
(in kelvins) of such a gas are State Variables and are related by the ideal gas law
Bor PV=NkPV nRT T
where n is the number of moles of the gas where R is the gas constant
8314 Jmol KR
where N is the number of molecules and where kB is Boltzmannrsquos constant
231380 10 J KBk
It works well for air at atmosphere pressure and even better for partial vaccuums The relative
ease of measuring pressure and the linear relationship between pressure and temperature (if V
and n are held fixed) makes an ideal gas an ideal thermometer Thermometers can be based on
properties of solids or liquids but the behavior of these materials with temperature is more
complicated
11
Serway Chapter 20
Heat
Heat is energy that flows between a system and its environment because of a tempera-
ture difference between them The units of heat are Joules as expected for an energy
Unfortunately there are several competing units of energy They are related by
1 cal 4186 J 1 Cal 4186 J 1 Btu 1054 J
Heat Capacity
There is often a simple linear relation between the heat that flows in or out of part of a
system and the temperature change that results from this energy transfer When this
linear relation holds it is convenient to define the heat capacity C and the specific
heat c as follows
For the entire object the heat Q it absorbs is given by Q = C(Tf - Ti)
For a particular material the specific heat is defined by c = Cm which is the heat
capacity per unit mass so that
Q = mc(Tf ndash Ti)
Note C has units of energy (J or Cal)(Kelvin kg)
Heats of Transformation or Latent Heat Q = plusmn mL
When a substance changes phase from solid to liquid or from liquid to gas it absorbs
heat without a change in temperature The latent heat or heat of transformation is
usually given per unit mass of the substance For example for water the heat of fusion
(melting) is L = 333 kJkg while the heat of vaporization (boiling) is L = 2260 kJkg
Note that heat for boiling is considerably bigger than melting for water You have to
be careful with signs heat is given off (negative) if you go down in temperature and
condense steam
Work
In general the small amount of work done on a system as a force Fon is exerted on it
through a vector displacement dx is given by
on xdW d F
12
But if the displacement is done very slowly (as we always assume in thermodynamics)
then the force exerted on the system and the force exerted by the system are in
balance so the force exerted by the system is ndash Fon In thermodynamics it is more
convenient to talk about the force exerted by the system so we change the above
formula for the work done on the system to
xdW d F
where F is the force exerted by the system This has confused students for more than a
century now but this is the way your book and many other books do it so you are
stuck You will need to memorize the minus sign in this definition of the work to be
able to use your textbook
There are many chances to get signs wrong in this and the next two chapters (Mosiah
2 )
When an external agent changes the volume of a gas at pressure P by a small amount
dV the (small amount) of work done on the system is given by
dW PdV
Notice that this minus sign is just what we need to make dW be positive if the external
agent compresses the gas for then dV is negative If on the other hand the external
agent gives way allowing the gas to expand against it then dV is positive and we say
that the work done on the gas is negative
The work done on the system (eg by the gas in a cylinder) in a thermodynamic
process is the area under the curve in a PV diagram It is positive for compressions
and negative for expansions If the volume of gas remains constant in a process then
no work is done by the gas
Cyclic processes are important For cyclic processes represented by PV diagrams the
magnitude of the net work during one cycle is simply the area enclosed by the cycle
on the diagram Be careful to keep track of signs when you are calculating that
enclosed area In cyclic process Q = ndashW for a cycle Put a PV diagram in your notes
1
Path A-B B-C C-D D-A A to A net
Q
W
ΔU
ΔS
Internal Energy
The energy stored in a substance is called its internal energy Eint This energy may be
stored as random kinetic energy or as potential energy in each molecule (stretched
chemical bonds electrons in excited states etc) For ideal gases all states with the
same temperature will have the same Eint
First Law of Thermodynamics
The change ΔEint in the internal energy of a system is given by
intE Q W
where Q is the heat absorbed by the system and where W is the work done on the
system Hence if a system absorbs heat (and if Wge0) the internal energy increases
Likewise if the system does work (W on the system is negative) and if Qge0 the
internal energy decreases Potential Pitfall Many times people talk about work done
by the system It is the minus of W on the system Donrsquot get tripped up
Processes
Adiabatic process No heat is exchanged Q = 0 requiring that the system be insulated
from the environment A process may be approximately adiabatic if it happens so
rapidly that heat does not have time to enter or leave the system Work + or ndash is done
and ΔEint = W
Adiabatic free expansion A gas is allowed to expand into a vacuum without pushing
on anythingndashit just rushes into the vacuum In this process Q = W = 0 so the internal
energy and temperature does not change (Note the difference between an adiabatic
process and a free expansion is that NO work is done in the adiabatic free expansion)
Isobaric process The pressure is held fixed ΔP = 0 For example usually the
pressure increases when a gas is heated but if it were allowed to expand during the
2
heating process in just the right way its pressure could remain fixed In isobaric
processes both Q and W are non-zero (Work is eacy = ndashP(Vf ndash Vi)
Isovolumetric process The volume is held fixed ΔV = 0 The work done by the gas is
then zero and so we have ΔEint = Q
Isothermal process The temperature is held fixed ΔT = 0 There is no change in Eint
so Q = ndashW for isothermal processes For an ideal gas PV = nRT so PV = constant in
an isothermal process The work done on the gas is then given by
lnf
i
Vi
Vf
VW PdV nRT
V
Heat Conduction
The quantity P is defined to be the rate at which heat flows through an object and is a
power having units of watts It is analogous to electric current which is the rate at
which charge flows through an object If the flow of heat through a slab of length L
and cross-sectional area A is steady in time then P is given by the equation
h cT TdQkA
dt L
P =
where Th and Tc are the (hot and cold) temperatures of the two ends of the slab The
heat flows of course because of this temperature difference The quantity k is called
the thermal conductivity and is a constant that is characteristic of the material It is
analogous to the electrical conductivity h cT TL is sometimes called the temperature
gradient and is written dTd dTdx
R-Values
It is common to have the heat-conducting properties of materials described by their R-
values especially for insulating materials like fiberglass batting The connection
between k and R is R = Lk where L is the material thickness In this country R-values
always have units of 2ft F hourBtu
Polyurethane foam 6 (1 in thick) Air space 1 (35 in thick) Fiber glass batting 11
(35 in thick) Wood about 1 (1 in thick) Glass 09 (0125 in thick)
The heat flow rate through a slab of area A is given by
3
h cT TA
R
P
in units of Btuhour Note that A must be in square feet and the temperatures must be
in degrees Fahrenheit
Convection
Convection is the transfer of thermal energy by flow of material For instance a home
furnace doesnrsquot heat a house by waiting for the heat from the burner to slowly conduct
throughout the house instead it quickly pumps warm air to all of the rooms
Generally convection is a much faster way to transfer heat than conduction
Radiation
Electromagnetic radiation can also transfer heat When you warm yourself near a
campfire which has burned itself down into a bed of glowing embers you are
receiving radiant heat from the infrared portion of the electromagnetic spectrum The
rate at which an object emits radiant heat is given by Stefanrsquos law
4AeTP
where P is the radiated power in watts σ is a constant
8 2 45696 10 W m K
A is the surface area of the object in m2 and T is the temperature in kelvins The
constant e is called the emissivity and it varies from substance to substance A perfect
absorber (think black velvet) has e = 1 while a perfect reflector (think mirror) has e =
0 Hence black objects radiate very well while shiny ones do not Also an object that
is hotter than its surroundings radiates more energy than it absorbs whereas an object
that is cooler than its surroundings absorbs more energy than it radiates
Terminology
Transfer variables vs state variables
Energy transfer by heat as well as work done depends on the initial final and
intermediate states of the system They are transfer variables But their sum (Q + W =
4
Eint) is a state variable
Figure 205
5
Serway Chapter 21
Kinetic Theory
The ideal gas law works for all atoms and molecules at low pressure It is rather
amazing that it does Kinetic theory explains why The properties of an ideal gas can
be understood by thinking of it as N rapidly moving particles of mass m As these
particles collide with the container walls momentum is imparted to the walls which
we call the force of gas pressure In this picture the pressure is related to the average
of the square of the particle velocity 2v by
22 1( )
3 2
NP mv
V
Using the ideal gas law we obtain the average translational kinetic energy per
molecule
21 3
2 2 Bmv k T
The rms speed is then given by
2rms
3 3Bk T RTv v
m M
where M is the molecular mass in kgmol
Degrees of Freedom
Roughly speaking a degree of freedom is a way in which a molecule can store energy
For instance since there are three different directions in space along which a molecule
can move there are three degrees of freedom for the translational kinetic energy
There are also three different axes of rotation about which a polyatomic molecule can
spin so we say there are three degrees of freedom for the rotational kinetic energy
There are even degrees of freedom associated with the various ways in which a
molecule can vibrate and with the different energy levels in which the electrons of
the molecule can exist
Internal Energy and Degrees of Freedom The internal energy of an ideal gas made
up of molecules with J degrees of freedom is given by
int 2 2 B
J JE nRT Nk T
6
Heat Capacities of an Ideal Gas The heat capacity of a gas is described by means of
molar heat capacities CV and CP These are the heat capacities per mole and the
subscript V on CV means that the volume is being held constant while for CP the
pressure is held constant For example to raise the temperature of n moles of a gas
whose pressure is held constant by 10 K we would have to supply an amount of heat Q
= nCP (10) K
Molar Specific Heat of an Ideal Gas at Constant Volume
VQ nC T
3monatomic
2VC R
5diatomic
2VC R
5polyatomic
2VC
Real gases deviate from these formulas because in addition to the translational and ro-
tational degrees of freedom they also have vibrational and electronic degrees of
freedom These are unimportant at low temperatures due to quantum mechanical
effects but become increasingly important at higher temperatures The rough rule is
No of degrees of freedom
2VC R
Molar Specific Heat of an Ideal Gas at Constant Pressure
PQ nC T
P VC C R
The internal energy of an ideal gas depends only on the temperature
int VE nC T
Adiabatic Processes in an Ideal Gas
7
An adiabatic process is one in which no heat is exchanged between the system and the
environment When an ideal gas expands or contracts adiabatically not only does its
pressure change as expected from the ideal gas law but its temperature changes as
well Under these conditions the final pressure Pf can be computed from the initial
pressure Pi and from the final and initial volumes Vf and Vi by
or constantf f i iP V PV PV
where γ = CPCV The quantity γ is called the adiabatic exponent Note that this
doesnrsquot mean that the ideal gas law no longer holds it does and in fact it can be
combined with the adiabatic law for pressure given above to obtain the adiabatic law
for temperatures
1 constantTV
Compressions in sound waves are adiabatic because they happen too rapidly for any
appreciable amount of heat to flow This is why the adiabatic exponent appears in the
formula for the speed of sound in an ideal gas
RTv
M
Note that v depends only on T and not on P Because it depends only on the
temperature the speed of sound is the same in Provo as at sea level in spite of the
lower pressure here due to the difference in elevation
Equipartition of Energy
Every kind of molecule has a certain number of degrees of freedom which are
independent ways in which it can store energy Each such degree of freedom has
associated with it ndash on average ndash an energy of 12 Bk T per molecule (or 1
2 RT per mole)
(Note since a molecule has so many possible degrees of freedom it would seem that
there should be a lot of 12 sBk T to spread around But because energy is quantized
some of these degrees of freedom are not ldquoactiverdquo until the temperature becomes high
enough that 12 Bk T is as big as the lowest quantum of energy
8
Serway Chapter 22
Second Law of Thermodynamics
There are several equivalent forms of this important law
Kelvin It is not possible to change heat completely into work with no other change
taking place Or in other words there are no perfect heat engines
Clausius It is not possible for heat to flow from one body to another body at a higher
temperature with no other change taking place Or in other words there are no
perfect refrigerators
Entropy In any thermodynamic process that proceeds from one equilibrium state to
another the entropy of the system + environment either remains unchanged or
increases The total entropy never decreases This law is a bit of an oddity among the
laws of physics because it is not absolute Things are forbidden by the second law not
because it is impossible for them to happen but because it is extremely unlikely for
them to happen (See below for more information about entropy)
Reversible and Irreversible Processes
A reversible process is one which occurs so slowly that it is in thermal equilibrium (or
very nearly so) at all times A hallmark of such processes is that a motion picture of
them looks perfectly normal whether run forward or backward Imagine for instance
the slow expansion of a gas at constant temperature in a cylinder whose volume is being
increased by a slowly moving piston Run the movie backwards and what do you see
You see the slow compression of a gas at constant temperature which looks perfectly
normal
An irreversible process is one which occurs in such a way that thermal equilibrium is
not maintained throughout the process The mark of this kind of process is that a motion
picture of it looks very odd when run backward Imagine the sudden expansion of a gas
into a previously evacuated chamber because a hole was punched in the wall between a
pressurized chamber and the evacuated one Run the movie backward and what do you
see You see the gas in the soon-to-be-evacuated chamber gather itself together and
stream through a tiny hole into a chamber in which there is already plenty of gas If you
have ever seen this happen get in touch with the support group for those who have
witnessed the spontaneous combustion of large mammals by calling 1-800-PYROCOW
9
Heat Engines
Because of the vexing difference in sign between work done on and system and work done
by a system we will invent a new work variable Weng Heat engines do work and so the net
W for these engines is negative But in engineering applications hidden minus signs are
regarded as evil so for heat engines we donrsquot talk about W instead we talk about its
magnitude engW W So for heat engines the first law is
int engE Q W
But for heat pumps and refrigerators work is done on the system so we use the usual work
W when we talk about these systems
A heat engine is a machine that absorbs heat converts part of it to work and exhausts the
rest The heat must be absorbed at high temperature and exhausted at low temperature If the
absorbed heat is Qh the exhausted heat is Qc and the work done by the engine is Weng then
eng h cW Q Q
and the efficiency of the engine is defined to be
eng
h
We
Q
A perfect engine would convert the heat hQ completely into work Weng giving an effi-
ciency of e = 1 Energy conservation alone allows a perfect engine but the second law
requires e lt 1
Refrigerators and Heat Pumps
A refrigerator is a machine that absorbs heat at low temperature and exhausts it at high
temperature the ldquobackwardsrdquo heat transfer being driven by the work done on the machine
by some source of power A heat pump is a machine that either works like a refrigerator
keeping a place cold by transferring heat from this cold place to a higher temperature
environment (cooling mode like an air conditioner) or it functions as a heater
transferring heat into a warm place from a cooler one (heating mode like a window unit
that heats a house by extracting thermal energy from the cold outdoors) The coefficient
of performance of a refrigerator or of a heat pump in cooling mode is defined to be
COP cooling modecQ
W
10
For a heat pump in heating mode the coefficient of performance is
COP heating modehQ
W
Note that we donrsquot have to use the engineering work here because in these systems
work is done on the system and W is naturally positive
A perfect refrigerator would take in heat Qc and exhaust the same amount of heat without
doing any work giving an infinite coefficient of performance Energy conservation alone
allows a perfect refrigerator but the second law requires COP lt infin A perfect heat pump
in heating mode would transfer Qh into the house without doing any work and so would
also have an infinite coefficient of performance The second law forbids this too A good
coefficient of performance for a real device would be around 5 or 6
Carnot Cycle
The most efficient of all possible engines is one that uses the Carnot cycle This cycle
employs an ideal gas has no friction and operates very slowly so that the gas can be in
thermal equilibrium at all parts of the cycle This means of course that it canrsquot
possibly be built and even if it could be built it would not run fast enough to be useful
Nevertheless this cycle is very important because it gives an upper bound on the
efficiency of real engines There cannot possibly be an engine that is more efficient
than one based on the Carnot cycle This cycle consists of the following four steps
1 The ideal gas absorbs heat Qh at constant temperature Th while the gas increases its
volume The reason that heat is absorbed is that expansion tends to cool the gas but
thermal contact with the environment at Th keeps the temperature high by heat
conduction into the ideal gas
2 The ideal gas further increases its volume by an adiabatic expansion This expansion
causes the gas to cool so at the end of this part of the cycle the gas is at temperature Tc
3 The gas exhausts heat Qc at constant temperature Tc while the gas decreases in
volume The reason that heat is exhausted is that compression tends to heat the gas
but thermal contact with the environment at Tc keeps the temperature low by heat
conduction out of the ideal gas
4 The gas is adiabatically compressed back to its original volume (the volume it started
with in step 1) This compression heats the gas from Tc up to Th
11
The efficiency of a Carnot engine is given by the very simple formula
1 cC
h
Te
T
where the temperatures must in be Kelvin No real engine can be more efficient than
this
The coefficient of performance of a Carnot refrigerator or heat pump in cooling mode
is given by
COP cooling mode cC
h c
T
T T
and no real refrigerator can have a coefficient of performance greater than this
The coefficient of performance of a Carnot heat pump in heating mode is
COP heating mode hC
h c
T
T T
So why donrsquot we just use these wonderful Carnot engines and have perfect efficiency
To make the reversible steps in the cycle really reversible they would have to occur
infinitely slowly So the price you pay for making a perfect engine is that it takes
forever to get it to do any work
Entropy
The entropy of a system is defined in terms of its molecular makeup and measures
roughly the disorder of the system If the system is packed into a very small volume
then it is quite ordered and the entropy will be low If it occupies a large volume the
entropy is high (To see what this has to do with disorder note that socks in a drawer
occupy a small volume while socks on the bed in the corner by the door and
hanging from the chandelier occupy a large volume) If the system is very cold then
the molecules hardly move and may even reach out to each other and form a crystal
This is a highly ordered state and therefore has low entropy If the system is very hot
with rapidly speeding molecules crashing into the container walls and bouncing off
each other things are disordered and the entropy is high
It is possible to calculate the entropy of a system in terms of its macroscopic thermody-
namic properties ie pressure volume temperature number of moles etc The key to this
calculation is the concept of a reversible process A reversible process is one that is
12
carried out without friction and so slowly that the process can be reversed at any stage by
making an infinitesimal change in the environment of the system The slow expansion of the
gas at Th in the Carnot cycle with tiny amounts of heat being transferred to the environment
is reversible If we turned around and began slowly to compress the gas would just slowly
exhaust heat to the environment in the exact reverse way that it absorbed it during expansion
Most processes however are irreversible For example if a gas-filled box were suddenly
increased in size so that the particles were free to wander into the void created by the sudden
expansion then the gas would eventually fill the new volume uniformly at the same
temperature as before the expansion (The temperature is unchanged in this imaginary
process because the kinetic energy of the molecules would be unaffected by such an
instantaneous expansion of the container walls) This imaginary but highly thought-
stimulating process is called a free expansion and it is impossible to reverse it During the
expansion we didnrsquot push on any of the molecules so reversing this process would mean
making them go back into their original volume without pushing on them they simply will
not cooperate to this extent Another way to see that just pushing them back where they came
from does not reverse the free expansion is to think about what would happen if we just
compressed either adiabatically or isothermally An adiabatic compression back to the
original volume would heat the gas above its original temperature and an isothermal
compression would require that heat be exhausted to the environment But the free expansion
involved neither temperature changes nor heat exchanges so neither of these two processes
is the reverse of the free expansion It is simply impossible to reverse this rapid expansion in
a way that takes us back to the initial state
It is possible to calculate the change of entropy for both reversible and irreversible processes
Letrsquos consider a reversible process first In a reversible process the entropy change is given
by the formula
dQS
T
where dQ is the amount of heat added to the system during a small step of the process
The total energy change during the process may then simply be calculated by integration
f f
f i i i
dQS S S dS
T
13
(Just as in the case of energy we are mainly interested in differences rather than in
absolute magnitudes)
This integration method does not work for irreversible processes For instance in the
rapid free expansion discussed above no heat is added to the system but its disorder
obviously goes up We need to find some other way to calculate the entropy The key
is the fact that the entropy of a system depends only on its current state and not at all
on how it arrived there So to calculate the entropy change in an irreversible process
first find out what the initial and final conditions of the process are Then invent a
reversible process that takes the system from the initial state to the final state Since
the entropy depends only on the state of system and not on the process the entropy
change for the reversible process is the same as that for the irreversible process
Entropy is thus like pressure or temperature if the pressure changes from Pl to to P2
then the pressure difference is P2 - Pl regardless of how the pressure change was
made This seems obvious for pressure but not for entropy only because you arent
comfortable with entropy yet
Entropy of an Ideal Gas
For n moles of an ideal gas the difference in entropy between a state with temperature
T and volume V and some standard state with temperature To and volume Vo is given
by
ln lno o Vo o
T VS T V S T V nC nR
T V
Entropy in a Phase Change
Calculating entropy change in a phase change like melting or boiling is easy It is
QT Where Q is in the latent heat for example the flows in to cause the melting and
T is the temperature of the phase change There is NO integral to do
14
Serway Chapter 35
Angle of Reflection
If plane waves are incident on a reflecting surface with the propagation direction of
the waves making angle θ1 with the normal direction to the surface then the reflected
angle θ1prime relative to the surface normal is simply
1 1
ie the incident angles and reflected angles are the same
Refraction
If plane waves traveling through medium 1 are incident on a plane interface between
medium 1 and medium 2 then the angle of incidence of the incoming wave θ1 and
the angle of refraction of the transmitted wave θ2 are related by Snellrsquos law
2 2
1 1
sin
sin
v
v
where v1 and v2 are the wave speeds in medium 1 and medium 2 The angle of
incidence and the angle of refraction are both measured between the wave propagation
direction and the normal to the interface In terms of indices of refraction in the case
of light waves Snellrsquos law takes the more familiar form
1 1 2 2sin sinn n
where ni = cvi
Total Internal Reflection
If a wave is incident from a medium of low wave speed into a medium of high wave
speed the law of refraction requires that the angle of refraction be greater than the
angle of incidence If the angle of refraction is required to be greater than 90deg then no
refracted wave can exist and total internal reflection occurs The critical incident
angle θc beyond which total internal reflection occurs is given by
2
1
sin c
n
n
15
Dispersion of Light
In addition to the speed of light varying from material to material it also varies with
wavelength within each material This means that the index of refraction is generally a
function of wavelength
c
nv
Since the wave speed is not constant such a medium is dispersive meaning in this
context that refraction actually disperses white light into its various colors because
Snellrsquos law gives a different angle for each wavelength In most materials the
variation with wavelength is quite small but this small effect is responsible for some
of the most spectacular color effects we ever see including rainbows a flashing
crystal chandelier and the colored fire of a diamond solitaire by candlelight
16
Serway Chapter 36
Real and Virtual Images
When light rays are focused at a certain plane producing an image if a sheet of white
paper is placed there we call the image a real image The images produced by film
projectors and overhead projectors are examples of real images
When light rays appear to come from a certain location but no image is produced
when a screen is placed there we say that there is a virtual image at that location For
instance when you look in a mirror it appears that someone is behind the mirror but
a screen placed back there in the dark would show nothing Your image in the mirror
is a virtual image
Ray Tracing
There are lots of rules about how to find the images in optical systems but the best
way to keep things straight is to learn how to draw the principal rays for curved
mirrors and lenses The rays for convex and concave mirrors are shown in Fig 3615
The rays for converging and diverging thin lenses are shown in Fig 3627 You should
memorize the principal rays and know how to use them to locate images
Curved Mirrors
The focal length of a curved mirror with radius of curvature R is given by
2
Rf
The relation between object distance p image distance q and focal length f is
1 1 1
p q f
If the mirror is a diverging mirror f should be negative and if q should turn out to be
negative the image is virtual
Lateral Magnification
The lateral magnification in an optical system is defined by the ratio of the image size
to the object size
17
Image height
Object height
qM
p
for curved mirrors
As usual there are sign conventions here too but this formula just gives the
magnitude It is better to keep keep track of upright images versus inverted images by
means of ray diagrams rather than by memorizing sign conventions
Thin Lenses
The focal length of a thin lens is related to the radii of curvature of the two faces R1
and R2 of the lens by
1 2
1 1 11n
f R R
Note that this formula differs from Eq (3611) in the text by not having a minus sign
between the two R-terms We like this form better because for a simple converging
lens like a magnifying glass we just use positive values of R for both surfaces If one
of the faces is concave producing divergence use a negative value for R And if a
surface is flat use R = infin If the face is flat the radius is infinite
The relation between the image object and focal distances for a thin lens is the same
as that for a curved mirror
1 1 1
q p f
Use a negative focal length if the lens is diverging
The lateral magnification for a thin lens is the same as for a curved mirror
Image height
Object height
qM
p
Camera
The lens system in a camera projects a real image of an object onto the film (or CCD
array in a digital camera) The position of the image is adjusted to be on the film by
moving the lens into or out of the camera
18
Eye
The eye is like a camera in that a real image is formed on the retina Unlike a camera
the image position is adjusted by changing the focal length of the lens This is done by
the ciliary muscle which squeezes the lens changing its shape
Near Point The near point is the closest distance from the eye for which the lens can
focus an image on the retina It is usually 18-25 cm for young persons
Far Point The far point is the greatest distance from the eye for which the lens can
focus an image on the retina For a person with normal vision the far point is at
infinity
Nearsightedness A person is nearsighted if their far point is at some finite distance
less than infinity This condition can be corrected with a lens that takes an object at
infinity and produces a virtual image at the personrsquos far point
Farsightedness and Presbyopia A person is farsighted if their near point is too far
away for comfortable near work like reading or knitting This can be corrected by a
lens which takes an object at a normal near point distance of 18-25 cm and produces a
virtual image at the personrsquos natural near point Presbyopia involves a similar
problem which nearly all people experience as they age The ciliary muscle becomes
too weak and the lens becomes too stiff to allow the eye to provide for both near and
far vision The solution for this problem is either reading glasses or bifocal lenses
Reading classes are just weak magnifying glasses mounted on eyeglass frames
Bifocal lenses are split into upper and lower halves The lower half is a lens which
gives the proper correction for near work and the upper half is a different lens for
proper focusing at infinity
Angular Size
When an object is brought closer to the eye it appears to be larger because the image
on the retina is larger The size of this image is directly proportional to the objectrsquos
angular size which is the angle subtended by the object measured from the center of
the lens of the eye In optical instruments which are to be used with the eye the
angular size of the final image is whatrsquos important because it determines how large
the image will appear to the viewer
19
Simple Magnifier
A simple magnifier is a single converging lens or magnifying glass It takes an object
closer to the eye than a normal near point and produces a virtual image at or beyond
this near point The angular magnification is defined to be the ratio of the angular size
when viewed through the lens to the angular size of the object when viewed at the
normal near point (without aid of the lens)
Microscope
This instrument has two lenses (1) The objective is near the object being viewed and
produces a greatly magnified real image (2) The eyepiece is a simple magnifier which
the viewer uses to closely examine the image from (1)
Telescope
This instrument also has two lenses (1) The objective at the front of the telescope
takes light from a distant object and produces a real inverted image (which is rather
small) near its focal point (2) This small real image is then examined by the eyepiece
functioning as a simple magnifier to produce a virtual image with a larger angular
size
20
Serway Chapter 37
Two-Slit Interference
If light is incident on two closely spaced narrow slits a pattern of light and dark
stripes is produced beyond the slits The bright stripes or fringes are caused by
constructive interference of the two waves coming from the slits Constructive
interference occurs whenever two waves arrive at a location in phase with each other
This occurs when the distance x1 from slit 1 to a point P on the screen and the
distance x2 from slit 2 to point P differ by in integral number of wavelengths
1 2 where 0 1 2x x m m
where λ is the wavelength of the light When x1 and x2 are much larger than the slit
spacing d this condition reduces to
sind m
where θ is the angle between the direction of the incident light and the direction of the
light arriving at the screen
Thin Films
When light is partially reflected and partially transmitted by a thin film of transparent
material it is possible to have interference between the wave reflected from the front
of the film and light reflected from the back of the film (The colored reflections from
the thin film of oil on the water in a rain-soaked parking lot are an example of this
effect) It is difficult to write down formulas that will work in all cases so we will
just review the important principles here
1 If the two reflected waves are in phase with each other the film has enhanced
reflection (constructive interference) but if the two reflected waves are out of phase
with each other reflection is diminished (destructive interference) Phase shifts occur
due to reflection and due to the extra path length through the film of the wave
reflected from the back of the film
2 The phase change due to reflection is determined by the difference in index of
refraction between the two media involved in the reflection If the wave is incident
21
from a medium with a low index of refraction into a medium with a high index of
refraction a phase change of 180deg occurs and the reflected wave is inverted If
incident from high to low no phase shift occurs and the reflected wave is non-
inverted
3 The extra path length through the film of the wave reflected from the back of the
film is equal to 2t where t is the thickness of the film (The incident light is assumed
to be normal to the surface of the film) The number of wavelengths contained in the
extra path length is equal to 2tλn where λn = λn is the wavelength of the light in the
film
4 Rules for reflection from thin films
If one ray is inverted and the other is not then we have
12 constructive
2 nt m
2 destructivent m
If either both rays are inverted or both are non-inverted we have
2 constructivent m
12 destructive
2 nt m
22
Serway Chapter 38
Diffraction Grating
A diffraction grating is simply a fancy version of two-slit interference with the two
slits replaced by thousands of slits Just as in the two-slit case the bright fringes
occur at angles given by
sin md
but in this case d the distance between neighboring slits is made to be very small
The effect of having many slits instead of two is to make each bright fringe highly
localized with wide dark regions between neighboring maxima
Single Slit Diffraction
When light passes through an opening in an opaque screen an interference pattern is
produced beyond the opening To understand why we may replace the single opening
by many small coherent sources of light These many sources interfere with each
other producing a pattern known as a diffraction pattern (Note that many authors do
not distinguish between interference and diffraction treating them as interchangeable
terms) If the opening is a slit of width a then the diffraction pattern far from the slit
will have a bright central maximum with a succession of minima and weaker maxima
on either side The angle between the incident direction and the minima is given by
sin ma
where m = plusmn1 plusmn2
If the opening is circular with diameter D the angle between the incident direction
and the first minimum is given by
sin 122D
Optical Resolution and Rayleighrsquos Criterion
Two point sources can just be resolved (distinguished from each other) if the peak of
the diffraction image of the first source overlies the first minimum of the diffraction
image of the second source For circular holes of the kind usually encountered in
23
optical devices this condition is approximately satisfied when the angular separation
between the two sources as viewed from the optical instrument is greater than or equal
to the critical angle
min 122D
where λ is the wavelength of the light and where D is the diameter of the aperture in
the instrument
Polarization
We say that an electromagnetic wave is polarized if its electric field vector doesnrsquot
change direction in a random fashion The simplest kind of polarization is linear
polarization in which the electric field vector oscillates back and forth along the
same axis in space Polarized light can be produced from normal unpolarized light by
selective absorption (as in Polaroid sunglasses) by passing light through crystals that
have different indices of refraction for different polarizations (double refraction) by
scattering (the blue sky is polarized) and by reflection (glare) Polarization by
reflection occurs when light reflects from a shiny insulating (non-metallic) surface
The amount of polarization is greatest for reflection at Brewsterrsquos angle
2
1
tan P
n
n
where θP is the incidence angle of light from medium 1 onto medium 2 and where n1
and n2 are the indices of refraction for the media (In our everyday experience n1 = 1
since the light comes in through the air and n2 is the index of refraction of the shiny
insulating material producing the glare eg water glass plastic paint etc)
Malusrsquos Law
The intensity of transmitted polarized light through a perfect polarizer is related to the
incident intensity of polarized light by Malusrsquos law
2cosoI I
where Io is the intensity of the incident light and where θ is the angle between the
electric field vector in the incident wave and the transmission axis the polarizer
24
Serway Chapter 39
Principles of Relativity
All of the weirdness of relativity flows from two simple principles
(1) The laws of physics must be the same in all inertial (non-accelerating) reference
frames
(2) The speed of light in vacuum has the same value in all inertial reference frames
Note that (1) seems reasonable but (2) is very odd It says that if two space ships are
approaching each other at nearly the speed of light and a laser pulse is shot from ship
1 toward ship 2 then when the people on ship 2 measure the speed of the pulse as it
goes by it is moving at 3 times 108 ms the same as if the ships were stationary
Simultaneity
If observer 1 sees two events in her own frame as simultaneous at two different
locations a moving observer 2 will see these two events happening at different times
Relativistic gamma
The factor γ (gamma) appears regularly in the formulas of relativity
2
2
1
1 vc
where v is the relative speed between two inertial frames
Time Dilation
If observer 1 sees two events at the same location in space separated by time Δtp in his
own frame then observer 2 moving at speed v relative to observer 1 will see these
two events separated by a longer time Δt
pt t
Moving clocks run slow
Length Contraction
If observer 1 measures the length of an object along the x-direction at some instant of
time in her frame to be Lp then observer 2 moving in the x-direction at speed v will
measure the length L of the object to be shorter
pLL
25
Moving meter sticks are short
Velocity Addition
If a particle is moving at velocity vac relative to frame c and if frame c is moving at
speed vcb with respect to frame b (with both velocities directed along the same line)
what would be the velocity of the particle in frame b In classical physics the answer is
ab ac cbv v v
but if any of these velocities are an appreciable fraction of the speed of light the
answer changes
21ac cb
abac cb
v Vv
v v c
In using this equation make sure that vac has the same sign in the numerator and in the
denominator and that vbc also has the same sign in both places Note that this equation
is different (and in our opinion easier to understand) than the equations in the text
Momentum and Energy
The momentum and total energy of a particle moving at velocity v (its speed is v v )
are given by
22
2 2
22
1 1 vvc c
m mcm E mc
vp v
Rest Energy
When a particle is at rest (v = 0 and γ = 1) then E = mc2 is called the rest energy It
includes the potential energy of the particle so that if we change the potential energy
of a particle we change its rest mass
Kinetic Energy The kinetic energy is the difference between the total energy and the
rest energy
2 2K mc mc
For v ltlt c this reduces to K = mv22
General Relativity
Einsteinrsquos principle of equivalence in the theory of general relativity is In the vicinity
of a point a gravitational field is equivalent to an accelerated frame of reference in
26
the absence of gravitational effects For example there is no difference whatever in
the physics experienced by an astronaut on earth and one who is in interstellar space
far away from any stars or planets but who has just turned on her spaceshiprsquos engines
and is accelerating at 98 ms2
27
Serway Chapter 40
Photons
We now know that even though light behaves like a wave it is better described as
consisting of small packets of energy called photons The energy of a photon is related
to its frequency by
E hf
where h is Planckrsquos constant
346626 10 J sh
Photons also carry momentum given by
hf hp
c
Since the amount of energy in each photon is fixed the difference between dim light
and intense light (of the same wavelength) is that dim light consists of fewer photons
than intense light
Photoelectric Effect
In the photoelectric effect light shining onto a metal surface gives the electrons in the
metal enough energy to escape and be detected It requires a certain amount of energy
(called the work function typically = 3 ndash 5 eV) just to kick the electron out of the
metal so the light must deliver at least this much energy to an electron to produce the
effect It is observed that red light no matter how intense never produces electrons
But ultraviolet light even if quite dim will eject electrons from the metal Since red
light consists of 2 eV photons and ultraviolet light has photons with energies around
4-6 eV the photon idea explains the behavior of the photoelectric effect The
maximum energy that an ejected electron can have is
maxK hf
where f is the photon frequency The minimum frequency that light can have and cause
electrons to be ejected is called the cutoff energy and is found by setting Kmax = 0
cf h
28
Compton Effect
When high-frequency light interacts with free electrons the scattered light does not
have the same wavelength as the incident light contrary to what classical
electromagnetism would predict But the photon picture in which photons have
momentum and energy allows us to treat scattering as a collision between the photon
and the electron In this collision the electron and the scattered photon both have a
different momentum and energy than they did before And since p = hλ for a photon
if the momentum is different the wavelength will also be different
after before 1 cose
h
m c
where the angle θ is the angle between the incoming photon direction and the
direction of the scattered photon
Particles are Waves
Since photons behave like particles sometimes it is not surprising that elementary
particles can behave like waves sometimes The wavelength of a particle with
momentum p is given by
h
p
Wave-particle Duality
Both photons and elementary particles have a dual nature sometimes they behave like
particles and sometimes they behave like waves It is hard for us to comprehend the
nature of such an object by making mental pictures but experiments definitively show
that this is the case Since we have no direct experience with photons and elementary
particles (because their energies are so much smaller than the energies of the everyday
objects we encounter) it is perhaps not surprising that we have a hard time forming a
mental picture of how they behave
Electron Interference
Since an electron is both a particle and a wave just like a photon it should be able to
produce an interference pattern This is observed If an electron beam is shot at two
closely-spaced slits and if the electrons that pass through the slits are detected
downstream it is found that there are some locations where electrons are never detected
29
and others where lots of electrons are detected The pattern is exactly the same as the
one observed for light waves provided that we use the electron wavelength λ = hp in
place of the wavelength of light This pattern is observed even though each electron is
detected as a single dot on the screen Only after many such dots are collected does the
pattern emerge And if we try to understand how this effect could possibly work by
looking closely at each slit to see which one the electron came through the pattern
disappears the act of measurement destroys the interference This means that each
single electron somehow comes through both slits (as a wave would)
Uncertainty Principle
In classical physics we always imagine that the positions and momenta of moving
particles have definite values It might be hard to measure them but surely at each
instant of time a particle should be precisely located at some point in space and have a
similarly precise momentum This turns out not to be true Instead both position and
momentum are required to be uncertain with their uncertainties Δx and Δp satisfying
the Heisenberg uncertainty relation
2x p
So if the particle were known to be precisely at some particular location (so that Δx =
0) we couldnrsquot know anything about its momentum (Δp = infin) and if its momentum
were exactly known we couldnrsquot know its position
There is a similar relation involving the particlersquos energy E and the time interval Δt
over which this energy is measured
2E t
This means that energy is not actually conserved if we are considering very small time
intervals and this brief non-conservation of energy has been observed
30
Serway Chapter 41
Wave Function ψ and Probability
Quantum mechanics does not predict exactly what an electron or a photon will do Instead
it specifies the wave function or probability amplitude ψ of an electron or a photon
This wave function is a complex-valued function of space and time whose squared
magnitude is the probability density P for finding a particle at a particular place in
space at a certain time
2P
where is the complex conjugate of ψ
For example the wave function of an electron with perfectly specified momentum p
would have a wavelength given by
h
p
and its probability amplitude would be proportional to
2i x ipxe e
The corresponding probability density would then be
21ipx ip ipxP e e e
which means that the electron is equally probable to be anywhere along the x axis
This is in accord with the uncertainty principle since we specified the momentum
precisely we canrsquot have any idea about the position of the electron
As another example you have probably seen ldquofuzzy ballrdquo drawings of electron
orbitals in a chemistry book These fuzzy balls are meant to indicate the distribution
of the probability density 2
P in the orbital
Particle in a Box
A simple example in which we can calculate the wave function is the case of a particle of
mass m constrained to be inside a 1-dimensional box between x = 0 and x = L In this
case the general wave function is a linear superposition of wave functions ψn of the form
sin niE tn
n xx t A e
L
31
where A is a positive constant where n = 1 2 3 and where the energy associated
with each of the quantum states ψn is given by
22
28n
hE n
mL
This wave function is zero at x = 0 and x = L which means that the particle will never
be found at the walls of the box The wave function has maximum values in the
interior and at these places the particle is most likely to be found and it also has
places where it is zero and at these places the particle will also never be found as
expressed by the formula
2sinn x
PL
The particle in the box is interfering with itself producing a probability interference
pattern across the box just like the interference patterns we studied with light and
sound
32
Serway Chapter 42
Atoms
Many people picture an atom as a miniature solar system where electrons orbit around
a massive nucleus at the center This picture is misleading because in atoms the wave
nature of electrons dominates The electrons form 3-dimensional standing waves
(called orbitals) centered at the nucleus And if you ask what is it thatrsquos waving the
answer is ψ the probability amplitude (see Chapter 41)
Standing waves on a string can occur only for certain frequencies (the fundamental
and higher harmonics) Similarly atomic orbitals occur only for certain energies For
the hydrogen atom the energies of the orbitals have the a particularly simple form
2
1136 eVnE
n
where n = 1 2 3 is called the principal quantum number For other atoms the
determination of the orbital energies requires numerical calculation by computers
Atomic Spectra
If an electron is somehow given extra extra energy (we say that it is excited) so that it
occupies a higher orbital it will eventually ldquofallrdquo back down to a lower orbital Each
time an electron falls to a lower orbital it loses the difference in energy between the
two orbitals in the form of a photon Since the orbital energies are discrete so are the
energy differences and so are the wavelengths of the emitted light The entire set of
these discrete wavelengths is called the atomic spectrum and it is unique to each type
of atom For hydrogen the spectrum can be simply written as
H 2 2
1 1 1
f i
Rn n
where RH = 10973732 times 107 m-1 For other atoms the spectrum cannot be expressed as
a formula
Orbital Angular Momentum Electrons in atoms also have quantized values of
angular momentum The orbital quantum number ℓ specifies the value of this
quantized angular momentum through the formula
1L
33
If we want to know the value of the angular momentum along some direction in space
say the z direction the answer is not L but rather
zL m
where mℓ is another quantum number which runs from
1 1m
This quantum number is important when an atom sits in a magnetic field
Spin Angular Momentum It has been found experimentally that electrons and other
charged particles also carry internal angular momentum which we call spin
Electrons have an intrinsic spin angular momentum s along a specified axis that is
extremely quantized it can only take on 2 possible values
1
2zs s
We thus say that electrons have spin s = 12 or that electrons are rdquospin one-half parti-
cles Since s is an angular momentum it obeys the same rule as that for orbital angular
momentum namely that if its value along some axis is s then its total magnitude is
given by
31
2S s s
I know this seems weird but quantum mechanics is weird The only excuse for this
bizarre way of looking at the world is that it predicts what happens in experiments
Exclusion Principle The answer to the question of how many electrons (or any other
spin one-half particle) can be in one particular quantum state was discovered by
Wolfgang Pauli and is called the exclusion principle
ldquoNo two electrons can ever be in the same quantum state therefore no two electrons
in the same atom can have the same set of quantum numbersrdquo
This is the reason that we have atoms with different properties instead of every atom
simple having all of its electrons in the ground state All of the variety we see around
us in the world is the result of chemical differences and these differences would not
exist unless electrons obeyed this important principle The entire structure of the
periodic table (see pages 1377-1379 in Serway) is an expression of this principle
34
Serway Chapter 44
Nuclear Properties
The nucleus is composed of protons with charge +e and mass mp = 1007226 u and of
neutrons with zero charge and mass mn = 1008665 u where u is one atomic mass unit
-271 u=1660540 10 kg
The atomic number Z counts the number of protons in a nucleus while the neutron
number N counts the number of neutrons The mass number A is the sum of the two
A N Z
Protons and nuetrons have about the same mass and some times called baryons (heavy
ones) or nucleons (partcles in the nucleus) A is sometimes called the baryon
number The chemical elements are distinguished by Z ie hydrogen has Z = 1 iron
has Z = 26 uranium has Z = 92 etc But for a given Z there might be several nuclei
with different numbers of neutrons These nuclear siblings are called isotopes
These numbers are used to label nuclei according to the pattern
5626 ie FeA
Z X
denotes the isotope of iron with N = 56 ndash 26 = 30 neutrons
The nucleus is roughly spherical with a radius given approximately by
1 3 150 0where 12 10 mr r A r
Nuclear Stability
Because the positively charged protons electrically repel each other with an enormous
force at distances as small as 10-15 m there must be some really strong force that that
overcomes electrical repulsion to hold protons and neutrons together This force is
called with some lack of imagination the strong force It is a very short-range force
(it only acts over a distance of about 2 x 10-15 m) and attracts protons to protons
neutrons to neutrons and neutrons to protons But in spite of this strong nuclear force
the coulomb repulsion of the protons is still present so anything that might keep the
protons from being right next to each other would help keep the nucleus from
35
exploding This role is played by the neutrons and for nuclei with Z le 20 the stable
nuclei roughly have N = Z
For Z gt 20 the coulomb repulsion force is more powerful and more neutrons are
needed to dilute it up to about N = 15Z around Z = 80 For Z gt 83 no amount of
neutrons can help and these nuclei are unstable (radioactive) There is an ldquoisland of
(relative) stabilityrdquo around Z=90-92 (thorium and uranium respectively) These
elements have one or two isotopes with half-lifes of billions of years so there are
substantial amounts of such elements on Earth This fact makes it possible to have
practical fission devices
Radioactive Decay
There are three types of radioactive decay
Alpha decay the nucleus kicks out a helium nucleus (N = 2 Z = 2 A = 4) So if the
original nucleus is called X and the new nucleus is called Y then the decay would look
like this
4 42 2X Y+ HeA A
Z Z
Beta decay the nucleus either kicks out an electron (endash) or its positively-charged
antimatter twin the positron (e+) plus either an electron neutrino v or an electron
anti-neutrino v
1X Y+e electron decayA AZ Z v
1X Y+e positron decayA AZ Z v
A neutrino is a particle with no charge hardly any mass (much less than the electron
mass) and interacts so weakly with matter that most neutrinos upon encountering the
planet earth just pass right through it as if it werenrsquot there
Gamma decay the nucleons in the nucleus X are in an excited energy state X
(perhaps as a result of having undergone alpha or beta decay) and they drop down to a
lower energy state shedding the energy as a high frequency photon
X XA AZ Z
36
This process is exactly analogous to the way that the electrons in atoms emit photons
Decay Rate and Half Life
There is no way to predict exactly when an unstable or excited nucleus will decay but
there is an average rate at which this decay occurs called the decay constant λ The
meaning of this constant is that if there are a large number N of nuclei in a sample
then the number of decays per second that will be observed (called the decay rate R)
is R = λN In mathematical language
dMR N
dt
This simple differential equation has for its solution
0tN t N e
where N0 is the number of nuclei in the sample at time t = 0
The half-life is the time it takes for half of the nuclei in the sample to decay and is
related to the decay constant by
1 2
ln 2 0693T
Disintegration Energy
When a nucleus decays it is making a transition to an overall state of lower energy
which means according to Einsteinrsquos famous formula E = mc2 that the sum of the
masses after the decay must be less than the mass before with the lost mass appearing
as kinetic energy among the decay products For example in alpha decay this kinetic
energy called the disintegration energy Q is given by
2X YQ M M M c
37
Serway Chapter 45
Nuclear energy
There are two ways to extract energy from the nucleus fission and fusion For nuclei
with Z greater than 26 breaking the nucleus apart into pieces leads to a lower
2mc energy than the original nucleus so energy can be extracted by fission For
nuclei with Z less than 26 a lower energy is achieved by combining nuclei so energy
can be extracted by fusion
Fission
Since neutrons have no charge they are not repelled from nuclei as protons are for
this reason their behavior is the key to understanding how fission works
Fast neutrons mostly bounce off other nuclei with each collision slowing the neutron
down This elastic energy loss is most effective if the other nuclei have low mass (like
hydrogen) and these materials are called moderators because of their ability to slow
down fast neutrons
The reason that slowing neutrons is important is that slow neutrons are much more
likely to be absorbed by a nucleus which then leads to nuclear reactions of various
kinds For a few very large nuclei like uranium-235 and some plutonium isotopes
absorption of a slow neutron causes the nucleus to split into two large fragments plus
2 or 3 fast neutrons (energy is released in the process too) Thus one slow neutron
can produce energy plus 2 or 3 more neutrons which if moderated from fast to slow
can split 2 or 3 more nuclei leading to a runaway chain reaction If the reaction is
allowed to proceed unchecked it produces a large explosion If a neutron absorbing
material is added to the mix (like the cadmium in reactor control rods) it is possible to
keep the reaction under control and to extract the released energy as heat to drive
steam turbines and produce electricity
Fusion
Fusion involves mashing two nuclei together and since they are both charged and repel
each other this reaction is much harder to make go The nuclei must have enough energy to
overcome the coulomb repulsion which is why this reaction requires a high temperature
(hundreds of millions of degrees K) like that in the sun (which is burning hydrogen to
helium via fusion) or like that in the center of the fission explosion that is used to detonate
38
a hydrogen bomb
This reaction is of interest for power production in spite of this difficult temperature
requirement because of the abundance of fusion fuel on the planet There are about
012 g of deuterium ( 21H ) in every gallon of water on earth and it only costs about 4
cents to extract it The fusion energy available from this minuscule amount of
deuterium would run a 1000-Megawatt power plant for 10 seconds Water is so
abundant on earth that if fusion were to work we would have an essentially
inexhaustible source of energy
So why donrsquot we have fusion power plants Well the fuel is cheap but the match is
incredibly expensive The only way we know to control this difficult high-temperature
reaction is with large and expensive pieces of equipment involving either large
magnetic fields and complex high-power electromagnetic antennas or with gigantic
(football-fieldsized) laser facilities involving more than a hundred of the highest-
energy lasers ever built Power plants based on these current methods for controlling
fusion are unattractive to the fiscally-minded people who run the electric power
industry Hopefully better designs will be discovered as experiments continue
39
c actinium 68 Er erbium 101 Md mendelevium 104 Rf rutherfordium l aluminum 63 Eu europium 80 Hg mercury 62 Sm samarium m americium 100 Fm fermium 42 Mo molybdenum 21 Sc scandium b antimony 9 F fluorine 60 Nd neodymium 106 Sg seaborgium r argon 87 Fr francium 10 Ne neon 34 Se selenium s arsenic 64 Gd gadolinium 93 Np neptunium 14 Si silicon t astatine 31 Ga gallium 28 Ni nickel 47 Ag silver a barium 32 Ge germanium 41 Nb niobium 11 Na sodium k berkelium 79 Au gold 7 N nitrogen 38 Sr strontium e beryllium 72 Hf hafnium 102 No nobelium 16 S sulfur i bismuth 108 Hs hassium 76 Os osmium 73 Ta tantalum h bohrium 2 He helium 8 O oxygen 43 Tc technetium
boron 67 Ho holmium 46 Pd palladium 52 Te tellurium r bromine 1 H hydrogen 15 P phosphorus 65 Tb terbium d cadmium 49 In indium 78 Pt platinum 81 Tl thallium a calcium 53 I iodine 94 Pu plutonium 90 Th thorium f californium 77 Ir iridium 84 Po polonium 69 Tm thulium
carbon 26 Fe iron 19 K potassium 50 Sn tin e cerium 36 Kr krypton 59 Pr praseodymium 22 Ti titanium s cesium 57 La lanthanum 61 Pm promethium 74 W tungsten l chlorine 103 Lr lawrencium 91 Pa protactinium 92 U uranium r chromium 82 Pb lead 88 Ra radium 23 V vanadium o cobalt 3 Li lithium 86 Rn radon 54 Xe xenon u copper 71 Lu lutetium 75 Re rhenium 70 Yb ytterbium m curium 12 Mg magnesium 45 Rh rhodium 39 Y yttrium b dubnium 25 Mn manganese 37 Rb rubidium 30 Zn zinc y dysprosium 109 Mt meitnerium 44 Ru ruthenium 40 Zr zirconium s einsteinium
40
UNITS The SI units are given in parentheses Other commonly used units are given in terms of the SI units Symbols conform with the recommendations of the American National Standards Institute (ANSI) and the American Institute of Physics (AIP) Commonly used multiples of SI units
prefix symbol factor tera T 1012 giga G 109
mega M 106 kilo k 103
centi c 10-2 milli m 10-3
micro μ 10-6 nano n 10-9 pico p 10-12
femto f 10-15
length meter (m) angstrom 1 Ǻ = 10-10 m inch 1 in = 254 x 10-2 m foot 1 ft = 03048 m mile 1 mi = 1609 m
mass kilogram (kg) atomic mass unit 1 u = 1661 x 10-27 kgslug 1 slug = 1459 kg
force newton (N = kg bull ms2) dyne 1 dyn = 10-5 N pound 1 lb = 4448 N
pressure Pascal (Pa = kgm bull s2) atmosphere 1 atm = 1013 x 105 Pa poundssquare inch 1 psi = 6895 Pa cm of mercury 1 cm Hg = 1333 Pa bar 1 bar = 1000 x 105 Pa torr 1 torr = 1333 Pa
time second (s) minute 1 min = 60 s hour 1 h =3600 s
frequency hertz (Hz = s-1) radianssecond 1 rads = 12π Hz
energy joule (J = kg bull m2s2) erg 1 erg = 10-7 J electron volt 1 eV = 1602 x 10-19 J calorie 1 cal = 4187 J kilowatt-hour 1 kWh = 36 x 106 J British thermal unit 1 Btu = 1055 J
power watt (W = kg bull m2s3) horsepower 1 hp = 7457 W
charge coulomb (C = A bull s)
electric potential volt (V = kg bull m2 s3 bull A)
current ampere (A)
resistance ohm (Ω = kg bull m2s3 bull A2)
capacitance farad (F = s4 bull A2kg bull m2)
magnetic field tesla (T = kgs2 bull A) gauss 1 G = 10-4 T
magnetic flux weber (Wb = kg m2s2 bull A) maxwell 1 Mx = 10-8 Wb
magnetic inductance henry (H = kg bull m2s2 bullA2)
temperature kelvin (K) degrees Celsius 0degC = 27315 K
angle radian (rad) degree 1 deg = π180 rad revolution 1 rev = 2π rad
41
Some Physical Constantsa Quantity Symbol Valueb Atomic mass unit μ 1660 538 73 (13) x 10-27 kg
931494 013 (37) MeVc2 Avogadros number NA 6022 141 99 (47) x 1023 particlesmol Avogadrorsquos number 6022 x 1023 mol Bohr magneton
2B e
em 9274 008 99 (37) x 10-24 JT
Bohr magneton μB 927 x 10-24 JT Bohr radius
0
2
2e em e k
a 5291 772 083 (19) x 10-11 m
Boltzmanns constant B A
RNk 1380 650 3 (24) x 10-23 JK
Boltzmannrsquos constant kB 1380 x 10-23 JK Compton wavelength
C eh
m c 2426 310 215 (18) x 10-12 m
Deuteron mass md 3343 583 09 (26) x 10-27 kg 2013 553 212 71 (35) u
electron charge e 1602 x 10-19 C Electron mass me 9109 381 88 (72) x 10-31 kg
5485 799 110 (12) x 10-4 u 0510 998 902 (21) MeVc2
electron mass 911 x 10-31 kg Electron volt eV 1602 176 462 (63) x 10-19 J Elementary charge e 1602 176 462 (63) x 10-19 C Gas constant R 8314 472 (15) JKmol Gravitational constant G 6673 (10) x 10-11 Nm2kg2 Neutron mass mn 1674 927 16 (13) x 10-27 kg
1008 664 915 78 (55) u 939565 330 (38) MeVc2
neutron mass 1675 x 10-27 kg Nuclear magneton
2n p
em 5050 783 17 (20) x 10-27 JT
permeability constant μ0 1257 x 10-6 Hm permittivity constant ε0 8854 x 10-12 Fm Plancks constant h
2h
6626 068 76 (52) x 10-34 Js 1054 571 596 (82) x 10-34 Js
Planckrsquos constant h ħ
6626 x 10-34 Js 1055 x 10-34 Js
Proton mass mp 1672621 58 (13) x 10-27 kg 1007 276 466 88 (13) u 938271 998 (38) MeVc2
proton mass 1673 x 10-27 kg Rydberg constant RH 1097 373 156 854 9 (83) x 107 m-1 Speed of light in vacuum c 2997 92458 x 108 ms (exact) speed of light c 300 x 108 ms a
These constants are the values recommended in 1998 by CODATA based on a least-squares adjustment of data from different measurements For a more complete list see P J Mohr and B N Taylor Rev Mod Phys 72351 2000 b The numbers in parentheses for the values above represent the uncertainties of the last two digits
NOTE The ones in red are the ones from Appendix 13 The ones in yellow are the ones that are from the Serway book table but that were already on the Appendix 13 table
42
INDEX
Absolute zero 9
Adiabatic 17
Adiabatic exponent 18
Adiabatic process 17
Alpha decay 46
Amplitude 4
Angle of reflection 25
Angular frequency 3
Angular magnification 30
Angular momentum atomic 43
Angular size 29
Archimedes Principle 1
Atomic spectra 43
Atoms 43
Avogadrorsquos number 10
Beats 8
Bernoullirsquos Equation 2
Beta decay 46
Bifocals 29
Brewsterrsquos angle 34
British Thermal Unit Btu 11
Buoyancy 2
calorie 11
Calorie 11
Camera 28
Carnot cycle 21
Carnot efficiency 22
Celsius scale 9
Ciliary muscle 29
Coefficient of performance 21
Compton effect 39
Constructive interference 7
Continuity equation of 2
Contraction length 35
Convection 14
Curved mirrors 27
Decay constant 46
Decay rate radioactive 46
Decibel scale 5
Degrees of freedom 16
Density 1
Destructive interference 7
Diffraction single slit 33
Diffraction grating 33
Dilation time 35
Disintegration energy 47
Dispersion of light 26
Doppler effect 6
Double slit interference 31
Efficiency 20
Electron interference 39
Emissivity 15
Energy nuclear decay 47
Energy relativistic 36
43
Engineering work 20
Entropy 22
Entropy ideal gas 24
Equipartition of energy 18
Equivalence principle 36
Exclusion principle 44
Expansion thermal 9
Expansion coefficient linear 9
Eye 29
Fahrenheit scale 9
Far point 29
Farsightedness 29
First Law of Thermodynamics 12
Fission 48
Flux volume 2
Free expansion 23
Fringes 31
Fusion 48
Fusion heat of 11
Gamma relativistic 35
Gamma decay 46
General relativity 36
Half-life 46
Heat 11
Heat capacity 11
Heat conduction 13
Heat engine 20
Heat of fusion 11
Heat of vaporization 11
Heat pump 20
Heats of transformation 11
Hydrostatics 1
Ideal Gas Law 10
Images real and virtual 27
Intensity sound 5
Interference 7
Interference two-slit 31
Internal energy 12
Internal energy degrees of freedom 16
Irreversible process 19 23
Isotopes 45
Joule 11
Kelvin scale 9
Kinetic energy relativity 36
Kinetic theory 16
Latent heat 11
Length contraction 35
Linear expansion coefficient 9
Linear polarization 34
Linear superposition 7
Longitudinal wave 3
Loudness 5
Magnification lateral 27
Magnifying glass 30
Malusrsquos law 34
Microscope 30
Momentum relativistic 36
Muscle ciliary 29
44
Musical instruments 8
Musical scale 8
Near point 29
Nearsightedness 29
Nonlinear 7
Nuclear energy 48
Nuclear properties 45
Nuclear stability 45
Octave 8
Optical resolution 33
Orbital quantum number ℓ 43
Orbitals 43
Particle in a Box 41
Particles are waves 39
Pascalrsquos Principle 1
Period 3
Photoelectric effect 38
Photons 38
Pitch 8
Polarization 34
Power sound 5
Presbyopia 29
Pressure 1
Principal quantum number n 43
Principle of equivalence 36
Principle of linear superposition 7
Probability amplitude ψ 41
Processes thermodynamic 13
R-value 14
Radiation thermal 14
Radioactive decay 46
Radioactive decay rate 46
Ray tracing 27
Rayleighrsquos criterion 33
Reading glasses 29
Real image 27
Refraction 25
Refrigerator 20
Relativistic gamma 35
Relativity principles 35
Resolved for light sources 33
Rest energy 36
Reversible process 19 22
Rope wave speed 4
Second Law of Thermodynamics 19
Shock waves 6
Simple magnifier 30
Simultaneity 35
Single slit diffraction 33
Snellrsquos law 25
Sound speed 5
Specific heat 11
Spin Angular momentum 44
Standing waves 7
Stefanrsquos law 14
Strong force 45
Telescope 30
Temperature 9
45
Temperature Scales 9
Thermal conductivity 14
Thermal energy 12
Thermal expansion 9
Thin film interference 31
Thin lenses 28
Time dilation 35
Tone musical 8
Total internal reflection 25
Transverse wave 3
Traveling Waves 3
Two-slit interference 31
Uncertainty principle 40
Vaporization heat of 11
Velocity addition relativity 36
Virtual image 27
Volume flux 2
Wave function 41
Wave Function ψ and Probability 41
Wave speed 3
Wave-particle duality 39
Wavelength 3
Wavenumber 3
Wien Displacement Law 16
Work 11
7
Serway Chapter 18
Principle of Linear Superposition
We say that a system obeys the principle of linear superposition if two or more different motions
of the system can simply be added together to find the net motion of the system Light waves
obey this principle as they propagate through the air as can be seen by shining two flashlights so
that their beams cross The beams propagate along without affecting each other (Light sabers are
a spectacular but unfortunately fictional example of systems that do not obey the principle of
superposition) Wave pulses on an ideal rope also obey this principle two different pulses pass
through each other without change Standing waves are an example of this effect being simply
the linear superposition of two traveling waves of the same frequency but moving in opposite
directions Light waves in matter do not always obey this principle For instance two powerful
laser beams could be made to cross in a piece of glass in such a way that their combined heating
effect in the crossing region could melt the glass and scatter the beams in complicated ways This
is an example of a nonlinear effect
Interference
When two or more waves are present in the same medium at the same time their net effect may
often be obtained simply by adding them at each point in the medium according to the principle
of linear superposition (Note this wonrsquot work if the medium is nonlinear) When this addition
makes the total amplitude be greater than the individual amplitudes of the various waves we say
that the interference is constructive When the addition produces cancellation and an amplitude
less than the amplitudes of the separate waves we have destructive interference
Standing Waves
A standing wave is the superposition of two identical traveling waves moving in opposite
directions Nodes are places where the two waves perfectly destructively interfere to produce
zero amplitude at all times Anti-nodes are places where the two waves perfectly constructively
interfere to produce an amplitude maximum The distance between nodes is λ2 Standing waves
on a string fixed at both ends have nodes at each end of the string Standing waves in an air
column enclosed in a tube have displacement anti-nodes at open ends of the tube and
displacement nodes at closed ends The frequency of the standing wave with the lowest possible
frequency is called the fundamental frequency Standing waves on strings or in air columns all
have frequencies which are integer multiples of the fundamental frequency and are called
8
harmonics (The fundamental is called the ldquofirst harmonicrdquo)
Beats
Beats are heard when two waves with slightly different frequencies f1 and f2 are combined The
waves constructively interfere for a number of cycles then destructively interfere for a number
of cycles We hear a periodic ldquowah-wahrdquo frequency equal to the difference of the two wave
frequencies
1 2bf f f
Musical Instruments
Musical instruments produce tones by exciting standing waves on strings (violins piano) and in
tubes (trumpet organ) The fundamental frequency of the standing wave is called the pitch of the
tone The pitch of concert A is 440 Hz by definition Two tones are an octave apart if one pitch
has twice the frequency of the other In written music there are 12 intervals in each octave with
the ratio between successive intervals equal to 2112 = 105946 The ratios for each tone in an
octave starting at A and ending at the next higher A are
A A B C C D D E F F G G A
1 10595 11225 11892 12599 13348 14142 14983 15874 16818 17818 18877 2
A musical tone is actually a superposition of the fundamental frequency and the higher
harmonics The tone quality of a musical instrument is determined by the amplitudes of the
various harmonics that it produces A violin and a trumpet can play the same pitch but they
donrsquot sound at all alike to our ears The difference between them is in the various amplitudes of
their harmonics
9
Serway Chapter 19
Temperature
Formally temperature is what is measured by a thermometer Roughly high temperature is what
we call hot and low temperature is what we call cold On the atomic level temperature refers to
the kinetic energy of the molecules A collection of molecules is called ldquohotrdquo if the molecules
have rapid random motion while a collection of molecules is called ldquocoldrdquo if the random motion
is slow When two bodies are placed in close contact with each other they exchange molecular
kinetic energy until they come to the same temperature This is the microscopic picture of the
Zeroth Law of Thermodynamics
Absolute Zero
Absolute zero is the lowest possible temperature that any object can have This is the temperature
at which all of the energy than can be removed an object has been removed (This removable
energy we call thermal energy) There is still motion at absolute zero Electrons continue to orbit
around atomic nuclei and even atoms continue to move about with a small amount of kinetic
energy but this small energy cannot be removed from the object For example at absolute zero
helium is a liquid whose atoms still move and slide past each other
Temperature Scales
Kelvin Scale Absolute zero is at T = 0 K water freezes at T = 27315 K room temperature is
around T = 295 K and water boils at T = 373 K Note we donrsquot use a deg symbol Kelvin is
prefered SI Unit
Celsius Scale Absolute zero is at T = -273degC water freezes at T = 0degC room temperature is
around T = 22degC and water boils at T = 100degC
Fahrenheit Scale Absolute zero is at T = -459degF water freezes at TF = 32degF room temperature
is around T = 72degF and water boils at TF = 212degF TF =18TC + 32 Notice that temperature
differences are the same for the Kelvin and Celsius scales
Thermal Expansion
When materials are heated they usually expand and when they are cooled they usually contract
(Water near freezing is a spectacular counterexample it works the other way around) The
coefficient of linear expansion is defined by the relation
1
i
L
L T
10
where Li is the initial length of a rod of the material and ΔL is the change in its length due to a
small temperature change ΔT The coefficient of volume expansion is defined similarly
1
i
V
V T
where Vi is the initial volume of a piece of material and where ΔV is the change in its volume due
to a small temperature change ΔT
Avogadrorsquos Number (N or NA)
One mole of any substance corresponds to 6022 times 1023 molecules The atomic mass given on the
periodic table (p A32-33 in the text) is the mass in grams of one mole of atoms For example
the atomic mass of oxygen is 15999 gmol The mass of a single atom is given by the same
number in units of atomic mass units (u) For example the mass of an oxygen atom is 15999 u
(on the average)
Ideal Gas Law (an example of an equation of state)
When the molecules of a gas are sufficiently inert and widely separated that interactions between
them are negligible we say that it is an ideal gas The pressure P volume V and temperature T
(in kelvins) of such a gas are State Variables and are related by the ideal gas law
Bor PV=NkPV nRT T
where n is the number of moles of the gas where R is the gas constant
8314 Jmol KR
where N is the number of molecules and where kB is Boltzmannrsquos constant
231380 10 J KBk
It works well for air at atmosphere pressure and even better for partial vaccuums The relative
ease of measuring pressure and the linear relationship between pressure and temperature (if V
and n are held fixed) makes an ideal gas an ideal thermometer Thermometers can be based on
properties of solids or liquids but the behavior of these materials with temperature is more
complicated
11
Serway Chapter 20
Heat
Heat is energy that flows between a system and its environment because of a tempera-
ture difference between them The units of heat are Joules as expected for an energy
Unfortunately there are several competing units of energy They are related by
1 cal 4186 J 1 Cal 4186 J 1 Btu 1054 J
Heat Capacity
There is often a simple linear relation between the heat that flows in or out of part of a
system and the temperature change that results from this energy transfer When this
linear relation holds it is convenient to define the heat capacity C and the specific
heat c as follows
For the entire object the heat Q it absorbs is given by Q = C(Tf - Ti)
For a particular material the specific heat is defined by c = Cm which is the heat
capacity per unit mass so that
Q = mc(Tf ndash Ti)
Note C has units of energy (J or Cal)(Kelvin kg)
Heats of Transformation or Latent Heat Q = plusmn mL
When a substance changes phase from solid to liquid or from liquid to gas it absorbs
heat without a change in temperature The latent heat or heat of transformation is
usually given per unit mass of the substance For example for water the heat of fusion
(melting) is L = 333 kJkg while the heat of vaporization (boiling) is L = 2260 kJkg
Note that heat for boiling is considerably bigger than melting for water You have to
be careful with signs heat is given off (negative) if you go down in temperature and
condense steam
Work
In general the small amount of work done on a system as a force Fon is exerted on it
through a vector displacement dx is given by
on xdW d F
12
But if the displacement is done very slowly (as we always assume in thermodynamics)
then the force exerted on the system and the force exerted by the system are in
balance so the force exerted by the system is ndash Fon In thermodynamics it is more
convenient to talk about the force exerted by the system so we change the above
formula for the work done on the system to
xdW d F
where F is the force exerted by the system This has confused students for more than a
century now but this is the way your book and many other books do it so you are
stuck You will need to memorize the minus sign in this definition of the work to be
able to use your textbook
There are many chances to get signs wrong in this and the next two chapters (Mosiah
2 )
When an external agent changes the volume of a gas at pressure P by a small amount
dV the (small amount) of work done on the system is given by
dW PdV
Notice that this minus sign is just what we need to make dW be positive if the external
agent compresses the gas for then dV is negative If on the other hand the external
agent gives way allowing the gas to expand against it then dV is positive and we say
that the work done on the gas is negative
The work done on the system (eg by the gas in a cylinder) in a thermodynamic
process is the area under the curve in a PV diagram It is positive for compressions
and negative for expansions If the volume of gas remains constant in a process then
no work is done by the gas
Cyclic processes are important For cyclic processes represented by PV diagrams the
magnitude of the net work during one cycle is simply the area enclosed by the cycle
on the diagram Be careful to keep track of signs when you are calculating that
enclosed area In cyclic process Q = ndashW for a cycle Put a PV diagram in your notes
1
Path A-B B-C C-D D-A A to A net
Q
W
ΔU
ΔS
Internal Energy
The energy stored in a substance is called its internal energy Eint This energy may be
stored as random kinetic energy or as potential energy in each molecule (stretched
chemical bonds electrons in excited states etc) For ideal gases all states with the
same temperature will have the same Eint
First Law of Thermodynamics
The change ΔEint in the internal energy of a system is given by
intE Q W
where Q is the heat absorbed by the system and where W is the work done on the
system Hence if a system absorbs heat (and if Wge0) the internal energy increases
Likewise if the system does work (W on the system is negative) and if Qge0 the
internal energy decreases Potential Pitfall Many times people talk about work done
by the system It is the minus of W on the system Donrsquot get tripped up
Processes
Adiabatic process No heat is exchanged Q = 0 requiring that the system be insulated
from the environment A process may be approximately adiabatic if it happens so
rapidly that heat does not have time to enter or leave the system Work + or ndash is done
and ΔEint = W
Adiabatic free expansion A gas is allowed to expand into a vacuum without pushing
on anythingndashit just rushes into the vacuum In this process Q = W = 0 so the internal
energy and temperature does not change (Note the difference between an adiabatic
process and a free expansion is that NO work is done in the adiabatic free expansion)
Isobaric process The pressure is held fixed ΔP = 0 For example usually the
pressure increases when a gas is heated but if it were allowed to expand during the
2
heating process in just the right way its pressure could remain fixed In isobaric
processes both Q and W are non-zero (Work is eacy = ndashP(Vf ndash Vi)
Isovolumetric process The volume is held fixed ΔV = 0 The work done by the gas is
then zero and so we have ΔEint = Q
Isothermal process The temperature is held fixed ΔT = 0 There is no change in Eint
so Q = ndashW for isothermal processes For an ideal gas PV = nRT so PV = constant in
an isothermal process The work done on the gas is then given by
lnf
i
Vi
Vf
VW PdV nRT
V
Heat Conduction
The quantity P is defined to be the rate at which heat flows through an object and is a
power having units of watts It is analogous to electric current which is the rate at
which charge flows through an object If the flow of heat through a slab of length L
and cross-sectional area A is steady in time then P is given by the equation
h cT TdQkA
dt L
P =
where Th and Tc are the (hot and cold) temperatures of the two ends of the slab The
heat flows of course because of this temperature difference The quantity k is called
the thermal conductivity and is a constant that is characteristic of the material It is
analogous to the electrical conductivity h cT TL is sometimes called the temperature
gradient and is written dTd dTdx
R-Values
It is common to have the heat-conducting properties of materials described by their R-
values especially for insulating materials like fiberglass batting The connection
between k and R is R = Lk where L is the material thickness In this country R-values
always have units of 2ft F hourBtu
Polyurethane foam 6 (1 in thick) Air space 1 (35 in thick) Fiber glass batting 11
(35 in thick) Wood about 1 (1 in thick) Glass 09 (0125 in thick)
The heat flow rate through a slab of area A is given by
3
h cT TA
R
P
in units of Btuhour Note that A must be in square feet and the temperatures must be
in degrees Fahrenheit
Convection
Convection is the transfer of thermal energy by flow of material For instance a home
furnace doesnrsquot heat a house by waiting for the heat from the burner to slowly conduct
throughout the house instead it quickly pumps warm air to all of the rooms
Generally convection is a much faster way to transfer heat than conduction
Radiation
Electromagnetic radiation can also transfer heat When you warm yourself near a
campfire which has burned itself down into a bed of glowing embers you are
receiving radiant heat from the infrared portion of the electromagnetic spectrum The
rate at which an object emits radiant heat is given by Stefanrsquos law
4AeTP
where P is the radiated power in watts σ is a constant
8 2 45696 10 W m K
A is the surface area of the object in m2 and T is the temperature in kelvins The
constant e is called the emissivity and it varies from substance to substance A perfect
absorber (think black velvet) has e = 1 while a perfect reflector (think mirror) has e =
0 Hence black objects radiate very well while shiny ones do not Also an object that
is hotter than its surroundings radiates more energy than it absorbs whereas an object
that is cooler than its surroundings absorbs more energy than it radiates
Terminology
Transfer variables vs state variables
Energy transfer by heat as well as work done depends on the initial final and
intermediate states of the system They are transfer variables But their sum (Q + W =
4
Eint) is a state variable
Figure 205
5
Serway Chapter 21
Kinetic Theory
The ideal gas law works for all atoms and molecules at low pressure It is rather
amazing that it does Kinetic theory explains why The properties of an ideal gas can
be understood by thinking of it as N rapidly moving particles of mass m As these
particles collide with the container walls momentum is imparted to the walls which
we call the force of gas pressure In this picture the pressure is related to the average
of the square of the particle velocity 2v by
22 1( )
3 2
NP mv
V
Using the ideal gas law we obtain the average translational kinetic energy per
molecule
21 3
2 2 Bmv k T
The rms speed is then given by
2rms
3 3Bk T RTv v
m M
where M is the molecular mass in kgmol
Degrees of Freedom
Roughly speaking a degree of freedom is a way in which a molecule can store energy
For instance since there are three different directions in space along which a molecule
can move there are three degrees of freedom for the translational kinetic energy
There are also three different axes of rotation about which a polyatomic molecule can
spin so we say there are three degrees of freedom for the rotational kinetic energy
There are even degrees of freedom associated with the various ways in which a
molecule can vibrate and with the different energy levels in which the electrons of
the molecule can exist
Internal Energy and Degrees of Freedom The internal energy of an ideal gas made
up of molecules with J degrees of freedom is given by
int 2 2 B
J JE nRT Nk T
6
Heat Capacities of an Ideal Gas The heat capacity of a gas is described by means of
molar heat capacities CV and CP These are the heat capacities per mole and the
subscript V on CV means that the volume is being held constant while for CP the
pressure is held constant For example to raise the temperature of n moles of a gas
whose pressure is held constant by 10 K we would have to supply an amount of heat Q
= nCP (10) K
Molar Specific Heat of an Ideal Gas at Constant Volume
VQ nC T
3monatomic
2VC R
5diatomic
2VC R
5polyatomic
2VC
Real gases deviate from these formulas because in addition to the translational and ro-
tational degrees of freedom they also have vibrational and electronic degrees of
freedom These are unimportant at low temperatures due to quantum mechanical
effects but become increasingly important at higher temperatures The rough rule is
No of degrees of freedom
2VC R
Molar Specific Heat of an Ideal Gas at Constant Pressure
PQ nC T
P VC C R
The internal energy of an ideal gas depends only on the temperature
int VE nC T
Adiabatic Processes in an Ideal Gas
7
An adiabatic process is one in which no heat is exchanged between the system and the
environment When an ideal gas expands or contracts adiabatically not only does its
pressure change as expected from the ideal gas law but its temperature changes as
well Under these conditions the final pressure Pf can be computed from the initial
pressure Pi and from the final and initial volumes Vf and Vi by
or constantf f i iP V PV PV
where γ = CPCV The quantity γ is called the adiabatic exponent Note that this
doesnrsquot mean that the ideal gas law no longer holds it does and in fact it can be
combined with the adiabatic law for pressure given above to obtain the adiabatic law
for temperatures
1 constantTV
Compressions in sound waves are adiabatic because they happen too rapidly for any
appreciable amount of heat to flow This is why the adiabatic exponent appears in the
formula for the speed of sound in an ideal gas
RTv
M
Note that v depends only on T and not on P Because it depends only on the
temperature the speed of sound is the same in Provo as at sea level in spite of the
lower pressure here due to the difference in elevation
Equipartition of Energy
Every kind of molecule has a certain number of degrees of freedom which are
independent ways in which it can store energy Each such degree of freedom has
associated with it ndash on average ndash an energy of 12 Bk T per molecule (or 1
2 RT per mole)
(Note since a molecule has so many possible degrees of freedom it would seem that
there should be a lot of 12 sBk T to spread around But because energy is quantized
some of these degrees of freedom are not ldquoactiverdquo until the temperature becomes high
enough that 12 Bk T is as big as the lowest quantum of energy
8
Serway Chapter 22
Second Law of Thermodynamics
There are several equivalent forms of this important law
Kelvin It is not possible to change heat completely into work with no other change
taking place Or in other words there are no perfect heat engines
Clausius It is not possible for heat to flow from one body to another body at a higher
temperature with no other change taking place Or in other words there are no
perfect refrigerators
Entropy In any thermodynamic process that proceeds from one equilibrium state to
another the entropy of the system + environment either remains unchanged or
increases The total entropy never decreases This law is a bit of an oddity among the
laws of physics because it is not absolute Things are forbidden by the second law not
because it is impossible for them to happen but because it is extremely unlikely for
them to happen (See below for more information about entropy)
Reversible and Irreversible Processes
A reversible process is one which occurs so slowly that it is in thermal equilibrium (or
very nearly so) at all times A hallmark of such processes is that a motion picture of
them looks perfectly normal whether run forward or backward Imagine for instance
the slow expansion of a gas at constant temperature in a cylinder whose volume is being
increased by a slowly moving piston Run the movie backwards and what do you see
You see the slow compression of a gas at constant temperature which looks perfectly
normal
An irreversible process is one which occurs in such a way that thermal equilibrium is
not maintained throughout the process The mark of this kind of process is that a motion
picture of it looks very odd when run backward Imagine the sudden expansion of a gas
into a previously evacuated chamber because a hole was punched in the wall between a
pressurized chamber and the evacuated one Run the movie backward and what do you
see You see the gas in the soon-to-be-evacuated chamber gather itself together and
stream through a tiny hole into a chamber in which there is already plenty of gas If you
have ever seen this happen get in touch with the support group for those who have
witnessed the spontaneous combustion of large mammals by calling 1-800-PYROCOW
9
Heat Engines
Because of the vexing difference in sign between work done on and system and work done
by a system we will invent a new work variable Weng Heat engines do work and so the net
W for these engines is negative But in engineering applications hidden minus signs are
regarded as evil so for heat engines we donrsquot talk about W instead we talk about its
magnitude engW W So for heat engines the first law is
int engE Q W
But for heat pumps and refrigerators work is done on the system so we use the usual work
W when we talk about these systems
A heat engine is a machine that absorbs heat converts part of it to work and exhausts the
rest The heat must be absorbed at high temperature and exhausted at low temperature If the
absorbed heat is Qh the exhausted heat is Qc and the work done by the engine is Weng then
eng h cW Q Q
and the efficiency of the engine is defined to be
eng
h
We
Q
A perfect engine would convert the heat hQ completely into work Weng giving an effi-
ciency of e = 1 Energy conservation alone allows a perfect engine but the second law
requires e lt 1
Refrigerators and Heat Pumps
A refrigerator is a machine that absorbs heat at low temperature and exhausts it at high
temperature the ldquobackwardsrdquo heat transfer being driven by the work done on the machine
by some source of power A heat pump is a machine that either works like a refrigerator
keeping a place cold by transferring heat from this cold place to a higher temperature
environment (cooling mode like an air conditioner) or it functions as a heater
transferring heat into a warm place from a cooler one (heating mode like a window unit
that heats a house by extracting thermal energy from the cold outdoors) The coefficient
of performance of a refrigerator or of a heat pump in cooling mode is defined to be
COP cooling modecQ
W
10
For a heat pump in heating mode the coefficient of performance is
COP heating modehQ
W
Note that we donrsquot have to use the engineering work here because in these systems
work is done on the system and W is naturally positive
A perfect refrigerator would take in heat Qc and exhaust the same amount of heat without
doing any work giving an infinite coefficient of performance Energy conservation alone
allows a perfect refrigerator but the second law requires COP lt infin A perfect heat pump
in heating mode would transfer Qh into the house without doing any work and so would
also have an infinite coefficient of performance The second law forbids this too A good
coefficient of performance for a real device would be around 5 or 6
Carnot Cycle
The most efficient of all possible engines is one that uses the Carnot cycle This cycle
employs an ideal gas has no friction and operates very slowly so that the gas can be in
thermal equilibrium at all parts of the cycle This means of course that it canrsquot
possibly be built and even if it could be built it would not run fast enough to be useful
Nevertheless this cycle is very important because it gives an upper bound on the
efficiency of real engines There cannot possibly be an engine that is more efficient
than one based on the Carnot cycle This cycle consists of the following four steps
1 The ideal gas absorbs heat Qh at constant temperature Th while the gas increases its
volume The reason that heat is absorbed is that expansion tends to cool the gas but
thermal contact with the environment at Th keeps the temperature high by heat
conduction into the ideal gas
2 The ideal gas further increases its volume by an adiabatic expansion This expansion
causes the gas to cool so at the end of this part of the cycle the gas is at temperature Tc
3 The gas exhausts heat Qc at constant temperature Tc while the gas decreases in
volume The reason that heat is exhausted is that compression tends to heat the gas
but thermal contact with the environment at Tc keeps the temperature low by heat
conduction out of the ideal gas
4 The gas is adiabatically compressed back to its original volume (the volume it started
with in step 1) This compression heats the gas from Tc up to Th
11
The efficiency of a Carnot engine is given by the very simple formula
1 cC
h
Te
T
where the temperatures must in be Kelvin No real engine can be more efficient than
this
The coefficient of performance of a Carnot refrigerator or heat pump in cooling mode
is given by
COP cooling mode cC
h c
T
T T
and no real refrigerator can have a coefficient of performance greater than this
The coefficient of performance of a Carnot heat pump in heating mode is
COP heating mode hC
h c
T
T T
So why donrsquot we just use these wonderful Carnot engines and have perfect efficiency
To make the reversible steps in the cycle really reversible they would have to occur
infinitely slowly So the price you pay for making a perfect engine is that it takes
forever to get it to do any work
Entropy
The entropy of a system is defined in terms of its molecular makeup and measures
roughly the disorder of the system If the system is packed into a very small volume
then it is quite ordered and the entropy will be low If it occupies a large volume the
entropy is high (To see what this has to do with disorder note that socks in a drawer
occupy a small volume while socks on the bed in the corner by the door and
hanging from the chandelier occupy a large volume) If the system is very cold then
the molecules hardly move and may even reach out to each other and form a crystal
This is a highly ordered state and therefore has low entropy If the system is very hot
with rapidly speeding molecules crashing into the container walls and bouncing off
each other things are disordered and the entropy is high
It is possible to calculate the entropy of a system in terms of its macroscopic thermody-
namic properties ie pressure volume temperature number of moles etc The key to this
calculation is the concept of a reversible process A reversible process is one that is
12
carried out without friction and so slowly that the process can be reversed at any stage by
making an infinitesimal change in the environment of the system The slow expansion of the
gas at Th in the Carnot cycle with tiny amounts of heat being transferred to the environment
is reversible If we turned around and began slowly to compress the gas would just slowly
exhaust heat to the environment in the exact reverse way that it absorbed it during expansion
Most processes however are irreversible For example if a gas-filled box were suddenly
increased in size so that the particles were free to wander into the void created by the sudden
expansion then the gas would eventually fill the new volume uniformly at the same
temperature as before the expansion (The temperature is unchanged in this imaginary
process because the kinetic energy of the molecules would be unaffected by such an
instantaneous expansion of the container walls) This imaginary but highly thought-
stimulating process is called a free expansion and it is impossible to reverse it During the
expansion we didnrsquot push on any of the molecules so reversing this process would mean
making them go back into their original volume without pushing on them they simply will
not cooperate to this extent Another way to see that just pushing them back where they came
from does not reverse the free expansion is to think about what would happen if we just
compressed either adiabatically or isothermally An adiabatic compression back to the
original volume would heat the gas above its original temperature and an isothermal
compression would require that heat be exhausted to the environment But the free expansion
involved neither temperature changes nor heat exchanges so neither of these two processes
is the reverse of the free expansion It is simply impossible to reverse this rapid expansion in
a way that takes us back to the initial state
It is possible to calculate the change of entropy for both reversible and irreversible processes
Letrsquos consider a reversible process first In a reversible process the entropy change is given
by the formula
dQS
T
where dQ is the amount of heat added to the system during a small step of the process
The total energy change during the process may then simply be calculated by integration
f f
f i i i
dQS S S dS
T
13
(Just as in the case of energy we are mainly interested in differences rather than in
absolute magnitudes)
This integration method does not work for irreversible processes For instance in the
rapid free expansion discussed above no heat is added to the system but its disorder
obviously goes up We need to find some other way to calculate the entropy The key
is the fact that the entropy of a system depends only on its current state and not at all
on how it arrived there So to calculate the entropy change in an irreversible process
first find out what the initial and final conditions of the process are Then invent a
reversible process that takes the system from the initial state to the final state Since
the entropy depends only on the state of system and not on the process the entropy
change for the reversible process is the same as that for the irreversible process
Entropy is thus like pressure or temperature if the pressure changes from Pl to to P2
then the pressure difference is P2 - Pl regardless of how the pressure change was
made This seems obvious for pressure but not for entropy only because you arent
comfortable with entropy yet
Entropy of an Ideal Gas
For n moles of an ideal gas the difference in entropy between a state with temperature
T and volume V and some standard state with temperature To and volume Vo is given
by
ln lno o Vo o
T VS T V S T V nC nR
T V
Entropy in a Phase Change
Calculating entropy change in a phase change like melting or boiling is easy It is
QT Where Q is in the latent heat for example the flows in to cause the melting and
T is the temperature of the phase change There is NO integral to do
14
Serway Chapter 35
Angle of Reflection
If plane waves are incident on a reflecting surface with the propagation direction of
the waves making angle θ1 with the normal direction to the surface then the reflected
angle θ1prime relative to the surface normal is simply
1 1
ie the incident angles and reflected angles are the same
Refraction
If plane waves traveling through medium 1 are incident on a plane interface between
medium 1 and medium 2 then the angle of incidence of the incoming wave θ1 and
the angle of refraction of the transmitted wave θ2 are related by Snellrsquos law
2 2
1 1
sin
sin
v
v
where v1 and v2 are the wave speeds in medium 1 and medium 2 The angle of
incidence and the angle of refraction are both measured between the wave propagation
direction and the normal to the interface In terms of indices of refraction in the case
of light waves Snellrsquos law takes the more familiar form
1 1 2 2sin sinn n
where ni = cvi
Total Internal Reflection
If a wave is incident from a medium of low wave speed into a medium of high wave
speed the law of refraction requires that the angle of refraction be greater than the
angle of incidence If the angle of refraction is required to be greater than 90deg then no
refracted wave can exist and total internal reflection occurs The critical incident
angle θc beyond which total internal reflection occurs is given by
2
1
sin c
n
n
15
Dispersion of Light
In addition to the speed of light varying from material to material it also varies with
wavelength within each material This means that the index of refraction is generally a
function of wavelength
c
nv
Since the wave speed is not constant such a medium is dispersive meaning in this
context that refraction actually disperses white light into its various colors because
Snellrsquos law gives a different angle for each wavelength In most materials the
variation with wavelength is quite small but this small effect is responsible for some
of the most spectacular color effects we ever see including rainbows a flashing
crystal chandelier and the colored fire of a diamond solitaire by candlelight
16
Serway Chapter 36
Real and Virtual Images
When light rays are focused at a certain plane producing an image if a sheet of white
paper is placed there we call the image a real image The images produced by film
projectors and overhead projectors are examples of real images
When light rays appear to come from a certain location but no image is produced
when a screen is placed there we say that there is a virtual image at that location For
instance when you look in a mirror it appears that someone is behind the mirror but
a screen placed back there in the dark would show nothing Your image in the mirror
is a virtual image
Ray Tracing
There are lots of rules about how to find the images in optical systems but the best
way to keep things straight is to learn how to draw the principal rays for curved
mirrors and lenses The rays for convex and concave mirrors are shown in Fig 3615
The rays for converging and diverging thin lenses are shown in Fig 3627 You should
memorize the principal rays and know how to use them to locate images
Curved Mirrors
The focal length of a curved mirror with radius of curvature R is given by
2
Rf
The relation between object distance p image distance q and focal length f is
1 1 1
p q f
If the mirror is a diverging mirror f should be negative and if q should turn out to be
negative the image is virtual
Lateral Magnification
The lateral magnification in an optical system is defined by the ratio of the image size
to the object size
17
Image height
Object height
qM
p
for curved mirrors
As usual there are sign conventions here too but this formula just gives the
magnitude It is better to keep keep track of upright images versus inverted images by
means of ray diagrams rather than by memorizing sign conventions
Thin Lenses
The focal length of a thin lens is related to the radii of curvature of the two faces R1
and R2 of the lens by
1 2
1 1 11n
f R R
Note that this formula differs from Eq (3611) in the text by not having a minus sign
between the two R-terms We like this form better because for a simple converging
lens like a magnifying glass we just use positive values of R for both surfaces If one
of the faces is concave producing divergence use a negative value for R And if a
surface is flat use R = infin If the face is flat the radius is infinite
The relation between the image object and focal distances for a thin lens is the same
as that for a curved mirror
1 1 1
q p f
Use a negative focal length if the lens is diverging
The lateral magnification for a thin lens is the same as for a curved mirror
Image height
Object height
qM
p
Camera
The lens system in a camera projects a real image of an object onto the film (or CCD
array in a digital camera) The position of the image is adjusted to be on the film by
moving the lens into or out of the camera
18
Eye
The eye is like a camera in that a real image is formed on the retina Unlike a camera
the image position is adjusted by changing the focal length of the lens This is done by
the ciliary muscle which squeezes the lens changing its shape
Near Point The near point is the closest distance from the eye for which the lens can
focus an image on the retina It is usually 18-25 cm for young persons
Far Point The far point is the greatest distance from the eye for which the lens can
focus an image on the retina For a person with normal vision the far point is at
infinity
Nearsightedness A person is nearsighted if their far point is at some finite distance
less than infinity This condition can be corrected with a lens that takes an object at
infinity and produces a virtual image at the personrsquos far point
Farsightedness and Presbyopia A person is farsighted if their near point is too far
away for comfortable near work like reading or knitting This can be corrected by a
lens which takes an object at a normal near point distance of 18-25 cm and produces a
virtual image at the personrsquos natural near point Presbyopia involves a similar
problem which nearly all people experience as they age The ciliary muscle becomes
too weak and the lens becomes too stiff to allow the eye to provide for both near and
far vision The solution for this problem is either reading glasses or bifocal lenses
Reading classes are just weak magnifying glasses mounted on eyeglass frames
Bifocal lenses are split into upper and lower halves The lower half is a lens which
gives the proper correction for near work and the upper half is a different lens for
proper focusing at infinity
Angular Size
When an object is brought closer to the eye it appears to be larger because the image
on the retina is larger The size of this image is directly proportional to the objectrsquos
angular size which is the angle subtended by the object measured from the center of
the lens of the eye In optical instruments which are to be used with the eye the
angular size of the final image is whatrsquos important because it determines how large
the image will appear to the viewer
19
Simple Magnifier
A simple magnifier is a single converging lens or magnifying glass It takes an object
closer to the eye than a normal near point and produces a virtual image at or beyond
this near point The angular magnification is defined to be the ratio of the angular size
when viewed through the lens to the angular size of the object when viewed at the
normal near point (without aid of the lens)
Microscope
This instrument has two lenses (1) The objective is near the object being viewed and
produces a greatly magnified real image (2) The eyepiece is a simple magnifier which
the viewer uses to closely examine the image from (1)
Telescope
This instrument also has two lenses (1) The objective at the front of the telescope
takes light from a distant object and produces a real inverted image (which is rather
small) near its focal point (2) This small real image is then examined by the eyepiece
functioning as a simple magnifier to produce a virtual image with a larger angular
size
20
Serway Chapter 37
Two-Slit Interference
If light is incident on two closely spaced narrow slits a pattern of light and dark
stripes is produced beyond the slits The bright stripes or fringes are caused by
constructive interference of the two waves coming from the slits Constructive
interference occurs whenever two waves arrive at a location in phase with each other
This occurs when the distance x1 from slit 1 to a point P on the screen and the
distance x2 from slit 2 to point P differ by in integral number of wavelengths
1 2 where 0 1 2x x m m
where λ is the wavelength of the light When x1 and x2 are much larger than the slit
spacing d this condition reduces to
sind m
where θ is the angle between the direction of the incident light and the direction of the
light arriving at the screen
Thin Films
When light is partially reflected and partially transmitted by a thin film of transparent
material it is possible to have interference between the wave reflected from the front
of the film and light reflected from the back of the film (The colored reflections from
the thin film of oil on the water in a rain-soaked parking lot are an example of this
effect) It is difficult to write down formulas that will work in all cases so we will
just review the important principles here
1 If the two reflected waves are in phase with each other the film has enhanced
reflection (constructive interference) but if the two reflected waves are out of phase
with each other reflection is diminished (destructive interference) Phase shifts occur
due to reflection and due to the extra path length through the film of the wave
reflected from the back of the film
2 The phase change due to reflection is determined by the difference in index of
refraction between the two media involved in the reflection If the wave is incident
21
from a medium with a low index of refraction into a medium with a high index of
refraction a phase change of 180deg occurs and the reflected wave is inverted If
incident from high to low no phase shift occurs and the reflected wave is non-
inverted
3 The extra path length through the film of the wave reflected from the back of the
film is equal to 2t where t is the thickness of the film (The incident light is assumed
to be normal to the surface of the film) The number of wavelengths contained in the
extra path length is equal to 2tλn where λn = λn is the wavelength of the light in the
film
4 Rules for reflection from thin films
If one ray is inverted and the other is not then we have
12 constructive
2 nt m
2 destructivent m
If either both rays are inverted or both are non-inverted we have
2 constructivent m
12 destructive
2 nt m
22
Serway Chapter 38
Diffraction Grating
A diffraction grating is simply a fancy version of two-slit interference with the two
slits replaced by thousands of slits Just as in the two-slit case the bright fringes
occur at angles given by
sin md
but in this case d the distance between neighboring slits is made to be very small
The effect of having many slits instead of two is to make each bright fringe highly
localized with wide dark regions between neighboring maxima
Single Slit Diffraction
When light passes through an opening in an opaque screen an interference pattern is
produced beyond the opening To understand why we may replace the single opening
by many small coherent sources of light These many sources interfere with each
other producing a pattern known as a diffraction pattern (Note that many authors do
not distinguish between interference and diffraction treating them as interchangeable
terms) If the opening is a slit of width a then the diffraction pattern far from the slit
will have a bright central maximum with a succession of minima and weaker maxima
on either side The angle between the incident direction and the minima is given by
sin ma
where m = plusmn1 plusmn2
If the opening is circular with diameter D the angle between the incident direction
and the first minimum is given by
sin 122D
Optical Resolution and Rayleighrsquos Criterion
Two point sources can just be resolved (distinguished from each other) if the peak of
the diffraction image of the first source overlies the first minimum of the diffraction
image of the second source For circular holes of the kind usually encountered in
23
optical devices this condition is approximately satisfied when the angular separation
between the two sources as viewed from the optical instrument is greater than or equal
to the critical angle
min 122D
where λ is the wavelength of the light and where D is the diameter of the aperture in
the instrument
Polarization
We say that an electromagnetic wave is polarized if its electric field vector doesnrsquot
change direction in a random fashion The simplest kind of polarization is linear
polarization in which the electric field vector oscillates back and forth along the
same axis in space Polarized light can be produced from normal unpolarized light by
selective absorption (as in Polaroid sunglasses) by passing light through crystals that
have different indices of refraction for different polarizations (double refraction) by
scattering (the blue sky is polarized) and by reflection (glare) Polarization by
reflection occurs when light reflects from a shiny insulating (non-metallic) surface
The amount of polarization is greatest for reflection at Brewsterrsquos angle
2
1
tan P
n
n
where θP is the incidence angle of light from medium 1 onto medium 2 and where n1
and n2 are the indices of refraction for the media (In our everyday experience n1 = 1
since the light comes in through the air and n2 is the index of refraction of the shiny
insulating material producing the glare eg water glass plastic paint etc)
Malusrsquos Law
The intensity of transmitted polarized light through a perfect polarizer is related to the
incident intensity of polarized light by Malusrsquos law
2cosoI I
where Io is the intensity of the incident light and where θ is the angle between the
electric field vector in the incident wave and the transmission axis the polarizer
24
Serway Chapter 39
Principles of Relativity
All of the weirdness of relativity flows from two simple principles
(1) The laws of physics must be the same in all inertial (non-accelerating) reference
frames
(2) The speed of light in vacuum has the same value in all inertial reference frames
Note that (1) seems reasonable but (2) is very odd It says that if two space ships are
approaching each other at nearly the speed of light and a laser pulse is shot from ship
1 toward ship 2 then when the people on ship 2 measure the speed of the pulse as it
goes by it is moving at 3 times 108 ms the same as if the ships were stationary
Simultaneity
If observer 1 sees two events in her own frame as simultaneous at two different
locations a moving observer 2 will see these two events happening at different times
Relativistic gamma
The factor γ (gamma) appears regularly in the formulas of relativity
2
2
1
1 vc
where v is the relative speed between two inertial frames
Time Dilation
If observer 1 sees two events at the same location in space separated by time Δtp in his
own frame then observer 2 moving at speed v relative to observer 1 will see these
two events separated by a longer time Δt
pt t
Moving clocks run slow
Length Contraction
If observer 1 measures the length of an object along the x-direction at some instant of
time in her frame to be Lp then observer 2 moving in the x-direction at speed v will
measure the length L of the object to be shorter
pLL
25
Moving meter sticks are short
Velocity Addition
If a particle is moving at velocity vac relative to frame c and if frame c is moving at
speed vcb with respect to frame b (with both velocities directed along the same line)
what would be the velocity of the particle in frame b In classical physics the answer is
ab ac cbv v v
but if any of these velocities are an appreciable fraction of the speed of light the
answer changes
21ac cb
abac cb
v Vv
v v c
In using this equation make sure that vac has the same sign in the numerator and in the
denominator and that vbc also has the same sign in both places Note that this equation
is different (and in our opinion easier to understand) than the equations in the text
Momentum and Energy
The momentum and total energy of a particle moving at velocity v (its speed is v v )
are given by
22
2 2
22
1 1 vvc c
m mcm E mc
vp v
Rest Energy
When a particle is at rest (v = 0 and γ = 1) then E = mc2 is called the rest energy It
includes the potential energy of the particle so that if we change the potential energy
of a particle we change its rest mass
Kinetic Energy The kinetic energy is the difference between the total energy and the
rest energy
2 2K mc mc
For v ltlt c this reduces to K = mv22
General Relativity
Einsteinrsquos principle of equivalence in the theory of general relativity is In the vicinity
of a point a gravitational field is equivalent to an accelerated frame of reference in
26
the absence of gravitational effects For example there is no difference whatever in
the physics experienced by an astronaut on earth and one who is in interstellar space
far away from any stars or planets but who has just turned on her spaceshiprsquos engines
and is accelerating at 98 ms2
27
Serway Chapter 40
Photons
We now know that even though light behaves like a wave it is better described as
consisting of small packets of energy called photons The energy of a photon is related
to its frequency by
E hf
where h is Planckrsquos constant
346626 10 J sh
Photons also carry momentum given by
hf hp
c
Since the amount of energy in each photon is fixed the difference between dim light
and intense light (of the same wavelength) is that dim light consists of fewer photons
than intense light
Photoelectric Effect
In the photoelectric effect light shining onto a metal surface gives the electrons in the
metal enough energy to escape and be detected It requires a certain amount of energy
(called the work function typically = 3 ndash 5 eV) just to kick the electron out of the
metal so the light must deliver at least this much energy to an electron to produce the
effect It is observed that red light no matter how intense never produces electrons
But ultraviolet light even if quite dim will eject electrons from the metal Since red
light consists of 2 eV photons and ultraviolet light has photons with energies around
4-6 eV the photon idea explains the behavior of the photoelectric effect The
maximum energy that an ejected electron can have is
maxK hf
where f is the photon frequency The minimum frequency that light can have and cause
electrons to be ejected is called the cutoff energy and is found by setting Kmax = 0
cf h
28
Compton Effect
When high-frequency light interacts with free electrons the scattered light does not
have the same wavelength as the incident light contrary to what classical
electromagnetism would predict But the photon picture in which photons have
momentum and energy allows us to treat scattering as a collision between the photon
and the electron In this collision the electron and the scattered photon both have a
different momentum and energy than they did before And since p = hλ for a photon
if the momentum is different the wavelength will also be different
after before 1 cose
h
m c
where the angle θ is the angle between the incoming photon direction and the
direction of the scattered photon
Particles are Waves
Since photons behave like particles sometimes it is not surprising that elementary
particles can behave like waves sometimes The wavelength of a particle with
momentum p is given by
h
p
Wave-particle Duality
Both photons and elementary particles have a dual nature sometimes they behave like
particles and sometimes they behave like waves It is hard for us to comprehend the
nature of such an object by making mental pictures but experiments definitively show
that this is the case Since we have no direct experience with photons and elementary
particles (because their energies are so much smaller than the energies of the everyday
objects we encounter) it is perhaps not surprising that we have a hard time forming a
mental picture of how they behave
Electron Interference
Since an electron is both a particle and a wave just like a photon it should be able to
produce an interference pattern This is observed If an electron beam is shot at two
closely-spaced slits and if the electrons that pass through the slits are detected
downstream it is found that there are some locations where electrons are never detected
29
and others where lots of electrons are detected The pattern is exactly the same as the
one observed for light waves provided that we use the electron wavelength λ = hp in
place of the wavelength of light This pattern is observed even though each electron is
detected as a single dot on the screen Only after many such dots are collected does the
pattern emerge And if we try to understand how this effect could possibly work by
looking closely at each slit to see which one the electron came through the pattern
disappears the act of measurement destroys the interference This means that each
single electron somehow comes through both slits (as a wave would)
Uncertainty Principle
In classical physics we always imagine that the positions and momenta of moving
particles have definite values It might be hard to measure them but surely at each
instant of time a particle should be precisely located at some point in space and have a
similarly precise momentum This turns out not to be true Instead both position and
momentum are required to be uncertain with their uncertainties Δx and Δp satisfying
the Heisenberg uncertainty relation
2x p
So if the particle were known to be precisely at some particular location (so that Δx =
0) we couldnrsquot know anything about its momentum (Δp = infin) and if its momentum
were exactly known we couldnrsquot know its position
There is a similar relation involving the particlersquos energy E and the time interval Δt
over which this energy is measured
2E t
This means that energy is not actually conserved if we are considering very small time
intervals and this brief non-conservation of energy has been observed
30
Serway Chapter 41
Wave Function ψ and Probability
Quantum mechanics does not predict exactly what an electron or a photon will do Instead
it specifies the wave function or probability amplitude ψ of an electron or a photon
This wave function is a complex-valued function of space and time whose squared
magnitude is the probability density P for finding a particle at a particular place in
space at a certain time
2P
where is the complex conjugate of ψ
For example the wave function of an electron with perfectly specified momentum p
would have a wavelength given by
h
p
and its probability amplitude would be proportional to
2i x ipxe e
The corresponding probability density would then be
21ipx ip ipxP e e e
which means that the electron is equally probable to be anywhere along the x axis
This is in accord with the uncertainty principle since we specified the momentum
precisely we canrsquot have any idea about the position of the electron
As another example you have probably seen ldquofuzzy ballrdquo drawings of electron
orbitals in a chemistry book These fuzzy balls are meant to indicate the distribution
of the probability density 2
P in the orbital
Particle in a Box
A simple example in which we can calculate the wave function is the case of a particle of
mass m constrained to be inside a 1-dimensional box between x = 0 and x = L In this
case the general wave function is a linear superposition of wave functions ψn of the form
sin niE tn
n xx t A e
L
31
where A is a positive constant where n = 1 2 3 and where the energy associated
with each of the quantum states ψn is given by
22
28n
hE n
mL
This wave function is zero at x = 0 and x = L which means that the particle will never
be found at the walls of the box The wave function has maximum values in the
interior and at these places the particle is most likely to be found and it also has
places where it is zero and at these places the particle will also never be found as
expressed by the formula
2sinn x
PL
The particle in the box is interfering with itself producing a probability interference
pattern across the box just like the interference patterns we studied with light and
sound
32
Serway Chapter 42
Atoms
Many people picture an atom as a miniature solar system where electrons orbit around
a massive nucleus at the center This picture is misleading because in atoms the wave
nature of electrons dominates The electrons form 3-dimensional standing waves
(called orbitals) centered at the nucleus And if you ask what is it thatrsquos waving the
answer is ψ the probability amplitude (see Chapter 41)
Standing waves on a string can occur only for certain frequencies (the fundamental
and higher harmonics) Similarly atomic orbitals occur only for certain energies For
the hydrogen atom the energies of the orbitals have the a particularly simple form
2
1136 eVnE
n
where n = 1 2 3 is called the principal quantum number For other atoms the
determination of the orbital energies requires numerical calculation by computers
Atomic Spectra
If an electron is somehow given extra extra energy (we say that it is excited) so that it
occupies a higher orbital it will eventually ldquofallrdquo back down to a lower orbital Each
time an electron falls to a lower orbital it loses the difference in energy between the
two orbitals in the form of a photon Since the orbital energies are discrete so are the
energy differences and so are the wavelengths of the emitted light The entire set of
these discrete wavelengths is called the atomic spectrum and it is unique to each type
of atom For hydrogen the spectrum can be simply written as
H 2 2
1 1 1
f i
Rn n
where RH = 10973732 times 107 m-1 For other atoms the spectrum cannot be expressed as
a formula
Orbital Angular Momentum Electrons in atoms also have quantized values of
angular momentum The orbital quantum number ℓ specifies the value of this
quantized angular momentum through the formula
1L
33
If we want to know the value of the angular momentum along some direction in space
say the z direction the answer is not L but rather
zL m
where mℓ is another quantum number which runs from
1 1m
This quantum number is important when an atom sits in a magnetic field
Spin Angular Momentum It has been found experimentally that electrons and other
charged particles also carry internal angular momentum which we call spin
Electrons have an intrinsic spin angular momentum s along a specified axis that is
extremely quantized it can only take on 2 possible values
1
2zs s
We thus say that electrons have spin s = 12 or that electrons are rdquospin one-half parti-
cles Since s is an angular momentum it obeys the same rule as that for orbital angular
momentum namely that if its value along some axis is s then its total magnitude is
given by
31
2S s s
I know this seems weird but quantum mechanics is weird The only excuse for this
bizarre way of looking at the world is that it predicts what happens in experiments
Exclusion Principle The answer to the question of how many electrons (or any other
spin one-half particle) can be in one particular quantum state was discovered by
Wolfgang Pauli and is called the exclusion principle
ldquoNo two electrons can ever be in the same quantum state therefore no two electrons
in the same atom can have the same set of quantum numbersrdquo
This is the reason that we have atoms with different properties instead of every atom
simple having all of its electrons in the ground state All of the variety we see around
us in the world is the result of chemical differences and these differences would not
exist unless electrons obeyed this important principle The entire structure of the
periodic table (see pages 1377-1379 in Serway) is an expression of this principle
34
Serway Chapter 44
Nuclear Properties
The nucleus is composed of protons with charge +e and mass mp = 1007226 u and of
neutrons with zero charge and mass mn = 1008665 u where u is one atomic mass unit
-271 u=1660540 10 kg
The atomic number Z counts the number of protons in a nucleus while the neutron
number N counts the number of neutrons The mass number A is the sum of the two
A N Z
Protons and nuetrons have about the same mass and some times called baryons (heavy
ones) or nucleons (partcles in the nucleus) A is sometimes called the baryon
number The chemical elements are distinguished by Z ie hydrogen has Z = 1 iron
has Z = 26 uranium has Z = 92 etc But for a given Z there might be several nuclei
with different numbers of neutrons These nuclear siblings are called isotopes
These numbers are used to label nuclei according to the pattern
5626 ie FeA
Z X
denotes the isotope of iron with N = 56 ndash 26 = 30 neutrons
The nucleus is roughly spherical with a radius given approximately by
1 3 150 0where 12 10 mr r A r
Nuclear Stability
Because the positively charged protons electrically repel each other with an enormous
force at distances as small as 10-15 m there must be some really strong force that that
overcomes electrical repulsion to hold protons and neutrons together This force is
called with some lack of imagination the strong force It is a very short-range force
(it only acts over a distance of about 2 x 10-15 m) and attracts protons to protons
neutrons to neutrons and neutrons to protons But in spite of this strong nuclear force
the coulomb repulsion of the protons is still present so anything that might keep the
protons from being right next to each other would help keep the nucleus from
35
exploding This role is played by the neutrons and for nuclei with Z le 20 the stable
nuclei roughly have N = Z
For Z gt 20 the coulomb repulsion force is more powerful and more neutrons are
needed to dilute it up to about N = 15Z around Z = 80 For Z gt 83 no amount of
neutrons can help and these nuclei are unstable (radioactive) There is an ldquoisland of
(relative) stabilityrdquo around Z=90-92 (thorium and uranium respectively) These
elements have one or two isotopes with half-lifes of billions of years so there are
substantial amounts of such elements on Earth This fact makes it possible to have
practical fission devices
Radioactive Decay
There are three types of radioactive decay
Alpha decay the nucleus kicks out a helium nucleus (N = 2 Z = 2 A = 4) So if the
original nucleus is called X and the new nucleus is called Y then the decay would look
like this
4 42 2X Y+ HeA A
Z Z
Beta decay the nucleus either kicks out an electron (endash) or its positively-charged
antimatter twin the positron (e+) plus either an electron neutrino v or an electron
anti-neutrino v
1X Y+e electron decayA AZ Z v
1X Y+e positron decayA AZ Z v
A neutrino is a particle with no charge hardly any mass (much less than the electron
mass) and interacts so weakly with matter that most neutrinos upon encountering the
planet earth just pass right through it as if it werenrsquot there
Gamma decay the nucleons in the nucleus X are in an excited energy state X
(perhaps as a result of having undergone alpha or beta decay) and they drop down to a
lower energy state shedding the energy as a high frequency photon
X XA AZ Z
36
This process is exactly analogous to the way that the electrons in atoms emit photons
Decay Rate and Half Life
There is no way to predict exactly when an unstable or excited nucleus will decay but
there is an average rate at which this decay occurs called the decay constant λ The
meaning of this constant is that if there are a large number N of nuclei in a sample
then the number of decays per second that will be observed (called the decay rate R)
is R = λN In mathematical language
dMR N
dt
This simple differential equation has for its solution
0tN t N e
where N0 is the number of nuclei in the sample at time t = 0
The half-life is the time it takes for half of the nuclei in the sample to decay and is
related to the decay constant by
1 2
ln 2 0693T
Disintegration Energy
When a nucleus decays it is making a transition to an overall state of lower energy
which means according to Einsteinrsquos famous formula E = mc2 that the sum of the
masses after the decay must be less than the mass before with the lost mass appearing
as kinetic energy among the decay products For example in alpha decay this kinetic
energy called the disintegration energy Q is given by
2X YQ M M M c
37
Serway Chapter 45
Nuclear energy
There are two ways to extract energy from the nucleus fission and fusion For nuclei
with Z greater than 26 breaking the nucleus apart into pieces leads to a lower
2mc energy than the original nucleus so energy can be extracted by fission For
nuclei with Z less than 26 a lower energy is achieved by combining nuclei so energy
can be extracted by fusion
Fission
Since neutrons have no charge they are not repelled from nuclei as protons are for
this reason their behavior is the key to understanding how fission works
Fast neutrons mostly bounce off other nuclei with each collision slowing the neutron
down This elastic energy loss is most effective if the other nuclei have low mass (like
hydrogen) and these materials are called moderators because of their ability to slow
down fast neutrons
The reason that slowing neutrons is important is that slow neutrons are much more
likely to be absorbed by a nucleus which then leads to nuclear reactions of various
kinds For a few very large nuclei like uranium-235 and some plutonium isotopes
absorption of a slow neutron causes the nucleus to split into two large fragments plus
2 or 3 fast neutrons (energy is released in the process too) Thus one slow neutron
can produce energy plus 2 or 3 more neutrons which if moderated from fast to slow
can split 2 or 3 more nuclei leading to a runaway chain reaction If the reaction is
allowed to proceed unchecked it produces a large explosion If a neutron absorbing
material is added to the mix (like the cadmium in reactor control rods) it is possible to
keep the reaction under control and to extract the released energy as heat to drive
steam turbines and produce electricity
Fusion
Fusion involves mashing two nuclei together and since they are both charged and repel
each other this reaction is much harder to make go The nuclei must have enough energy to
overcome the coulomb repulsion which is why this reaction requires a high temperature
(hundreds of millions of degrees K) like that in the sun (which is burning hydrogen to
helium via fusion) or like that in the center of the fission explosion that is used to detonate
38
a hydrogen bomb
This reaction is of interest for power production in spite of this difficult temperature
requirement because of the abundance of fusion fuel on the planet There are about
012 g of deuterium ( 21H ) in every gallon of water on earth and it only costs about 4
cents to extract it The fusion energy available from this minuscule amount of
deuterium would run a 1000-Megawatt power plant for 10 seconds Water is so
abundant on earth that if fusion were to work we would have an essentially
inexhaustible source of energy
So why donrsquot we have fusion power plants Well the fuel is cheap but the match is
incredibly expensive The only way we know to control this difficult high-temperature
reaction is with large and expensive pieces of equipment involving either large
magnetic fields and complex high-power electromagnetic antennas or with gigantic
(football-fieldsized) laser facilities involving more than a hundred of the highest-
energy lasers ever built Power plants based on these current methods for controlling
fusion are unattractive to the fiscally-minded people who run the electric power
industry Hopefully better designs will be discovered as experiments continue
39
c actinium 68 Er erbium 101 Md mendelevium 104 Rf rutherfordium l aluminum 63 Eu europium 80 Hg mercury 62 Sm samarium m americium 100 Fm fermium 42 Mo molybdenum 21 Sc scandium b antimony 9 F fluorine 60 Nd neodymium 106 Sg seaborgium r argon 87 Fr francium 10 Ne neon 34 Se selenium s arsenic 64 Gd gadolinium 93 Np neptunium 14 Si silicon t astatine 31 Ga gallium 28 Ni nickel 47 Ag silver a barium 32 Ge germanium 41 Nb niobium 11 Na sodium k berkelium 79 Au gold 7 N nitrogen 38 Sr strontium e beryllium 72 Hf hafnium 102 No nobelium 16 S sulfur i bismuth 108 Hs hassium 76 Os osmium 73 Ta tantalum h bohrium 2 He helium 8 O oxygen 43 Tc technetium
boron 67 Ho holmium 46 Pd palladium 52 Te tellurium r bromine 1 H hydrogen 15 P phosphorus 65 Tb terbium d cadmium 49 In indium 78 Pt platinum 81 Tl thallium a calcium 53 I iodine 94 Pu plutonium 90 Th thorium f californium 77 Ir iridium 84 Po polonium 69 Tm thulium
carbon 26 Fe iron 19 K potassium 50 Sn tin e cerium 36 Kr krypton 59 Pr praseodymium 22 Ti titanium s cesium 57 La lanthanum 61 Pm promethium 74 W tungsten l chlorine 103 Lr lawrencium 91 Pa protactinium 92 U uranium r chromium 82 Pb lead 88 Ra radium 23 V vanadium o cobalt 3 Li lithium 86 Rn radon 54 Xe xenon u copper 71 Lu lutetium 75 Re rhenium 70 Yb ytterbium m curium 12 Mg magnesium 45 Rh rhodium 39 Y yttrium b dubnium 25 Mn manganese 37 Rb rubidium 30 Zn zinc y dysprosium 109 Mt meitnerium 44 Ru ruthenium 40 Zr zirconium s einsteinium
40
UNITS The SI units are given in parentheses Other commonly used units are given in terms of the SI units Symbols conform with the recommendations of the American National Standards Institute (ANSI) and the American Institute of Physics (AIP) Commonly used multiples of SI units
prefix symbol factor tera T 1012 giga G 109
mega M 106 kilo k 103
centi c 10-2 milli m 10-3
micro μ 10-6 nano n 10-9 pico p 10-12
femto f 10-15
length meter (m) angstrom 1 Ǻ = 10-10 m inch 1 in = 254 x 10-2 m foot 1 ft = 03048 m mile 1 mi = 1609 m
mass kilogram (kg) atomic mass unit 1 u = 1661 x 10-27 kgslug 1 slug = 1459 kg
force newton (N = kg bull ms2) dyne 1 dyn = 10-5 N pound 1 lb = 4448 N
pressure Pascal (Pa = kgm bull s2) atmosphere 1 atm = 1013 x 105 Pa poundssquare inch 1 psi = 6895 Pa cm of mercury 1 cm Hg = 1333 Pa bar 1 bar = 1000 x 105 Pa torr 1 torr = 1333 Pa
time second (s) minute 1 min = 60 s hour 1 h =3600 s
frequency hertz (Hz = s-1) radianssecond 1 rads = 12π Hz
energy joule (J = kg bull m2s2) erg 1 erg = 10-7 J electron volt 1 eV = 1602 x 10-19 J calorie 1 cal = 4187 J kilowatt-hour 1 kWh = 36 x 106 J British thermal unit 1 Btu = 1055 J
power watt (W = kg bull m2s3) horsepower 1 hp = 7457 W
charge coulomb (C = A bull s)
electric potential volt (V = kg bull m2 s3 bull A)
current ampere (A)
resistance ohm (Ω = kg bull m2s3 bull A2)
capacitance farad (F = s4 bull A2kg bull m2)
magnetic field tesla (T = kgs2 bull A) gauss 1 G = 10-4 T
magnetic flux weber (Wb = kg m2s2 bull A) maxwell 1 Mx = 10-8 Wb
magnetic inductance henry (H = kg bull m2s2 bullA2)
temperature kelvin (K) degrees Celsius 0degC = 27315 K
angle radian (rad) degree 1 deg = π180 rad revolution 1 rev = 2π rad
41
Some Physical Constantsa Quantity Symbol Valueb Atomic mass unit μ 1660 538 73 (13) x 10-27 kg
931494 013 (37) MeVc2 Avogadros number NA 6022 141 99 (47) x 1023 particlesmol Avogadrorsquos number 6022 x 1023 mol Bohr magneton
2B e
em 9274 008 99 (37) x 10-24 JT
Bohr magneton μB 927 x 10-24 JT Bohr radius
0
2
2e em e k
a 5291 772 083 (19) x 10-11 m
Boltzmanns constant B A
RNk 1380 650 3 (24) x 10-23 JK
Boltzmannrsquos constant kB 1380 x 10-23 JK Compton wavelength
C eh
m c 2426 310 215 (18) x 10-12 m
Deuteron mass md 3343 583 09 (26) x 10-27 kg 2013 553 212 71 (35) u
electron charge e 1602 x 10-19 C Electron mass me 9109 381 88 (72) x 10-31 kg
5485 799 110 (12) x 10-4 u 0510 998 902 (21) MeVc2
electron mass 911 x 10-31 kg Electron volt eV 1602 176 462 (63) x 10-19 J Elementary charge e 1602 176 462 (63) x 10-19 C Gas constant R 8314 472 (15) JKmol Gravitational constant G 6673 (10) x 10-11 Nm2kg2 Neutron mass mn 1674 927 16 (13) x 10-27 kg
1008 664 915 78 (55) u 939565 330 (38) MeVc2
neutron mass 1675 x 10-27 kg Nuclear magneton
2n p
em 5050 783 17 (20) x 10-27 JT
permeability constant μ0 1257 x 10-6 Hm permittivity constant ε0 8854 x 10-12 Fm Plancks constant h
2h
6626 068 76 (52) x 10-34 Js 1054 571 596 (82) x 10-34 Js
Planckrsquos constant h ħ
6626 x 10-34 Js 1055 x 10-34 Js
Proton mass mp 1672621 58 (13) x 10-27 kg 1007 276 466 88 (13) u 938271 998 (38) MeVc2
proton mass 1673 x 10-27 kg Rydberg constant RH 1097 373 156 854 9 (83) x 107 m-1 Speed of light in vacuum c 2997 92458 x 108 ms (exact) speed of light c 300 x 108 ms a
These constants are the values recommended in 1998 by CODATA based on a least-squares adjustment of data from different measurements For a more complete list see P J Mohr and B N Taylor Rev Mod Phys 72351 2000 b The numbers in parentheses for the values above represent the uncertainties of the last two digits
NOTE The ones in red are the ones from Appendix 13 The ones in yellow are the ones that are from the Serway book table but that were already on the Appendix 13 table
42
INDEX
Absolute zero 9
Adiabatic 17
Adiabatic exponent 18
Adiabatic process 17
Alpha decay 46
Amplitude 4
Angle of reflection 25
Angular frequency 3
Angular magnification 30
Angular momentum atomic 43
Angular size 29
Archimedes Principle 1
Atomic spectra 43
Atoms 43
Avogadrorsquos number 10
Beats 8
Bernoullirsquos Equation 2
Beta decay 46
Bifocals 29
Brewsterrsquos angle 34
British Thermal Unit Btu 11
Buoyancy 2
calorie 11
Calorie 11
Camera 28
Carnot cycle 21
Carnot efficiency 22
Celsius scale 9
Ciliary muscle 29
Coefficient of performance 21
Compton effect 39
Constructive interference 7
Continuity equation of 2
Contraction length 35
Convection 14
Curved mirrors 27
Decay constant 46
Decay rate radioactive 46
Decibel scale 5
Degrees of freedom 16
Density 1
Destructive interference 7
Diffraction single slit 33
Diffraction grating 33
Dilation time 35
Disintegration energy 47
Dispersion of light 26
Doppler effect 6
Double slit interference 31
Efficiency 20
Electron interference 39
Emissivity 15
Energy nuclear decay 47
Energy relativistic 36
43
Engineering work 20
Entropy 22
Entropy ideal gas 24
Equipartition of energy 18
Equivalence principle 36
Exclusion principle 44
Expansion thermal 9
Expansion coefficient linear 9
Eye 29
Fahrenheit scale 9
Far point 29
Farsightedness 29
First Law of Thermodynamics 12
Fission 48
Flux volume 2
Free expansion 23
Fringes 31
Fusion 48
Fusion heat of 11
Gamma relativistic 35
Gamma decay 46
General relativity 36
Half-life 46
Heat 11
Heat capacity 11
Heat conduction 13
Heat engine 20
Heat of fusion 11
Heat of vaporization 11
Heat pump 20
Heats of transformation 11
Hydrostatics 1
Ideal Gas Law 10
Images real and virtual 27
Intensity sound 5
Interference 7
Interference two-slit 31
Internal energy 12
Internal energy degrees of freedom 16
Irreversible process 19 23
Isotopes 45
Joule 11
Kelvin scale 9
Kinetic energy relativity 36
Kinetic theory 16
Latent heat 11
Length contraction 35
Linear expansion coefficient 9
Linear polarization 34
Linear superposition 7
Longitudinal wave 3
Loudness 5
Magnification lateral 27
Magnifying glass 30
Malusrsquos law 34
Microscope 30
Momentum relativistic 36
Muscle ciliary 29
44
Musical instruments 8
Musical scale 8
Near point 29
Nearsightedness 29
Nonlinear 7
Nuclear energy 48
Nuclear properties 45
Nuclear stability 45
Octave 8
Optical resolution 33
Orbital quantum number ℓ 43
Orbitals 43
Particle in a Box 41
Particles are waves 39
Pascalrsquos Principle 1
Period 3
Photoelectric effect 38
Photons 38
Pitch 8
Polarization 34
Power sound 5
Presbyopia 29
Pressure 1
Principal quantum number n 43
Principle of equivalence 36
Principle of linear superposition 7
Probability amplitude ψ 41
Processes thermodynamic 13
R-value 14
Radiation thermal 14
Radioactive decay 46
Radioactive decay rate 46
Ray tracing 27
Rayleighrsquos criterion 33
Reading glasses 29
Real image 27
Refraction 25
Refrigerator 20
Relativistic gamma 35
Relativity principles 35
Resolved for light sources 33
Rest energy 36
Reversible process 19 22
Rope wave speed 4
Second Law of Thermodynamics 19
Shock waves 6
Simple magnifier 30
Simultaneity 35
Single slit diffraction 33
Snellrsquos law 25
Sound speed 5
Specific heat 11
Spin Angular momentum 44
Standing waves 7
Stefanrsquos law 14
Strong force 45
Telescope 30
Temperature 9
45
Temperature Scales 9
Thermal conductivity 14
Thermal energy 12
Thermal expansion 9
Thin film interference 31
Thin lenses 28
Time dilation 35
Tone musical 8
Total internal reflection 25
Transverse wave 3
Traveling Waves 3
Two-slit interference 31
Uncertainty principle 40
Vaporization heat of 11
Velocity addition relativity 36
Virtual image 27
Volume flux 2
Wave function 41
Wave Function ψ and Probability 41
Wave speed 3
Wave-particle duality 39
Wavelength 3
Wavenumber 3
Wien Displacement Law 16
Work 11
8
harmonics (The fundamental is called the ldquofirst harmonicrdquo)
Beats
Beats are heard when two waves with slightly different frequencies f1 and f2 are combined The
waves constructively interfere for a number of cycles then destructively interfere for a number
of cycles We hear a periodic ldquowah-wahrdquo frequency equal to the difference of the two wave
frequencies
1 2bf f f
Musical Instruments
Musical instruments produce tones by exciting standing waves on strings (violins piano) and in
tubes (trumpet organ) The fundamental frequency of the standing wave is called the pitch of the
tone The pitch of concert A is 440 Hz by definition Two tones are an octave apart if one pitch
has twice the frequency of the other In written music there are 12 intervals in each octave with
the ratio between successive intervals equal to 2112 = 105946 The ratios for each tone in an
octave starting at A and ending at the next higher A are
A A B C C D D E F F G G A
1 10595 11225 11892 12599 13348 14142 14983 15874 16818 17818 18877 2
A musical tone is actually a superposition of the fundamental frequency and the higher
harmonics The tone quality of a musical instrument is determined by the amplitudes of the
various harmonics that it produces A violin and a trumpet can play the same pitch but they
donrsquot sound at all alike to our ears The difference between them is in the various amplitudes of
their harmonics
9
Serway Chapter 19
Temperature
Formally temperature is what is measured by a thermometer Roughly high temperature is what
we call hot and low temperature is what we call cold On the atomic level temperature refers to
the kinetic energy of the molecules A collection of molecules is called ldquohotrdquo if the molecules
have rapid random motion while a collection of molecules is called ldquocoldrdquo if the random motion
is slow When two bodies are placed in close contact with each other they exchange molecular
kinetic energy until they come to the same temperature This is the microscopic picture of the
Zeroth Law of Thermodynamics
Absolute Zero
Absolute zero is the lowest possible temperature that any object can have This is the temperature
at which all of the energy than can be removed an object has been removed (This removable
energy we call thermal energy) There is still motion at absolute zero Electrons continue to orbit
around atomic nuclei and even atoms continue to move about with a small amount of kinetic
energy but this small energy cannot be removed from the object For example at absolute zero
helium is a liquid whose atoms still move and slide past each other
Temperature Scales
Kelvin Scale Absolute zero is at T = 0 K water freezes at T = 27315 K room temperature is
around T = 295 K and water boils at T = 373 K Note we donrsquot use a deg symbol Kelvin is
prefered SI Unit
Celsius Scale Absolute zero is at T = -273degC water freezes at T = 0degC room temperature is
around T = 22degC and water boils at T = 100degC
Fahrenheit Scale Absolute zero is at T = -459degF water freezes at TF = 32degF room temperature
is around T = 72degF and water boils at TF = 212degF TF =18TC + 32 Notice that temperature
differences are the same for the Kelvin and Celsius scales
Thermal Expansion
When materials are heated they usually expand and when they are cooled they usually contract
(Water near freezing is a spectacular counterexample it works the other way around) The
coefficient of linear expansion is defined by the relation
1
i
L
L T
10
where Li is the initial length of a rod of the material and ΔL is the change in its length due to a
small temperature change ΔT The coefficient of volume expansion is defined similarly
1
i
V
V T
where Vi is the initial volume of a piece of material and where ΔV is the change in its volume due
to a small temperature change ΔT
Avogadrorsquos Number (N or NA)
One mole of any substance corresponds to 6022 times 1023 molecules The atomic mass given on the
periodic table (p A32-33 in the text) is the mass in grams of one mole of atoms For example
the atomic mass of oxygen is 15999 gmol The mass of a single atom is given by the same
number in units of atomic mass units (u) For example the mass of an oxygen atom is 15999 u
(on the average)
Ideal Gas Law (an example of an equation of state)
When the molecules of a gas are sufficiently inert and widely separated that interactions between
them are negligible we say that it is an ideal gas The pressure P volume V and temperature T
(in kelvins) of such a gas are State Variables and are related by the ideal gas law
Bor PV=NkPV nRT T
where n is the number of moles of the gas where R is the gas constant
8314 Jmol KR
where N is the number of molecules and where kB is Boltzmannrsquos constant
231380 10 J KBk
It works well for air at atmosphere pressure and even better for partial vaccuums The relative
ease of measuring pressure and the linear relationship between pressure and temperature (if V
and n are held fixed) makes an ideal gas an ideal thermometer Thermometers can be based on
properties of solids or liquids but the behavior of these materials with temperature is more
complicated
11
Serway Chapter 20
Heat
Heat is energy that flows between a system and its environment because of a tempera-
ture difference between them The units of heat are Joules as expected for an energy
Unfortunately there are several competing units of energy They are related by
1 cal 4186 J 1 Cal 4186 J 1 Btu 1054 J
Heat Capacity
There is often a simple linear relation between the heat that flows in or out of part of a
system and the temperature change that results from this energy transfer When this
linear relation holds it is convenient to define the heat capacity C and the specific
heat c as follows
For the entire object the heat Q it absorbs is given by Q = C(Tf - Ti)
For a particular material the specific heat is defined by c = Cm which is the heat
capacity per unit mass so that
Q = mc(Tf ndash Ti)
Note C has units of energy (J or Cal)(Kelvin kg)
Heats of Transformation or Latent Heat Q = plusmn mL
When a substance changes phase from solid to liquid or from liquid to gas it absorbs
heat without a change in temperature The latent heat or heat of transformation is
usually given per unit mass of the substance For example for water the heat of fusion
(melting) is L = 333 kJkg while the heat of vaporization (boiling) is L = 2260 kJkg
Note that heat for boiling is considerably bigger than melting for water You have to
be careful with signs heat is given off (negative) if you go down in temperature and
condense steam
Work
In general the small amount of work done on a system as a force Fon is exerted on it
through a vector displacement dx is given by
on xdW d F
12
But if the displacement is done very slowly (as we always assume in thermodynamics)
then the force exerted on the system and the force exerted by the system are in
balance so the force exerted by the system is ndash Fon In thermodynamics it is more
convenient to talk about the force exerted by the system so we change the above
formula for the work done on the system to
xdW d F
where F is the force exerted by the system This has confused students for more than a
century now but this is the way your book and many other books do it so you are
stuck You will need to memorize the minus sign in this definition of the work to be
able to use your textbook
There are many chances to get signs wrong in this and the next two chapters (Mosiah
2 )
When an external agent changes the volume of a gas at pressure P by a small amount
dV the (small amount) of work done on the system is given by
dW PdV
Notice that this minus sign is just what we need to make dW be positive if the external
agent compresses the gas for then dV is negative If on the other hand the external
agent gives way allowing the gas to expand against it then dV is positive and we say
that the work done on the gas is negative
The work done on the system (eg by the gas in a cylinder) in a thermodynamic
process is the area under the curve in a PV diagram It is positive for compressions
and negative for expansions If the volume of gas remains constant in a process then
no work is done by the gas
Cyclic processes are important For cyclic processes represented by PV diagrams the
magnitude of the net work during one cycle is simply the area enclosed by the cycle
on the diagram Be careful to keep track of signs when you are calculating that
enclosed area In cyclic process Q = ndashW for a cycle Put a PV diagram in your notes
1
Path A-B B-C C-D D-A A to A net
Q
W
ΔU
ΔS
Internal Energy
The energy stored in a substance is called its internal energy Eint This energy may be
stored as random kinetic energy or as potential energy in each molecule (stretched
chemical bonds electrons in excited states etc) For ideal gases all states with the
same temperature will have the same Eint
First Law of Thermodynamics
The change ΔEint in the internal energy of a system is given by
intE Q W
where Q is the heat absorbed by the system and where W is the work done on the
system Hence if a system absorbs heat (and if Wge0) the internal energy increases
Likewise if the system does work (W on the system is negative) and if Qge0 the
internal energy decreases Potential Pitfall Many times people talk about work done
by the system It is the minus of W on the system Donrsquot get tripped up
Processes
Adiabatic process No heat is exchanged Q = 0 requiring that the system be insulated
from the environment A process may be approximately adiabatic if it happens so
rapidly that heat does not have time to enter or leave the system Work + or ndash is done
and ΔEint = W
Adiabatic free expansion A gas is allowed to expand into a vacuum without pushing
on anythingndashit just rushes into the vacuum In this process Q = W = 0 so the internal
energy and temperature does not change (Note the difference between an adiabatic
process and a free expansion is that NO work is done in the adiabatic free expansion)
Isobaric process The pressure is held fixed ΔP = 0 For example usually the
pressure increases when a gas is heated but if it were allowed to expand during the
2
heating process in just the right way its pressure could remain fixed In isobaric
processes both Q and W are non-zero (Work is eacy = ndashP(Vf ndash Vi)
Isovolumetric process The volume is held fixed ΔV = 0 The work done by the gas is
then zero and so we have ΔEint = Q
Isothermal process The temperature is held fixed ΔT = 0 There is no change in Eint
so Q = ndashW for isothermal processes For an ideal gas PV = nRT so PV = constant in
an isothermal process The work done on the gas is then given by
lnf
i
Vi
Vf
VW PdV nRT
V
Heat Conduction
The quantity P is defined to be the rate at which heat flows through an object and is a
power having units of watts It is analogous to electric current which is the rate at
which charge flows through an object If the flow of heat through a slab of length L
and cross-sectional area A is steady in time then P is given by the equation
h cT TdQkA
dt L
P =
where Th and Tc are the (hot and cold) temperatures of the two ends of the slab The
heat flows of course because of this temperature difference The quantity k is called
the thermal conductivity and is a constant that is characteristic of the material It is
analogous to the electrical conductivity h cT TL is sometimes called the temperature
gradient and is written dTd dTdx
R-Values
It is common to have the heat-conducting properties of materials described by their R-
values especially for insulating materials like fiberglass batting The connection
between k and R is R = Lk where L is the material thickness In this country R-values
always have units of 2ft F hourBtu
Polyurethane foam 6 (1 in thick) Air space 1 (35 in thick) Fiber glass batting 11
(35 in thick) Wood about 1 (1 in thick) Glass 09 (0125 in thick)
The heat flow rate through a slab of area A is given by
3
h cT TA
R
P
in units of Btuhour Note that A must be in square feet and the temperatures must be
in degrees Fahrenheit
Convection
Convection is the transfer of thermal energy by flow of material For instance a home
furnace doesnrsquot heat a house by waiting for the heat from the burner to slowly conduct
throughout the house instead it quickly pumps warm air to all of the rooms
Generally convection is a much faster way to transfer heat than conduction
Radiation
Electromagnetic radiation can also transfer heat When you warm yourself near a
campfire which has burned itself down into a bed of glowing embers you are
receiving radiant heat from the infrared portion of the electromagnetic spectrum The
rate at which an object emits radiant heat is given by Stefanrsquos law
4AeTP
where P is the radiated power in watts σ is a constant
8 2 45696 10 W m K
A is the surface area of the object in m2 and T is the temperature in kelvins The
constant e is called the emissivity and it varies from substance to substance A perfect
absorber (think black velvet) has e = 1 while a perfect reflector (think mirror) has e =
0 Hence black objects radiate very well while shiny ones do not Also an object that
is hotter than its surroundings radiates more energy than it absorbs whereas an object
that is cooler than its surroundings absorbs more energy than it radiates
Terminology
Transfer variables vs state variables
Energy transfer by heat as well as work done depends on the initial final and
intermediate states of the system They are transfer variables But their sum (Q + W =
4
Eint) is a state variable
Figure 205
5
Serway Chapter 21
Kinetic Theory
The ideal gas law works for all atoms and molecules at low pressure It is rather
amazing that it does Kinetic theory explains why The properties of an ideal gas can
be understood by thinking of it as N rapidly moving particles of mass m As these
particles collide with the container walls momentum is imparted to the walls which
we call the force of gas pressure In this picture the pressure is related to the average
of the square of the particle velocity 2v by
22 1( )
3 2
NP mv
V
Using the ideal gas law we obtain the average translational kinetic energy per
molecule
21 3
2 2 Bmv k T
The rms speed is then given by
2rms
3 3Bk T RTv v
m M
where M is the molecular mass in kgmol
Degrees of Freedom
Roughly speaking a degree of freedom is a way in which a molecule can store energy
For instance since there are three different directions in space along which a molecule
can move there are three degrees of freedom for the translational kinetic energy
There are also three different axes of rotation about which a polyatomic molecule can
spin so we say there are three degrees of freedom for the rotational kinetic energy
There are even degrees of freedom associated with the various ways in which a
molecule can vibrate and with the different energy levels in which the electrons of
the molecule can exist
Internal Energy and Degrees of Freedom The internal energy of an ideal gas made
up of molecules with J degrees of freedom is given by
int 2 2 B
J JE nRT Nk T
6
Heat Capacities of an Ideal Gas The heat capacity of a gas is described by means of
molar heat capacities CV and CP These are the heat capacities per mole and the
subscript V on CV means that the volume is being held constant while for CP the
pressure is held constant For example to raise the temperature of n moles of a gas
whose pressure is held constant by 10 K we would have to supply an amount of heat Q
= nCP (10) K
Molar Specific Heat of an Ideal Gas at Constant Volume
VQ nC T
3monatomic
2VC R
5diatomic
2VC R
5polyatomic
2VC
Real gases deviate from these formulas because in addition to the translational and ro-
tational degrees of freedom they also have vibrational and electronic degrees of
freedom These are unimportant at low temperatures due to quantum mechanical
effects but become increasingly important at higher temperatures The rough rule is
No of degrees of freedom
2VC R
Molar Specific Heat of an Ideal Gas at Constant Pressure
PQ nC T
P VC C R
The internal energy of an ideal gas depends only on the temperature
int VE nC T
Adiabatic Processes in an Ideal Gas
7
An adiabatic process is one in which no heat is exchanged between the system and the
environment When an ideal gas expands or contracts adiabatically not only does its
pressure change as expected from the ideal gas law but its temperature changes as
well Under these conditions the final pressure Pf can be computed from the initial
pressure Pi and from the final and initial volumes Vf and Vi by
or constantf f i iP V PV PV
where γ = CPCV The quantity γ is called the adiabatic exponent Note that this
doesnrsquot mean that the ideal gas law no longer holds it does and in fact it can be
combined with the adiabatic law for pressure given above to obtain the adiabatic law
for temperatures
1 constantTV
Compressions in sound waves are adiabatic because they happen too rapidly for any
appreciable amount of heat to flow This is why the adiabatic exponent appears in the
formula for the speed of sound in an ideal gas
RTv
M
Note that v depends only on T and not on P Because it depends only on the
temperature the speed of sound is the same in Provo as at sea level in spite of the
lower pressure here due to the difference in elevation
Equipartition of Energy
Every kind of molecule has a certain number of degrees of freedom which are
independent ways in which it can store energy Each such degree of freedom has
associated with it ndash on average ndash an energy of 12 Bk T per molecule (or 1
2 RT per mole)
(Note since a molecule has so many possible degrees of freedom it would seem that
there should be a lot of 12 sBk T to spread around But because energy is quantized
some of these degrees of freedom are not ldquoactiverdquo until the temperature becomes high
enough that 12 Bk T is as big as the lowest quantum of energy
8
Serway Chapter 22
Second Law of Thermodynamics
There are several equivalent forms of this important law
Kelvin It is not possible to change heat completely into work with no other change
taking place Or in other words there are no perfect heat engines
Clausius It is not possible for heat to flow from one body to another body at a higher
temperature with no other change taking place Or in other words there are no
perfect refrigerators
Entropy In any thermodynamic process that proceeds from one equilibrium state to
another the entropy of the system + environment either remains unchanged or
increases The total entropy never decreases This law is a bit of an oddity among the
laws of physics because it is not absolute Things are forbidden by the second law not
because it is impossible for them to happen but because it is extremely unlikely for
them to happen (See below for more information about entropy)
Reversible and Irreversible Processes
A reversible process is one which occurs so slowly that it is in thermal equilibrium (or
very nearly so) at all times A hallmark of such processes is that a motion picture of
them looks perfectly normal whether run forward or backward Imagine for instance
the slow expansion of a gas at constant temperature in a cylinder whose volume is being
increased by a slowly moving piston Run the movie backwards and what do you see
You see the slow compression of a gas at constant temperature which looks perfectly
normal
An irreversible process is one which occurs in such a way that thermal equilibrium is
not maintained throughout the process The mark of this kind of process is that a motion
picture of it looks very odd when run backward Imagine the sudden expansion of a gas
into a previously evacuated chamber because a hole was punched in the wall between a
pressurized chamber and the evacuated one Run the movie backward and what do you
see You see the gas in the soon-to-be-evacuated chamber gather itself together and
stream through a tiny hole into a chamber in which there is already plenty of gas If you
have ever seen this happen get in touch with the support group for those who have
witnessed the spontaneous combustion of large mammals by calling 1-800-PYROCOW
9
Heat Engines
Because of the vexing difference in sign between work done on and system and work done
by a system we will invent a new work variable Weng Heat engines do work and so the net
W for these engines is negative But in engineering applications hidden minus signs are
regarded as evil so for heat engines we donrsquot talk about W instead we talk about its
magnitude engW W So for heat engines the first law is
int engE Q W
But for heat pumps and refrigerators work is done on the system so we use the usual work
W when we talk about these systems
A heat engine is a machine that absorbs heat converts part of it to work and exhausts the
rest The heat must be absorbed at high temperature and exhausted at low temperature If the
absorbed heat is Qh the exhausted heat is Qc and the work done by the engine is Weng then
eng h cW Q Q
and the efficiency of the engine is defined to be
eng
h
We
Q
A perfect engine would convert the heat hQ completely into work Weng giving an effi-
ciency of e = 1 Energy conservation alone allows a perfect engine but the second law
requires e lt 1
Refrigerators and Heat Pumps
A refrigerator is a machine that absorbs heat at low temperature and exhausts it at high
temperature the ldquobackwardsrdquo heat transfer being driven by the work done on the machine
by some source of power A heat pump is a machine that either works like a refrigerator
keeping a place cold by transferring heat from this cold place to a higher temperature
environment (cooling mode like an air conditioner) or it functions as a heater
transferring heat into a warm place from a cooler one (heating mode like a window unit
that heats a house by extracting thermal energy from the cold outdoors) The coefficient
of performance of a refrigerator or of a heat pump in cooling mode is defined to be
COP cooling modecQ
W
10
For a heat pump in heating mode the coefficient of performance is
COP heating modehQ
W
Note that we donrsquot have to use the engineering work here because in these systems
work is done on the system and W is naturally positive
A perfect refrigerator would take in heat Qc and exhaust the same amount of heat without
doing any work giving an infinite coefficient of performance Energy conservation alone
allows a perfect refrigerator but the second law requires COP lt infin A perfect heat pump
in heating mode would transfer Qh into the house without doing any work and so would
also have an infinite coefficient of performance The second law forbids this too A good
coefficient of performance for a real device would be around 5 or 6
Carnot Cycle
The most efficient of all possible engines is one that uses the Carnot cycle This cycle
employs an ideal gas has no friction and operates very slowly so that the gas can be in
thermal equilibrium at all parts of the cycle This means of course that it canrsquot
possibly be built and even if it could be built it would not run fast enough to be useful
Nevertheless this cycle is very important because it gives an upper bound on the
efficiency of real engines There cannot possibly be an engine that is more efficient
than one based on the Carnot cycle This cycle consists of the following four steps
1 The ideal gas absorbs heat Qh at constant temperature Th while the gas increases its
volume The reason that heat is absorbed is that expansion tends to cool the gas but
thermal contact with the environment at Th keeps the temperature high by heat
conduction into the ideal gas
2 The ideal gas further increases its volume by an adiabatic expansion This expansion
causes the gas to cool so at the end of this part of the cycle the gas is at temperature Tc
3 The gas exhausts heat Qc at constant temperature Tc while the gas decreases in
volume The reason that heat is exhausted is that compression tends to heat the gas
but thermal contact with the environment at Tc keeps the temperature low by heat
conduction out of the ideal gas
4 The gas is adiabatically compressed back to its original volume (the volume it started
with in step 1) This compression heats the gas from Tc up to Th
11
The efficiency of a Carnot engine is given by the very simple formula
1 cC
h
Te
T
where the temperatures must in be Kelvin No real engine can be more efficient than
this
The coefficient of performance of a Carnot refrigerator or heat pump in cooling mode
is given by
COP cooling mode cC
h c
T
T T
and no real refrigerator can have a coefficient of performance greater than this
The coefficient of performance of a Carnot heat pump in heating mode is
COP heating mode hC
h c
T
T T
So why donrsquot we just use these wonderful Carnot engines and have perfect efficiency
To make the reversible steps in the cycle really reversible they would have to occur
infinitely slowly So the price you pay for making a perfect engine is that it takes
forever to get it to do any work
Entropy
The entropy of a system is defined in terms of its molecular makeup and measures
roughly the disorder of the system If the system is packed into a very small volume
then it is quite ordered and the entropy will be low If it occupies a large volume the
entropy is high (To see what this has to do with disorder note that socks in a drawer
occupy a small volume while socks on the bed in the corner by the door and
hanging from the chandelier occupy a large volume) If the system is very cold then
the molecules hardly move and may even reach out to each other and form a crystal
This is a highly ordered state and therefore has low entropy If the system is very hot
with rapidly speeding molecules crashing into the container walls and bouncing off
each other things are disordered and the entropy is high
It is possible to calculate the entropy of a system in terms of its macroscopic thermody-
namic properties ie pressure volume temperature number of moles etc The key to this
calculation is the concept of a reversible process A reversible process is one that is
12
carried out without friction and so slowly that the process can be reversed at any stage by
making an infinitesimal change in the environment of the system The slow expansion of the
gas at Th in the Carnot cycle with tiny amounts of heat being transferred to the environment
is reversible If we turned around and began slowly to compress the gas would just slowly
exhaust heat to the environment in the exact reverse way that it absorbed it during expansion
Most processes however are irreversible For example if a gas-filled box were suddenly
increased in size so that the particles were free to wander into the void created by the sudden
expansion then the gas would eventually fill the new volume uniformly at the same
temperature as before the expansion (The temperature is unchanged in this imaginary
process because the kinetic energy of the molecules would be unaffected by such an
instantaneous expansion of the container walls) This imaginary but highly thought-
stimulating process is called a free expansion and it is impossible to reverse it During the
expansion we didnrsquot push on any of the molecules so reversing this process would mean
making them go back into their original volume without pushing on them they simply will
not cooperate to this extent Another way to see that just pushing them back where they came
from does not reverse the free expansion is to think about what would happen if we just
compressed either adiabatically or isothermally An adiabatic compression back to the
original volume would heat the gas above its original temperature and an isothermal
compression would require that heat be exhausted to the environment But the free expansion
involved neither temperature changes nor heat exchanges so neither of these two processes
is the reverse of the free expansion It is simply impossible to reverse this rapid expansion in
a way that takes us back to the initial state
It is possible to calculate the change of entropy for both reversible and irreversible processes
Letrsquos consider a reversible process first In a reversible process the entropy change is given
by the formula
dQS
T
where dQ is the amount of heat added to the system during a small step of the process
The total energy change during the process may then simply be calculated by integration
f f
f i i i
dQS S S dS
T
13
(Just as in the case of energy we are mainly interested in differences rather than in
absolute magnitudes)
This integration method does not work for irreversible processes For instance in the
rapid free expansion discussed above no heat is added to the system but its disorder
obviously goes up We need to find some other way to calculate the entropy The key
is the fact that the entropy of a system depends only on its current state and not at all
on how it arrived there So to calculate the entropy change in an irreversible process
first find out what the initial and final conditions of the process are Then invent a
reversible process that takes the system from the initial state to the final state Since
the entropy depends only on the state of system and not on the process the entropy
change for the reversible process is the same as that for the irreversible process
Entropy is thus like pressure or temperature if the pressure changes from Pl to to P2
then the pressure difference is P2 - Pl regardless of how the pressure change was
made This seems obvious for pressure but not for entropy only because you arent
comfortable with entropy yet
Entropy of an Ideal Gas
For n moles of an ideal gas the difference in entropy between a state with temperature
T and volume V and some standard state with temperature To and volume Vo is given
by
ln lno o Vo o
T VS T V S T V nC nR
T V
Entropy in a Phase Change
Calculating entropy change in a phase change like melting or boiling is easy It is
QT Where Q is in the latent heat for example the flows in to cause the melting and
T is the temperature of the phase change There is NO integral to do
14
Serway Chapter 35
Angle of Reflection
If plane waves are incident on a reflecting surface with the propagation direction of
the waves making angle θ1 with the normal direction to the surface then the reflected
angle θ1prime relative to the surface normal is simply
1 1
ie the incident angles and reflected angles are the same
Refraction
If plane waves traveling through medium 1 are incident on a plane interface between
medium 1 and medium 2 then the angle of incidence of the incoming wave θ1 and
the angle of refraction of the transmitted wave θ2 are related by Snellrsquos law
2 2
1 1
sin
sin
v
v
where v1 and v2 are the wave speeds in medium 1 and medium 2 The angle of
incidence and the angle of refraction are both measured between the wave propagation
direction and the normal to the interface In terms of indices of refraction in the case
of light waves Snellrsquos law takes the more familiar form
1 1 2 2sin sinn n
where ni = cvi
Total Internal Reflection
If a wave is incident from a medium of low wave speed into a medium of high wave
speed the law of refraction requires that the angle of refraction be greater than the
angle of incidence If the angle of refraction is required to be greater than 90deg then no
refracted wave can exist and total internal reflection occurs The critical incident
angle θc beyond which total internal reflection occurs is given by
2
1
sin c
n
n
15
Dispersion of Light
In addition to the speed of light varying from material to material it also varies with
wavelength within each material This means that the index of refraction is generally a
function of wavelength
c
nv
Since the wave speed is not constant such a medium is dispersive meaning in this
context that refraction actually disperses white light into its various colors because
Snellrsquos law gives a different angle for each wavelength In most materials the
variation with wavelength is quite small but this small effect is responsible for some
of the most spectacular color effects we ever see including rainbows a flashing
crystal chandelier and the colored fire of a diamond solitaire by candlelight
16
Serway Chapter 36
Real and Virtual Images
When light rays are focused at a certain plane producing an image if a sheet of white
paper is placed there we call the image a real image The images produced by film
projectors and overhead projectors are examples of real images
When light rays appear to come from a certain location but no image is produced
when a screen is placed there we say that there is a virtual image at that location For
instance when you look in a mirror it appears that someone is behind the mirror but
a screen placed back there in the dark would show nothing Your image in the mirror
is a virtual image
Ray Tracing
There are lots of rules about how to find the images in optical systems but the best
way to keep things straight is to learn how to draw the principal rays for curved
mirrors and lenses The rays for convex and concave mirrors are shown in Fig 3615
The rays for converging and diverging thin lenses are shown in Fig 3627 You should
memorize the principal rays and know how to use them to locate images
Curved Mirrors
The focal length of a curved mirror with radius of curvature R is given by
2
Rf
The relation between object distance p image distance q and focal length f is
1 1 1
p q f
If the mirror is a diverging mirror f should be negative and if q should turn out to be
negative the image is virtual
Lateral Magnification
The lateral magnification in an optical system is defined by the ratio of the image size
to the object size
17
Image height
Object height
qM
p
for curved mirrors
As usual there are sign conventions here too but this formula just gives the
magnitude It is better to keep keep track of upright images versus inverted images by
means of ray diagrams rather than by memorizing sign conventions
Thin Lenses
The focal length of a thin lens is related to the radii of curvature of the two faces R1
and R2 of the lens by
1 2
1 1 11n
f R R
Note that this formula differs from Eq (3611) in the text by not having a minus sign
between the two R-terms We like this form better because for a simple converging
lens like a magnifying glass we just use positive values of R for both surfaces If one
of the faces is concave producing divergence use a negative value for R And if a
surface is flat use R = infin If the face is flat the radius is infinite
The relation between the image object and focal distances for a thin lens is the same
as that for a curved mirror
1 1 1
q p f
Use a negative focal length if the lens is diverging
The lateral magnification for a thin lens is the same as for a curved mirror
Image height
Object height
qM
p
Camera
The lens system in a camera projects a real image of an object onto the film (or CCD
array in a digital camera) The position of the image is adjusted to be on the film by
moving the lens into or out of the camera
18
Eye
The eye is like a camera in that a real image is formed on the retina Unlike a camera
the image position is adjusted by changing the focal length of the lens This is done by
the ciliary muscle which squeezes the lens changing its shape
Near Point The near point is the closest distance from the eye for which the lens can
focus an image on the retina It is usually 18-25 cm for young persons
Far Point The far point is the greatest distance from the eye for which the lens can
focus an image on the retina For a person with normal vision the far point is at
infinity
Nearsightedness A person is nearsighted if their far point is at some finite distance
less than infinity This condition can be corrected with a lens that takes an object at
infinity and produces a virtual image at the personrsquos far point
Farsightedness and Presbyopia A person is farsighted if their near point is too far
away for comfortable near work like reading or knitting This can be corrected by a
lens which takes an object at a normal near point distance of 18-25 cm and produces a
virtual image at the personrsquos natural near point Presbyopia involves a similar
problem which nearly all people experience as they age The ciliary muscle becomes
too weak and the lens becomes too stiff to allow the eye to provide for both near and
far vision The solution for this problem is either reading glasses or bifocal lenses
Reading classes are just weak magnifying glasses mounted on eyeglass frames
Bifocal lenses are split into upper and lower halves The lower half is a lens which
gives the proper correction for near work and the upper half is a different lens for
proper focusing at infinity
Angular Size
When an object is brought closer to the eye it appears to be larger because the image
on the retina is larger The size of this image is directly proportional to the objectrsquos
angular size which is the angle subtended by the object measured from the center of
the lens of the eye In optical instruments which are to be used with the eye the
angular size of the final image is whatrsquos important because it determines how large
the image will appear to the viewer
19
Simple Magnifier
A simple magnifier is a single converging lens or magnifying glass It takes an object
closer to the eye than a normal near point and produces a virtual image at or beyond
this near point The angular magnification is defined to be the ratio of the angular size
when viewed through the lens to the angular size of the object when viewed at the
normal near point (without aid of the lens)
Microscope
This instrument has two lenses (1) The objective is near the object being viewed and
produces a greatly magnified real image (2) The eyepiece is a simple magnifier which
the viewer uses to closely examine the image from (1)
Telescope
This instrument also has two lenses (1) The objective at the front of the telescope
takes light from a distant object and produces a real inverted image (which is rather
small) near its focal point (2) This small real image is then examined by the eyepiece
functioning as a simple magnifier to produce a virtual image with a larger angular
size
20
Serway Chapter 37
Two-Slit Interference
If light is incident on two closely spaced narrow slits a pattern of light and dark
stripes is produced beyond the slits The bright stripes or fringes are caused by
constructive interference of the two waves coming from the slits Constructive
interference occurs whenever two waves arrive at a location in phase with each other
This occurs when the distance x1 from slit 1 to a point P on the screen and the
distance x2 from slit 2 to point P differ by in integral number of wavelengths
1 2 where 0 1 2x x m m
where λ is the wavelength of the light When x1 and x2 are much larger than the slit
spacing d this condition reduces to
sind m
where θ is the angle between the direction of the incident light and the direction of the
light arriving at the screen
Thin Films
When light is partially reflected and partially transmitted by a thin film of transparent
material it is possible to have interference between the wave reflected from the front
of the film and light reflected from the back of the film (The colored reflections from
the thin film of oil on the water in a rain-soaked parking lot are an example of this
effect) It is difficult to write down formulas that will work in all cases so we will
just review the important principles here
1 If the two reflected waves are in phase with each other the film has enhanced
reflection (constructive interference) but if the two reflected waves are out of phase
with each other reflection is diminished (destructive interference) Phase shifts occur
due to reflection and due to the extra path length through the film of the wave
reflected from the back of the film
2 The phase change due to reflection is determined by the difference in index of
refraction between the two media involved in the reflection If the wave is incident
21
from a medium with a low index of refraction into a medium with a high index of
refraction a phase change of 180deg occurs and the reflected wave is inverted If
incident from high to low no phase shift occurs and the reflected wave is non-
inverted
3 The extra path length through the film of the wave reflected from the back of the
film is equal to 2t where t is the thickness of the film (The incident light is assumed
to be normal to the surface of the film) The number of wavelengths contained in the
extra path length is equal to 2tλn where λn = λn is the wavelength of the light in the
film
4 Rules for reflection from thin films
If one ray is inverted and the other is not then we have
12 constructive
2 nt m
2 destructivent m
If either both rays are inverted or both are non-inverted we have
2 constructivent m
12 destructive
2 nt m
22
Serway Chapter 38
Diffraction Grating
A diffraction grating is simply a fancy version of two-slit interference with the two
slits replaced by thousands of slits Just as in the two-slit case the bright fringes
occur at angles given by
sin md
but in this case d the distance between neighboring slits is made to be very small
The effect of having many slits instead of two is to make each bright fringe highly
localized with wide dark regions between neighboring maxima
Single Slit Diffraction
When light passes through an opening in an opaque screen an interference pattern is
produced beyond the opening To understand why we may replace the single opening
by many small coherent sources of light These many sources interfere with each
other producing a pattern known as a diffraction pattern (Note that many authors do
not distinguish between interference and diffraction treating them as interchangeable
terms) If the opening is a slit of width a then the diffraction pattern far from the slit
will have a bright central maximum with a succession of minima and weaker maxima
on either side The angle between the incident direction and the minima is given by
sin ma
where m = plusmn1 plusmn2
If the opening is circular with diameter D the angle between the incident direction
and the first minimum is given by
sin 122D
Optical Resolution and Rayleighrsquos Criterion
Two point sources can just be resolved (distinguished from each other) if the peak of
the diffraction image of the first source overlies the first minimum of the diffraction
image of the second source For circular holes of the kind usually encountered in
23
optical devices this condition is approximately satisfied when the angular separation
between the two sources as viewed from the optical instrument is greater than or equal
to the critical angle
min 122D
where λ is the wavelength of the light and where D is the diameter of the aperture in
the instrument
Polarization
We say that an electromagnetic wave is polarized if its electric field vector doesnrsquot
change direction in a random fashion The simplest kind of polarization is linear
polarization in which the electric field vector oscillates back and forth along the
same axis in space Polarized light can be produced from normal unpolarized light by
selective absorption (as in Polaroid sunglasses) by passing light through crystals that
have different indices of refraction for different polarizations (double refraction) by
scattering (the blue sky is polarized) and by reflection (glare) Polarization by
reflection occurs when light reflects from a shiny insulating (non-metallic) surface
The amount of polarization is greatest for reflection at Brewsterrsquos angle
2
1
tan P
n
n
where θP is the incidence angle of light from medium 1 onto medium 2 and where n1
and n2 are the indices of refraction for the media (In our everyday experience n1 = 1
since the light comes in through the air and n2 is the index of refraction of the shiny
insulating material producing the glare eg water glass plastic paint etc)
Malusrsquos Law
The intensity of transmitted polarized light through a perfect polarizer is related to the
incident intensity of polarized light by Malusrsquos law
2cosoI I
where Io is the intensity of the incident light and where θ is the angle between the
electric field vector in the incident wave and the transmission axis the polarizer
24
Serway Chapter 39
Principles of Relativity
All of the weirdness of relativity flows from two simple principles
(1) The laws of physics must be the same in all inertial (non-accelerating) reference
frames
(2) The speed of light in vacuum has the same value in all inertial reference frames
Note that (1) seems reasonable but (2) is very odd It says that if two space ships are
approaching each other at nearly the speed of light and a laser pulse is shot from ship
1 toward ship 2 then when the people on ship 2 measure the speed of the pulse as it
goes by it is moving at 3 times 108 ms the same as if the ships were stationary
Simultaneity
If observer 1 sees two events in her own frame as simultaneous at two different
locations a moving observer 2 will see these two events happening at different times
Relativistic gamma
The factor γ (gamma) appears regularly in the formulas of relativity
2
2
1
1 vc
where v is the relative speed between two inertial frames
Time Dilation
If observer 1 sees two events at the same location in space separated by time Δtp in his
own frame then observer 2 moving at speed v relative to observer 1 will see these
two events separated by a longer time Δt
pt t
Moving clocks run slow
Length Contraction
If observer 1 measures the length of an object along the x-direction at some instant of
time in her frame to be Lp then observer 2 moving in the x-direction at speed v will
measure the length L of the object to be shorter
pLL
25
Moving meter sticks are short
Velocity Addition
If a particle is moving at velocity vac relative to frame c and if frame c is moving at
speed vcb with respect to frame b (with both velocities directed along the same line)
what would be the velocity of the particle in frame b In classical physics the answer is
ab ac cbv v v
but if any of these velocities are an appreciable fraction of the speed of light the
answer changes
21ac cb
abac cb
v Vv
v v c
In using this equation make sure that vac has the same sign in the numerator and in the
denominator and that vbc also has the same sign in both places Note that this equation
is different (and in our opinion easier to understand) than the equations in the text
Momentum and Energy
The momentum and total energy of a particle moving at velocity v (its speed is v v )
are given by
22
2 2
22
1 1 vvc c
m mcm E mc
vp v
Rest Energy
When a particle is at rest (v = 0 and γ = 1) then E = mc2 is called the rest energy It
includes the potential energy of the particle so that if we change the potential energy
of a particle we change its rest mass
Kinetic Energy The kinetic energy is the difference between the total energy and the
rest energy
2 2K mc mc
For v ltlt c this reduces to K = mv22
General Relativity
Einsteinrsquos principle of equivalence in the theory of general relativity is In the vicinity
of a point a gravitational field is equivalent to an accelerated frame of reference in
26
the absence of gravitational effects For example there is no difference whatever in
the physics experienced by an astronaut on earth and one who is in interstellar space
far away from any stars or planets but who has just turned on her spaceshiprsquos engines
and is accelerating at 98 ms2
27
Serway Chapter 40
Photons
We now know that even though light behaves like a wave it is better described as
consisting of small packets of energy called photons The energy of a photon is related
to its frequency by
E hf
where h is Planckrsquos constant
346626 10 J sh
Photons also carry momentum given by
hf hp
c
Since the amount of energy in each photon is fixed the difference between dim light
and intense light (of the same wavelength) is that dim light consists of fewer photons
than intense light
Photoelectric Effect
In the photoelectric effect light shining onto a metal surface gives the electrons in the
metal enough energy to escape and be detected It requires a certain amount of energy
(called the work function typically = 3 ndash 5 eV) just to kick the electron out of the
metal so the light must deliver at least this much energy to an electron to produce the
effect It is observed that red light no matter how intense never produces electrons
But ultraviolet light even if quite dim will eject electrons from the metal Since red
light consists of 2 eV photons and ultraviolet light has photons with energies around
4-6 eV the photon idea explains the behavior of the photoelectric effect The
maximum energy that an ejected electron can have is
maxK hf
where f is the photon frequency The minimum frequency that light can have and cause
electrons to be ejected is called the cutoff energy and is found by setting Kmax = 0
cf h
28
Compton Effect
When high-frequency light interacts with free electrons the scattered light does not
have the same wavelength as the incident light contrary to what classical
electromagnetism would predict But the photon picture in which photons have
momentum and energy allows us to treat scattering as a collision between the photon
and the electron In this collision the electron and the scattered photon both have a
different momentum and energy than they did before And since p = hλ for a photon
if the momentum is different the wavelength will also be different
after before 1 cose
h
m c
where the angle θ is the angle between the incoming photon direction and the
direction of the scattered photon
Particles are Waves
Since photons behave like particles sometimes it is not surprising that elementary
particles can behave like waves sometimes The wavelength of a particle with
momentum p is given by
h
p
Wave-particle Duality
Both photons and elementary particles have a dual nature sometimes they behave like
particles and sometimes they behave like waves It is hard for us to comprehend the
nature of such an object by making mental pictures but experiments definitively show
that this is the case Since we have no direct experience with photons and elementary
particles (because their energies are so much smaller than the energies of the everyday
objects we encounter) it is perhaps not surprising that we have a hard time forming a
mental picture of how they behave
Electron Interference
Since an electron is both a particle and a wave just like a photon it should be able to
produce an interference pattern This is observed If an electron beam is shot at two
closely-spaced slits and if the electrons that pass through the slits are detected
downstream it is found that there are some locations where electrons are never detected
29
and others where lots of electrons are detected The pattern is exactly the same as the
one observed for light waves provided that we use the electron wavelength λ = hp in
place of the wavelength of light This pattern is observed even though each electron is
detected as a single dot on the screen Only after many such dots are collected does the
pattern emerge And if we try to understand how this effect could possibly work by
looking closely at each slit to see which one the electron came through the pattern
disappears the act of measurement destroys the interference This means that each
single electron somehow comes through both slits (as a wave would)
Uncertainty Principle
In classical physics we always imagine that the positions and momenta of moving
particles have definite values It might be hard to measure them but surely at each
instant of time a particle should be precisely located at some point in space and have a
similarly precise momentum This turns out not to be true Instead both position and
momentum are required to be uncertain with their uncertainties Δx and Δp satisfying
the Heisenberg uncertainty relation
2x p
So if the particle were known to be precisely at some particular location (so that Δx =
0) we couldnrsquot know anything about its momentum (Δp = infin) and if its momentum
were exactly known we couldnrsquot know its position
There is a similar relation involving the particlersquos energy E and the time interval Δt
over which this energy is measured
2E t
This means that energy is not actually conserved if we are considering very small time
intervals and this brief non-conservation of energy has been observed
30
Serway Chapter 41
Wave Function ψ and Probability
Quantum mechanics does not predict exactly what an electron or a photon will do Instead
it specifies the wave function or probability amplitude ψ of an electron or a photon
This wave function is a complex-valued function of space and time whose squared
magnitude is the probability density P for finding a particle at a particular place in
space at a certain time
2P
where is the complex conjugate of ψ
For example the wave function of an electron with perfectly specified momentum p
would have a wavelength given by
h
p
and its probability amplitude would be proportional to
2i x ipxe e
The corresponding probability density would then be
21ipx ip ipxP e e e
which means that the electron is equally probable to be anywhere along the x axis
This is in accord with the uncertainty principle since we specified the momentum
precisely we canrsquot have any idea about the position of the electron
As another example you have probably seen ldquofuzzy ballrdquo drawings of electron
orbitals in a chemistry book These fuzzy balls are meant to indicate the distribution
of the probability density 2
P in the orbital
Particle in a Box
A simple example in which we can calculate the wave function is the case of a particle of
mass m constrained to be inside a 1-dimensional box between x = 0 and x = L In this
case the general wave function is a linear superposition of wave functions ψn of the form
sin niE tn
n xx t A e
L
31
where A is a positive constant where n = 1 2 3 and where the energy associated
with each of the quantum states ψn is given by
22
28n
hE n
mL
This wave function is zero at x = 0 and x = L which means that the particle will never
be found at the walls of the box The wave function has maximum values in the
interior and at these places the particle is most likely to be found and it also has
places where it is zero and at these places the particle will also never be found as
expressed by the formula
2sinn x
PL
The particle in the box is interfering with itself producing a probability interference
pattern across the box just like the interference patterns we studied with light and
sound
32
Serway Chapter 42
Atoms
Many people picture an atom as a miniature solar system where electrons orbit around
a massive nucleus at the center This picture is misleading because in atoms the wave
nature of electrons dominates The electrons form 3-dimensional standing waves
(called orbitals) centered at the nucleus And if you ask what is it thatrsquos waving the
answer is ψ the probability amplitude (see Chapter 41)
Standing waves on a string can occur only for certain frequencies (the fundamental
and higher harmonics) Similarly atomic orbitals occur only for certain energies For
the hydrogen atom the energies of the orbitals have the a particularly simple form
2
1136 eVnE
n
where n = 1 2 3 is called the principal quantum number For other atoms the
determination of the orbital energies requires numerical calculation by computers
Atomic Spectra
If an electron is somehow given extra extra energy (we say that it is excited) so that it
occupies a higher orbital it will eventually ldquofallrdquo back down to a lower orbital Each
time an electron falls to a lower orbital it loses the difference in energy between the
two orbitals in the form of a photon Since the orbital energies are discrete so are the
energy differences and so are the wavelengths of the emitted light The entire set of
these discrete wavelengths is called the atomic spectrum and it is unique to each type
of atom For hydrogen the spectrum can be simply written as
H 2 2
1 1 1
f i
Rn n
where RH = 10973732 times 107 m-1 For other atoms the spectrum cannot be expressed as
a formula
Orbital Angular Momentum Electrons in atoms also have quantized values of
angular momentum The orbital quantum number ℓ specifies the value of this
quantized angular momentum through the formula
1L
33
If we want to know the value of the angular momentum along some direction in space
say the z direction the answer is not L but rather
zL m
where mℓ is another quantum number which runs from
1 1m
This quantum number is important when an atom sits in a magnetic field
Spin Angular Momentum It has been found experimentally that electrons and other
charged particles also carry internal angular momentum which we call spin
Electrons have an intrinsic spin angular momentum s along a specified axis that is
extremely quantized it can only take on 2 possible values
1
2zs s
We thus say that electrons have spin s = 12 or that electrons are rdquospin one-half parti-
cles Since s is an angular momentum it obeys the same rule as that for orbital angular
momentum namely that if its value along some axis is s then its total magnitude is
given by
31
2S s s
I know this seems weird but quantum mechanics is weird The only excuse for this
bizarre way of looking at the world is that it predicts what happens in experiments
Exclusion Principle The answer to the question of how many electrons (or any other
spin one-half particle) can be in one particular quantum state was discovered by
Wolfgang Pauli and is called the exclusion principle
ldquoNo two electrons can ever be in the same quantum state therefore no two electrons
in the same atom can have the same set of quantum numbersrdquo
This is the reason that we have atoms with different properties instead of every atom
simple having all of its electrons in the ground state All of the variety we see around
us in the world is the result of chemical differences and these differences would not
exist unless electrons obeyed this important principle The entire structure of the
periodic table (see pages 1377-1379 in Serway) is an expression of this principle
34
Serway Chapter 44
Nuclear Properties
The nucleus is composed of protons with charge +e and mass mp = 1007226 u and of
neutrons with zero charge and mass mn = 1008665 u where u is one atomic mass unit
-271 u=1660540 10 kg
The atomic number Z counts the number of protons in a nucleus while the neutron
number N counts the number of neutrons The mass number A is the sum of the two
A N Z
Protons and nuetrons have about the same mass and some times called baryons (heavy
ones) or nucleons (partcles in the nucleus) A is sometimes called the baryon
number The chemical elements are distinguished by Z ie hydrogen has Z = 1 iron
has Z = 26 uranium has Z = 92 etc But for a given Z there might be several nuclei
with different numbers of neutrons These nuclear siblings are called isotopes
These numbers are used to label nuclei according to the pattern
5626 ie FeA
Z X
denotes the isotope of iron with N = 56 ndash 26 = 30 neutrons
The nucleus is roughly spherical with a radius given approximately by
1 3 150 0where 12 10 mr r A r
Nuclear Stability
Because the positively charged protons electrically repel each other with an enormous
force at distances as small as 10-15 m there must be some really strong force that that
overcomes electrical repulsion to hold protons and neutrons together This force is
called with some lack of imagination the strong force It is a very short-range force
(it only acts over a distance of about 2 x 10-15 m) and attracts protons to protons
neutrons to neutrons and neutrons to protons But in spite of this strong nuclear force
the coulomb repulsion of the protons is still present so anything that might keep the
protons from being right next to each other would help keep the nucleus from
35
exploding This role is played by the neutrons and for nuclei with Z le 20 the stable
nuclei roughly have N = Z
For Z gt 20 the coulomb repulsion force is more powerful and more neutrons are
needed to dilute it up to about N = 15Z around Z = 80 For Z gt 83 no amount of
neutrons can help and these nuclei are unstable (radioactive) There is an ldquoisland of
(relative) stabilityrdquo around Z=90-92 (thorium and uranium respectively) These
elements have one or two isotopes with half-lifes of billions of years so there are
substantial amounts of such elements on Earth This fact makes it possible to have
practical fission devices
Radioactive Decay
There are three types of radioactive decay
Alpha decay the nucleus kicks out a helium nucleus (N = 2 Z = 2 A = 4) So if the
original nucleus is called X and the new nucleus is called Y then the decay would look
like this
4 42 2X Y+ HeA A
Z Z
Beta decay the nucleus either kicks out an electron (endash) or its positively-charged
antimatter twin the positron (e+) plus either an electron neutrino v or an electron
anti-neutrino v
1X Y+e electron decayA AZ Z v
1X Y+e positron decayA AZ Z v
A neutrino is a particle with no charge hardly any mass (much less than the electron
mass) and interacts so weakly with matter that most neutrinos upon encountering the
planet earth just pass right through it as if it werenrsquot there
Gamma decay the nucleons in the nucleus X are in an excited energy state X
(perhaps as a result of having undergone alpha or beta decay) and they drop down to a
lower energy state shedding the energy as a high frequency photon
X XA AZ Z
36
This process is exactly analogous to the way that the electrons in atoms emit photons
Decay Rate and Half Life
There is no way to predict exactly when an unstable or excited nucleus will decay but
there is an average rate at which this decay occurs called the decay constant λ The
meaning of this constant is that if there are a large number N of nuclei in a sample
then the number of decays per second that will be observed (called the decay rate R)
is R = λN In mathematical language
dMR N
dt
This simple differential equation has for its solution
0tN t N e
where N0 is the number of nuclei in the sample at time t = 0
The half-life is the time it takes for half of the nuclei in the sample to decay and is
related to the decay constant by
1 2
ln 2 0693T
Disintegration Energy
When a nucleus decays it is making a transition to an overall state of lower energy
which means according to Einsteinrsquos famous formula E = mc2 that the sum of the
masses after the decay must be less than the mass before with the lost mass appearing
as kinetic energy among the decay products For example in alpha decay this kinetic
energy called the disintegration energy Q is given by
2X YQ M M M c
37
Serway Chapter 45
Nuclear energy
There are two ways to extract energy from the nucleus fission and fusion For nuclei
with Z greater than 26 breaking the nucleus apart into pieces leads to a lower
2mc energy than the original nucleus so energy can be extracted by fission For
nuclei with Z less than 26 a lower energy is achieved by combining nuclei so energy
can be extracted by fusion
Fission
Since neutrons have no charge they are not repelled from nuclei as protons are for
this reason their behavior is the key to understanding how fission works
Fast neutrons mostly bounce off other nuclei with each collision slowing the neutron
down This elastic energy loss is most effective if the other nuclei have low mass (like
hydrogen) and these materials are called moderators because of their ability to slow
down fast neutrons
The reason that slowing neutrons is important is that slow neutrons are much more
likely to be absorbed by a nucleus which then leads to nuclear reactions of various
kinds For a few very large nuclei like uranium-235 and some plutonium isotopes
absorption of a slow neutron causes the nucleus to split into two large fragments plus
2 or 3 fast neutrons (energy is released in the process too) Thus one slow neutron
can produce energy plus 2 or 3 more neutrons which if moderated from fast to slow
can split 2 or 3 more nuclei leading to a runaway chain reaction If the reaction is
allowed to proceed unchecked it produces a large explosion If a neutron absorbing
material is added to the mix (like the cadmium in reactor control rods) it is possible to
keep the reaction under control and to extract the released energy as heat to drive
steam turbines and produce electricity
Fusion
Fusion involves mashing two nuclei together and since they are both charged and repel
each other this reaction is much harder to make go The nuclei must have enough energy to
overcome the coulomb repulsion which is why this reaction requires a high temperature
(hundreds of millions of degrees K) like that in the sun (which is burning hydrogen to
helium via fusion) or like that in the center of the fission explosion that is used to detonate
38
a hydrogen bomb
This reaction is of interest for power production in spite of this difficult temperature
requirement because of the abundance of fusion fuel on the planet There are about
012 g of deuterium ( 21H ) in every gallon of water on earth and it only costs about 4
cents to extract it The fusion energy available from this minuscule amount of
deuterium would run a 1000-Megawatt power plant for 10 seconds Water is so
abundant on earth that if fusion were to work we would have an essentially
inexhaustible source of energy
So why donrsquot we have fusion power plants Well the fuel is cheap but the match is
incredibly expensive The only way we know to control this difficult high-temperature
reaction is with large and expensive pieces of equipment involving either large
magnetic fields and complex high-power electromagnetic antennas or with gigantic
(football-fieldsized) laser facilities involving more than a hundred of the highest-
energy lasers ever built Power plants based on these current methods for controlling
fusion are unattractive to the fiscally-minded people who run the electric power
industry Hopefully better designs will be discovered as experiments continue
39
c actinium 68 Er erbium 101 Md mendelevium 104 Rf rutherfordium l aluminum 63 Eu europium 80 Hg mercury 62 Sm samarium m americium 100 Fm fermium 42 Mo molybdenum 21 Sc scandium b antimony 9 F fluorine 60 Nd neodymium 106 Sg seaborgium r argon 87 Fr francium 10 Ne neon 34 Se selenium s arsenic 64 Gd gadolinium 93 Np neptunium 14 Si silicon t astatine 31 Ga gallium 28 Ni nickel 47 Ag silver a barium 32 Ge germanium 41 Nb niobium 11 Na sodium k berkelium 79 Au gold 7 N nitrogen 38 Sr strontium e beryllium 72 Hf hafnium 102 No nobelium 16 S sulfur i bismuth 108 Hs hassium 76 Os osmium 73 Ta tantalum h bohrium 2 He helium 8 O oxygen 43 Tc technetium
boron 67 Ho holmium 46 Pd palladium 52 Te tellurium r bromine 1 H hydrogen 15 P phosphorus 65 Tb terbium d cadmium 49 In indium 78 Pt platinum 81 Tl thallium a calcium 53 I iodine 94 Pu plutonium 90 Th thorium f californium 77 Ir iridium 84 Po polonium 69 Tm thulium
carbon 26 Fe iron 19 K potassium 50 Sn tin e cerium 36 Kr krypton 59 Pr praseodymium 22 Ti titanium s cesium 57 La lanthanum 61 Pm promethium 74 W tungsten l chlorine 103 Lr lawrencium 91 Pa protactinium 92 U uranium r chromium 82 Pb lead 88 Ra radium 23 V vanadium o cobalt 3 Li lithium 86 Rn radon 54 Xe xenon u copper 71 Lu lutetium 75 Re rhenium 70 Yb ytterbium m curium 12 Mg magnesium 45 Rh rhodium 39 Y yttrium b dubnium 25 Mn manganese 37 Rb rubidium 30 Zn zinc y dysprosium 109 Mt meitnerium 44 Ru ruthenium 40 Zr zirconium s einsteinium
40
UNITS The SI units are given in parentheses Other commonly used units are given in terms of the SI units Symbols conform with the recommendations of the American National Standards Institute (ANSI) and the American Institute of Physics (AIP) Commonly used multiples of SI units
prefix symbol factor tera T 1012 giga G 109
mega M 106 kilo k 103
centi c 10-2 milli m 10-3
micro μ 10-6 nano n 10-9 pico p 10-12
femto f 10-15
length meter (m) angstrom 1 Ǻ = 10-10 m inch 1 in = 254 x 10-2 m foot 1 ft = 03048 m mile 1 mi = 1609 m
mass kilogram (kg) atomic mass unit 1 u = 1661 x 10-27 kgslug 1 slug = 1459 kg
force newton (N = kg bull ms2) dyne 1 dyn = 10-5 N pound 1 lb = 4448 N
pressure Pascal (Pa = kgm bull s2) atmosphere 1 atm = 1013 x 105 Pa poundssquare inch 1 psi = 6895 Pa cm of mercury 1 cm Hg = 1333 Pa bar 1 bar = 1000 x 105 Pa torr 1 torr = 1333 Pa
time second (s) minute 1 min = 60 s hour 1 h =3600 s
frequency hertz (Hz = s-1) radianssecond 1 rads = 12π Hz
energy joule (J = kg bull m2s2) erg 1 erg = 10-7 J electron volt 1 eV = 1602 x 10-19 J calorie 1 cal = 4187 J kilowatt-hour 1 kWh = 36 x 106 J British thermal unit 1 Btu = 1055 J
power watt (W = kg bull m2s3) horsepower 1 hp = 7457 W
charge coulomb (C = A bull s)
electric potential volt (V = kg bull m2 s3 bull A)
current ampere (A)
resistance ohm (Ω = kg bull m2s3 bull A2)
capacitance farad (F = s4 bull A2kg bull m2)
magnetic field tesla (T = kgs2 bull A) gauss 1 G = 10-4 T
magnetic flux weber (Wb = kg m2s2 bull A) maxwell 1 Mx = 10-8 Wb
magnetic inductance henry (H = kg bull m2s2 bullA2)
temperature kelvin (K) degrees Celsius 0degC = 27315 K
angle radian (rad) degree 1 deg = π180 rad revolution 1 rev = 2π rad
41
Some Physical Constantsa Quantity Symbol Valueb Atomic mass unit μ 1660 538 73 (13) x 10-27 kg
931494 013 (37) MeVc2 Avogadros number NA 6022 141 99 (47) x 1023 particlesmol Avogadrorsquos number 6022 x 1023 mol Bohr magneton
2B e
em 9274 008 99 (37) x 10-24 JT
Bohr magneton μB 927 x 10-24 JT Bohr radius
0
2
2e em e k
a 5291 772 083 (19) x 10-11 m
Boltzmanns constant B A
RNk 1380 650 3 (24) x 10-23 JK
Boltzmannrsquos constant kB 1380 x 10-23 JK Compton wavelength
C eh
m c 2426 310 215 (18) x 10-12 m
Deuteron mass md 3343 583 09 (26) x 10-27 kg 2013 553 212 71 (35) u
electron charge e 1602 x 10-19 C Electron mass me 9109 381 88 (72) x 10-31 kg
5485 799 110 (12) x 10-4 u 0510 998 902 (21) MeVc2
electron mass 911 x 10-31 kg Electron volt eV 1602 176 462 (63) x 10-19 J Elementary charge e 1602 176 462 (63) x 10-19 C Gas constant R 8314 472 (15) JKmol Gravitational constant G 6673 (10) x 10-11 Nm2kg2 Neutron mass mn 1674 927 16 (13) x 10-27 kg
1008 664 915 78 (55) u 939565 330 (38) MeVc2
neutron mass 1675 x 10-27 kg Nuclear magneton
2n p
em 5050 783 17 (20) x 10-27 JT
permeability constant μ0 1257 x 10-6 Hm permittivity constant ε0 8854 x 10-12 Fm Plancks constant h
2h
6626 068 76 (52) x 10-34 Js 1054 571 596 (82) x 10-34 Js
Planckrsquos constant h ħ
6626 x 10-34 Js 1055 x 10-34 Js
Proton mass mp 1672621 58 (13) x 10-27 kg 1007 276 466 88 (13) u 938271 998 (38) MeVc2
proton mass 1673 x 10-27 kg Rydberg constant RH 1097 373 156 854 9 (83) x 107 m-1 Speed of light in vacuum c 2997 92458 x 108 ms (exact) speed of light c 300 x 108 ms a
These constants are the values recommended in 1998 by CODATA based on a least-squares adjustment of data from different measurements For a more complete list see P J Mohr and B N Taylor Rev Mod Phys 72351 2000 b The numbers in parentheses for the values above represent the uncertainties of the last two digits
NOTE The ones in red are the ones from Appendix 13 The ones in yellow are the ones that are from the Serway book table but that were already on the Appendix 13 table
42
INDEX
Absolute zero 9
Adiabatic 17
Adiabatic exponent 18
Adiabatic process 17
Alpha decay 46
Amplitude 4
Angle of reflection 25
Angular frequency 3
Angular magnification 30
Angular momentum atomic 43
Angular size 29
Archimedes Principle 1
Atomic spectra 43
Atoms 43
Avogadrorsquos number 10
Beats 8
Bernoullirsquos Equation 2
Beta decay 46
Bifocals 29
Brewsterrsquos angle 34
British Thermal Unit Btu 11
Buoyancy 2
calorie 11
Calorie 11
Camera 28
Carnot cycle 21
Carnot efficiency 22
Celsius scale 9
Ciliary muscle 29
Coefficient of performance 21
Compton effect 39
Constructive interference 7
Continuity equation of 2
Contraction length 35
Convection 14
Curved mirrors 27
Decay constant 46
Decay rate radioactive 46
Decibel scale 5
Degrees of freedom 16
Density 1
Destructive interference 7
Diffraction single slit 33
Diffraction grating 33
Dilation time 35
Disintegration energy 47
Dispersion of light 26
Doppler effect 6
Double slit interference 31
Efficiency 20
Electron interference 39
Emissivity 15
Energy nuclear decay 47
Energy relativistic 36
43
Engineering work 20
Entropy 22
Entropy ideal gas 24
Equipartition of energy 18
Equivalence principle 36
Exclusion principle 44
Expansion thermal 9
Expansion coefficient linear 9
Eye 29
Fahrenheit scale 9
Far point 29
Farsightedness 29
First Law of Thermodynamics 12
Fission 48
Flux volume 2
Free expansion 23
Fringes 31
Fusion 48
Fusion heat of 11
Gamma relativistic 35
Gamma decay 46
General relativity 36
Half-life 46
Heat 11
Heat capacity 11
Heat conduction 13
Heat engine 20
Heat of fusion 11
Heat of vaporization 11
Heat pump 20
Heats of transformation 11
Hydrostatics 1
Ideal Gas Law 10
Images real and virtual 27
Intensity sound 5
Interference 7
Interference two-slit 31
Internal energy 12
Internal energy degrees of freedom 16
Irreversible process 19 23
Isotopes 45
Joule 11
Kelvin scale 9
Kinetic energy relativity 36
Kinetic theory 16
Latent heat 11
Length contraction 35
Linear expansion coefficient 9
Linear polarization 34
Linear superposition 7
Longitudinal wave 3
Loudness 5
Magnification lateral 27
Magnifying glass 30
Malusrsquos law 34
Microscope 30
Momentum relativistic 36
Muscle ciliary 29
44
Musical instruments 8
Musical scale 8
Near point 29
Nearsightedness 29
Nonlinear 7
Nuclear energy 48
Nuclear properties 45
Nuclear stability 45
Octave 8
Optical resolution 33
Orbital quantum number ℓ 43
Orbitals 43
Particle in a Box 41
Particles are waves 39
Pascalrsquos Principle 1
Period 3
Photoelectric effect 38
Photons 38
Pitch 8
Polarization 34
Power sound 5
Presbyopia 29
Pressure 1
Principal quantum number n 43
Principle of equivalence 36
Principle of linear superposition 7
Probability amplitude ψ 41
Processes thermodynamic 13
R-value 14
Radiation thermal 14
Radioactive decay 46
Radioactive decay rate 46
Ray tracing 27
Rayleighrsquos criterion 33
Reading glasses 29
Real image 27
Refraction 25
Refrigerator 20
Relativistic gamma 35
Relativity principles 35
Resolved for light sources 33
Rest energy 36
Reversible process 19 22
Rope wave speed 4
Second Law of Thermodynamics 19
Shock waves 6
Simple magnifier 30
Simultaneity 35
Single slit diffraction 33
Snellrsquos law 25
Sound speed 5
Specific heat 11
Spin Angular momentum 44
Standing waves 7
Stefanrsquos law 14
Strong force 45
Telescope 30
Temperature 9
45
Temperature Scales 9
Thermal conductivity 14
Thermal energy 12
Thermal expansion 9
Thin film interference 31
Thin lenses 28
Time dilation 35
Tone musical 8
Total internal reflection 25
Transverse wave 3
Traveling Waves 3
Two-slit interference 31
Uncertainty principle 40
Vaporization heat of 11
Velocity addition relativity 36
Virtual image 27
Volume flux 2
Wave function 41
Wave Function ψ and Probability 41
Wave speed 3
Wave-particle duality 39
Wavelength 3
Wavenumber 3
Wien Displacement Law 16
Work 11
9
Serway Chapter 19
Temperature
Formally temperature is what is measured by a thermometer Roughly high temperature is what
we call hot and low temperature is what we call cold On the atomic level temperature refers to
the kinetic energy of the molecules A collection of molecules is called ldquohotrdquo if the molecules
have rapid random motion while a collection of molecules is called ldquocoldrdquo if the random motion
is slow When two bodies are placed in close contact with each other they exchange molecular
kinetic energy until they come to the same temperature This is the microscopic picture of the
Zeroth Law of Thermodynamics
Absolute Zero
Absolute zero is the lowest possible temperature that any object can have This is the temperature
at which all of the energy than can be removed an object has been removed (This removable
energy we call thermal energy) There is still motion at absolute zero Electrons continue to orbit
around atomic nuclei and even atoms continue to move about with a small amount of kinetic
energy but this small energy cannot be removed from the object For example at absolute zero
helium is a liquid whose atoms still move and slide past each other
Temperature Scales
Kelvin Scale Absolute zero is at T = 0 K water freezes at T = 27315 K room temperature is
around T = 295 K and water boils at T = 373 K Note we donrsquot use a deg symbol Kelvin is
prefered SI Unit
Celsius Scale Absolute zero is at T = -273degC water freezes at T = 0degC room temperature is
around T = 22degC and water boils at T = 100degC
Fahrenheit Scale Absolute zero is at T = -459degF water freezes at TF = 32degF room temperature
is around T = 72degF and water boils at TF = 212degF TF =18TC + 32 Notice that temperature
differences are the same for the Kelvin and Celsius scales
Thermal Expansion
When materials are heated they usually expand and when they are cooled they usually contract
(Water near freezing is a spectacular counterexample it works the other way around) The
coefficient of linear expansion is defined by the relation
1
i
L
L T
10
where Li is the initial length of a rod of the material and ΔL is the change in its length due to a
small temperature change ΔT The coefficient of volume expansion is defined similarly
1
i
V
V T
where Vi is the initial volume of a piece of material and where ΔV is the change in its volume due
to a small temperature change ΔT
Avogadrorsquos Number (N or NA)
One mole of any substance corresponds to 6022 times 1023 molecules The atomic mass given on the
periodic table (p A32-33 in the text) is the mass in grams of one mole of atoms For example
the atomic mass of oxygen is 15999 gmol The mass of a single atom is given by the same
number in units of atomic mass units (u) For example the mass of an oxygen atom is 15999 u
(on the average)
Ideal Gas Law (an example of an equation of state)
When the molecules of a gas are sufficiently inert and widely separated that interactions between
them are negligible we say that it is an ideal gas The pressure P volume V and temperature T
(in kelvins) of such a gas are State Variables and are related by the ideal gas law
Bor PV=NkPV nRT T
where n is the number of moles of the gas where R is the gas constant
8314 Jmol KR
where N is the number of molecules and where kB is Boltzmannrsquos constant
231380 10 J KBk
It works well for air at atmosphere pressure and even better for partial vaccuums The relative
ease of measuring pressure and the linear relationship between pressure and temperature (if V
and n are held fixed) makes an ideal gas an ideal thermometer Thermometers can be based on
properties of solids or liquids but the behavior of these materials with temperature is more
complicated
11
Serway Chapter 20
Heat
Heat is energy that flows between a system and its environment because of a tempera-
ture difference between them The units of heat are Joules as expected for an energy
Unfortunately there are several competing units of energy They are related by
1 cal 4186 J 1 Cal 4186 J 1 Btu 1054 J
Heat Capacity
There is often a simple linear relation between the heat that flows in or out of part of a
system and the temperature change that results from this energy transfer When this
linear relation holds it is convenient to define the heat capacity C and the specific
heat c as follows
For the entire object the heat Q it absorbs is given by Q = C(Tf - Ti)
For a particular material the specific heat is defined by c = Cm which is the heat
capacity per unit mass so that
Q = mc(Tf ndash Ti)
Note C has units of energy (J or Cal)(Kelvin kg)
Heats of Transformation or Latent Heat Q = plusmn mL
When a substance changes phase from solid to liquid or from liquid to gas it absorbs
heat without a change in temperature The latent heat or heat of transformation is
usually given per unit mass of the substance For example for water the heat of fusion
(melting) is L = 333 kJkg while the heat of vaporization (boiling) is L = 2260 kJkg
Note that heat for boiling is considerably bigger than melting for water You have to
be careful with signs heat is given off (negative) if you go down in temperature and
condense steam
Work
In general the small amount of work done on a system as a force Fon is exerted on it
through a vector displacement dx is given by
on xdW d F
12
But if the displacement is done very slowly (as we always assume in thermodynamics)
then the force exerted on the system and the force exerted by the system are in
balance so the force exerted by the system is ndash Fon In thermodynamics it is more
convenient to talk about the force exerted by the system so we change the above
formula for the work done on the system to
xdW d F
where F is the force exerted by the system This has confused students for more than a
century now but this is the way your book and many other books do it so you are
stuck You will need to memorize the minus sign in this definition of the work to be
able to use your textbook
There are many chances to get signs wrong in this and the next two chapters (Mosiah
2 )
When an external agent changes the volume of a gas at pressure P by a small amount
dV the (small amount) of work done on the system is given by
dW PdV
Notice that this minus sign is just what we need to make dW be positive if the external
agent compresses the gas for then dV is negative If on the other hand the external
agent gives way allowing the gas to expand against it then dV is positive and we say
that the work done on the gas is negative
The work done on the system (eg by the gas in a cylinder) in a thermodynamic
process is the area under the curve in a PV diagram It is positive for compressions
and negative for expansions If the volume of gas remains constant in a process then
no work is done by the gas
Cyclic processes are important For cyclic processes represented by PV diagrams the
magnitude of the net work during one cycle is simply the area enclosed by the cycle
on the diagram Be careful to keep track of signs when you are calculating that
enclosed area In cyclic process Q = ndashW for a cycle Put a PV diagram in your notes
1
Path A-B B-C C-D D-A A to A net
Q
W
ΔU
ΔS
Internal Energy
The energy stored in a substance is called its internal energy Eint This energy may be
stored as random kinetic energy or as potential energy in each molecule (stretched
chemical bonds electrons in excited states etc) For ideal gases all states with the
same temperature will have the same Eint
First Law of Thermodynamics
The change ΔEint in the internal energy of a system is given by
intE Q W
where Q is the heat absorbed by the system and where W is the work done on the
system Hence if a system absorbs heat (and if Wge0) the internal energy increases
Likewise if the system does work (W on the system is negative) and if Qge0 the
internal energy decreases Potential Pitfall Many times people talk about work done
by the system It is the minus of W on the system Donrsquot get tripped up
Processes
Adiabatic process No heat is exchanged Q = 0 requiring that the system be insulated
from the environment A process may be approximately adiabatic if it happens so
rapidly that heat does not have time to enter or leave the system Work + or ndash is done
and ΔEint = W
Adiabatic free expansion A gas is allowed to expand into a vacuum without pushing
on anythingndashit just rushes into the vacuum In this process Q = W = 0 so the internal
energy and temperature does not change (Note the difference between an adiabatic
process and a free expansion is that NO work is done in the adiabatic free expansion)
Isobaric process The pressure is held fixed ΔP = 0 For example usually the
pressure increases when a gas is heated but if it were allowed to expand during the
2
heating process in just the right way its pressure could remain fixed In isobaric
processes both Q and W are non-zero (Work is eacy = ndashP(Vf ndash Vi)
Isovolumetric process The volume is held fixed ΔV = 0 The work done by the gas is
then zero and so we have ΔEint = Q
Isothermal process The temperature is held fixed ΔT = 0 There is no change in Eint
so Q = ndashW for isothermal processes For an ideal gas PV = nRT so PV = constant in
an isothermal process The work done on the gas is then given by
lnf
i
Vi
Vf
VW PdV nRT
V
Heat Conduction
The quantity P is defined to be the rate at which heat flows through an object and is a
power having units of watts It is analogous to electric current which is the rate at
which charge flows through an object If the flow of heat through a slab of length L
and cross-sectional area A is steady in time then P is given by the equation
h cT TdQkA
dt L
P =
where Th and Tc are the (hot and cold) temperatures of the two ends of the slab The
heat flows of course because of this temperature difference The quantity k is called
the thermal conductivity and is a constant that is characteristic of the material It is
analogous to the electrical conductivity h cT TL is sometimes called the temperature
gradient and is written dTd dTdx
R-Values
It is common to have the heat-conducting properties of materials described by their R-
values especially for insulating materials like fiberglass batting The connection
between k and R is R = Lk where L is the material thickness In this country R-values
always have units of 2ft F hourBtu
Polyurethane foam 6 (1 in thick) Air space 1 (35 in thick) Fiber glass batting 11
(35 in thick) Wood about 1 (1 in thick) Glass 09 (0125 in thick)
The heat flow rate through a slab of area A is given by
3
h cT TA
R
P
in units of Btuhour Note that A must be in square feet and the temperatures must be
in degrees Fahrenheit
Convection
Convection is the transfer of thermal energy by flow of material For instance a home
furnace doesnrsquot heat a house by waiting for the heat from the burner to slowly conduct
throughout the house instead it quickly pumps warm air to all of the rooms
Generally convection is a much faster way to transfer heat than conduction
Radiation
Electromagnetic radiation can also transfer heat When you warm yourself near a
campfire which has burned itself down into a bed of glowing embers you are
receiving radiant heat from the infrared portion of the electromagnetic spectrum The
rate at which an object emits radiant heat is given by Stefanrsquos law
4AeTP
where P is the radiated power in watts σ is a constant
8 2 45696 10 W m K
A is the surface area of the object in m2 and T is the temperature in kelvins The
constant e is called the emissivity and it varies from substance to substance A perfect
absorber (think black velvet) has e = 1 while a perfect reflector (think mirror) has e =
0 Hence black objects radiate very well while shiny ones do not Also an object that
is hotter than its surroundings radiates more energy than it absorbs whereas an object
that is cooler than its surroundings absorbs more energy than it radiates
Terminology
Transfer variables vs state variables
Energy transfer by heat as well as work done depends on the initial final and
intermediate states of the system They are transfer variables But their sum (Q + W =
4
Eint) is a state variable
Figure 205
5
Serway Chapter 21
Kinetic Theory
The ideal gas law works for all atoms and molecules at low pressure It is rather
amazing that it does Kinetic theory explains why The properties of an ideal gas can
be understood by thinking of it as N rapidly moving particles of mass m As these
particles collide with the container walls momentum is imparted to the walls which
we call the force of gas pressure In this picture the pressure is related to the average
of the square of the particle velocity 2v by
22 1( )
3 2
NP mv
V
Using the ideal gas law we obtain the average translational kinetic energy per
molecule
21 3
2 2 Bmv k T
The rms speed is then given by
2rms
3 3Bk T RTv v
m M
where M is the molecular mass in kgmol
Degrees of Freedom
Roughly speaking a degree of freedom is a way in which a molecule can store energy
For instance since there are three different directions in space along which a molecule
can move there are three degrees of freedom for the translational kinetic energy
There are also three different axes of rotation about which a polyatomic molecule can
spin so we say there are three degrees of freedom for the rotational kinetic energy
There are even degrees of freedom associated with the various ways in which a
molecule can vibrate and with the different energy levels in which the electrons of
the molecule can exist
Internal Energy and Degrees of Freedom The internal energy of an ideal gas made
up of molecules with J degrees of freedom is given by
int 2 2 B
J JE nRT Nk T
6
Heat Capacities of an Ideal Gas The heat capacity of a gas is described by means of
molar heat capacities CV and CP These are the heat capacities per mole and the
subscript V on CV means that the volume is being held constant while for CP the
pressure is held constant For example to raise the temperature of n moles of a gas
whose pressure is held constant by 10 K we would have to supply an amount of heat Q
= nCP (10) K
Molar Specific Heat of an Ideal Gas at Constant Volume
VQ nC T
3monatomic
2VC R
5diatomic
2VC R
5polyatomic
2VC
Real gases deviate from these formulas because in addition to the translational and ro-
tational degrees of freedom they also have vibrational and electronic degrees of
freedom These are unimportant at low temperatures due to quantum mechanical
effects but become increasingly important at higher temperatures The rough rule is
No of degrees of freedom
2VC R
Molar Specific Heat of an Ideal Gas at Constant Pressure
PQ nC T
P VC C R
The internal energy of an ideal gas depends only on the temperature
int VE nC T
Adiabatic Processes in an Ideal Gas
7
An adiabatic process is one in which no heat is exchanged between the system and the
environment When an ideal gas expands or contracts adiabatically not only does its
pressure change as expected from the ideal gas law but its temperature changes as
well Under these conditions the final pressure Pf can be computed from the initial
pressure Pi and from the final and initial volumes Vf and Vi by
or constantf f i iP V PV PV
where γ = CPCV The quantity γ is called the adiabatic exponent Note that this
doesnrsquot mean that the ideal gas law no longer holds it does and in fact it can be
combined with the adiabatic law for pressure given above to obtain the adiabatic law
for temperatures
1 constantTV
Compressions in sound waves are adiabatic because they happen too rapidly for any
appreciable amount of heat to flow This is why the adiabatic exponent appears in the
formula for the speed of sound in an ideal gas
RTv
M
Note that v depends only on T and not on P Because it depends only on the
temperature the speed of sound is the same in Provo as at sea level in spite of the
lower pressure here due to the difference in elevation
Equipartition of Energy
Every kind of molecule has a certain number of degrees of freedom which are
independent ways in which it can store energy Each such degree of freedom has
associated with it ndash on average ndash an energy of 12 Bk T per molecule (or 1
2 RT per mole)
(Note since a molecule has so many possible degrees of freedom it would seem that
there should be a lot of 12 sBk T to spread around But because energy is quantized
some of these degrees of freedom are not ldquoactiverdquo until the temperature becomes high
enough that 12 Bk T is as big as the lowest quantum of energy
8
Serway Chapter 22
Second Law of Thermodynamics
There are several equivalent forms of this important law
Kelvin It is not possible to change heat completely into work with no other change
taking place Or in other words there are no perfect heat engines
Clausius It is not possible for heat to flow from one body to another body at a higher
temperature with no other change taking place Or in other words there are no
perfect refrigerators
Entropy In any thermodynamic process that proceeds from one equilibrium state to
another the entropy of the system + environment either remains unchanged or
increases The total entropy never decreases This law is a bit of an oddity among the
laws of physics because it is not absolute Things are forbidden by the second law not
because it is impossible for them to happen but because it is extremely unlikely for
them to happen (See below for more information about entropy)
Reversible and Irreversible Processes
A reversible process is one which occurs so slowly that it is in thermal equilibrium (or
very nearly so) at all times A hallmark of such processes is that a motion picture of
them looks perfectly normal whether run forward or backward Imagine for instance
the slow expansion of a gas at constant temperature in a cylinder whose volume is being
increased by a slowly moving piston Run the movie backwards and what do you see
You see the slow compression of a gas at constant temperature which looks perfectly
normal
An irreversible process is one which occurs in such a way that thermal equilibrium is
not maintained throughout the process The mark of this kind of process is that a motion
picture of it looks very odd when run backward Imagine the sudden expansion of a gas
into a previously evacuated chamber because a hole was punched in the wall between a
pressurized chamber and the evacuated one Run the movie backward and what do you
see You see the gas in the soon-to-be-evacuated chamber gather itself together and
stream through a tiny hole into a chamber in which there is already plenty of gas If you
have ever seen this happen get in touch with the support group for those who have
witnessed the spontaneous combustion of large mammals by calling 1-800-PYROCOW
9
Heat Engines
Because of the vexing difference in sign between work done on and system and work done
by a system we will invent a new work variable Weng Heat engines do work and so the net
W for these engines is negative But in engineering applications hidden minus signs are
regarded as evil so for heat engines we donrsquot talk about W instead we talk about its
magnitude engW W So for heat engines the first law is
int engE Q W
But for heat pumps and refrigerators work is done on the system so we use the usual work
W when we talk about these systems
A heat engine is a machine that absorbs heat converts part of it to work and exhausts the
rest The heat must be absorbed at high temperature and exhausted at low temperature If the
absorbed heat is Qh the exhausted heat is Qc and the work done by the engine is Weng then
eng h cW Q Q
and the efficiency of the engine is defined to be
eng
h
We
Q
A perfect engine would convert the heat hQ completely into work Weng giving an effi-
ciency of e = 1 Energy conservation alone allows a perfect engine but the second law
requires e lt 1
Refrigerators and Heat Pumps
A refrigerator is a machine that absorbs heat at low temperature and exhausts it at high
temperature the ldquobackwardsrdquo heat transfer being driven by the work done on the machine
by some source of power A heat pump is a machine that either works like a refrigerator
keeping a place cold by transferring heat from this cold place to a higher temperature
environment (cooling mode like an air conditioner) or it functions as a heater
transferring heat into a warm place from a cooler one (heating mode like a window unit
that heats a house by extracting thermal energy from the cold outdoors) The coefficient
of performance of a refrigerator or of a heat pump in cooling mode is defined to be
COP cooling modecQ
W
10
For a heat pump in heating mode the coefficient of performance is
COP heating modehQ
W
Note that we donrsquot have to use the engineering work here because in these systems
work is done on the system and W is naturally positive
A perfect refrigerator would take in heat Qc and exhaust the same amount of heat without
doing any work giving an infinite coefficient of performance Energy conservation alone
allows a perfect refrigerator but the second law requires COP lt infin A perfect heat pump
in heating mode would transfer Qh into the house without doing any work and so would
also have an infinite coefficient of performance The second law forbids this too A good
coefficient of performance for a real device would be around 5 or 6
Carnot Cycle
The most efficient of all possible engines is one that uses the Carnot cycle This cycle
employs an ideal gas has no friction and operates very slowly so that the gas can be in
thermal equilibrium at all parts of the cycle This means of course that it canrsquot
possibly be built and even if it could be built it would not run fast enough to be useful
Nevertheless this cycle is very important because it gives an upper bound on the
efficiency of real engines There cannot possibly be an engine that is more efficient
than one based on the Carnot cycle This cycle consists of the following four steps
1 The ideal gas absorbs heat Qh at constant temperature Th while the gas increases its
volume The reason that heat is absorbed is that expansion tends to cool the gas but
thermal contact with the environment at Th keeps the temperature high by heat
conduction into the ideal gas
2 The ideal gas further increases its volume by an adiabatic expansion This expansion
causes the gas to cool so at the end of this part of the cycle the gas is at temperature Tc
3 The gas exhausts heat Qc at constant temperature Tc while the gas decreases in
volume The reason that heat is exhausted is that compression tends to heat the gas
but thermal contact with the environment at Tc keeps the temperature low by heat
conduction out of the ideal gas
4 The gas is adiabatically compressed back to its original volume (the volume it started
with in step 1) This compression heats the gas from Tc up to Th
11
The efficiency of a Carnot engine is given by the very simple formula
1 cC
h
Te
T
where the temperatures must in be Kelvin No real engine can be more efficient than
this
The coefficient of performance of a Carnot refrigerator or heat pump in cooling mode
is given by
COP cooling mode cC
h c
T
T T
and no real refrigerator can have a coefficient of performance greater than this
The coefficient of performance of a Carnot heat pump in heating mode is
COP heating mode hC
h c
T
T T
So why donrsquot we just use these wonderful Carnot engines and have perfect efficiency
To make the reversible steps in the cycle really reversible they would have to occur
infinitely slowly So the price you pay for making a perfect engine is that it takes
forever to get it to do any work
Entropy
The entropy of a system is defined in terms of its molecular makeup and measures
roughly the disorder of the system If the system is packed into a very small volume
then it is quite ordered and the entropy will be low If it occupies a large volume the
entropy is high (To see what this has to do with disorder note that socks in a drawer
occupy a small volume while socks on the bed in the corner by the door and
hanging from the chandelier occupy a large volume) If the system is very cold then
the molecules hardly move and may even reach out to each other and form a crystal
This is a highly ordered state and therefore has low entropy If the system is very hot
with rapidly speeding molecules crashing into the container walls and bouncing off
each other things are disordered and the entropy is high
It is possible to calculate the entropy of a system in terms of its macroscopic thermody-
namic properties ie pressure volume temperature number of moles etc The key to this
calculation is the concept of a reversible process A reversible process is one that is
12
carried out without friction and so slowly that the process can be reversed at any stage by
making an infinitesimal change in the environment of the system The slow expansion of the
gas at Th in the Carnot cycle with tiny amounts of heat being transferred to the environment
is reversible If we turned around and began slowly to compress the gas would just slowly
exhaust heat to the environment in the exact reverse way that it absorbed it during expansion
Most processes however are irreversible For example if a gas-filled box were suddenly
increased in size so that the particles were free to wander into the void created by the sudden
expansion then the gas would eventually fill the new volume uniformly at the same
temperature as before the expansion (The temperature is unchanged in this imaginary
process because the kinetic energy of the molecules would be unaffected by such an
instantaneous expansion of the container walls) This imaginary but highly thought-
stimulating process is called a free expansion and it is impossible to reverse it During the
expansion we didnrsquot push on any of the molecules so reversing this process would mean
making them go back into their original volume without pushing on them they simply will
not cooperate to this extent Another way to see that just pushing them back where they came
from does not reverse the free expansion is to think about what would happen if we just
compressed either adiabatically or isothermally An adiabatic compression back to the
original volume would heat the gas above its original temperature and an isothermal
compression would require that heat be exhausted to the environment But the free expansion
involved neither temperature changes nor heat exchanges so neither of these two processes
is the reverse of the free expansion It is simply impossible to reverse this rapid expansion in
a way that takes us back to the initial state
It is possible to calculate the change of entropy for both reversible and irreversible processes
Letrsquos consider a reversible process first In a reversible process the entropy change is given
by the formula
dQS
T
where dQ is the amount of heat added to the system during a small step of the process
The total energy change during the process may then simply be calculated by integration
f f
f i i i
dQS S S dS
T
13
(Just as in the case of energy we are mainly interested in differences rather than in
absolute magnitudes)
This integration method does not work for irreversible processes For instance in the
rapid free expansion discussed above no heat is added to the system but its disorder
obviously goes up We need to find some other way to calculate the entropy The key
is the fact that the entropy of a system depends only on its current state and not at all
on how it arrived there So to calculate the entropy change in an irreversible process
first find out what the initial and final conditions of the process are Then invent a
reversible process that takes the system from the initial state to the final state Since
the entropy depends only on the state of system and not on the process the entropy
change for the reversible process is the same as that for the irreversible process
Entropy is thus like pressure or temperature if the pressure changes from Pl to to P2
then the pressure difference is P2 - Pl regardless of how the pressure change was
made This seems obvious for pressure but not for entropy only because you arent
comfortable with entropy yet
Entropy of an Ideal Gas
For n moles of an ideal gas the difference in entropy between a state with temperature
T and volume V and some standard state with temperature To and volume Vo is given
by
ln lno o Vo o
T VS T V S T V nC nR
T V
Entropy in a Phase Change
Calculating entropy change in a phase change like melting or boiling is easy It is
QT Where Q is in the latent heat for example the flows in to cause the melting and
T is the temperature of the phase change There is NO integral to do
14
Serway Chapter 35
Angle of Reflection
If plane waves are incident on a reflecting surface with the propagation direction of
the waves making angle θ1 with the normal direction to the surface then the reflected
angle θ1prime relative to the surface normal is simply
1 1
ie the incident angles and reflected angles are the same
Refraction
If plane waves traveling through medium 1 are incident on a plane interface between
medium 1 and medium 2 then the angle of incidence of the incoming wave θ1 and
the angle of refraction of the transmitted wave θ2 are related by Snellrsquos law
2 2
1 1
sin
sin
v
v
where v1 and v2 are the wave speeds in medium 1 and medium 2 The angle of
incidence and the angle of refraction are both measured between the wave propagation
direction and the normal to the interface In terms of indices of refraction in the case
of light waves Snellrsquos law takes the more familiar form
1 1 2 2sin sinn n
where ni = cvi
Total Internal Reflection
If a wave is incident from a medium of low wave speed into a medium of high wave
speed the law of refraction requires that the angle of refraction be greater than the
angle of incidence If the angle of refraction is required to be greater than 90deg then no
refracted wave can exist and total internal reflection occurs The critical incident
angle θc beyond which total internal reflection occurs is given by
2
1
sin c
n
n
15
Dispersion of Light
In addition to the speed of light varying from material to material it also varies with
wavelength within each material This means that the index of refraction is generally a
function of wavelength
c
nv
Since the wave speed is not constant such a medium is dispersive meaning in this
context that refraction actually disperses white light into its various colors because
Snellrsquos law gives a different angle for each wavelength In most materials the
variation with wavelength is quite small but this small effect is responsible for some
of the most spectacular color effects we ever see including rainbows a flashing
crystal chandelier and the colored fire of a diamond solitaire by candlelight
16
Serway Chapter 36
Real and Virtual Images
When light rays are focused at a certain plane producing an image if a sheet of white
paper is placed there we call the image a real image The images produced by film
projectors and overhead projectors are examples of real images
When light rays appear to come from a certain location but no image is produced
when a screen is placed there we say that there is a virtual image at that location For
instance when you look in a mirror it appears that someone is behind the mirror but
a screen placed back there in the dark would show nothing Your image in the mirror
is a virtual image
Ray Tracing
There are lots of rules about how to find the images in optical systems but the best
way to keep things straight is to learn how to draw the principal rays for curved
mirrors and lenses The rays for convex and concave mirrors are shown in Fig 3615
The rays for converging and diverging thin lenses are shown in Fig 3627 You should
memorize the principal rays and know how to use them to locate images
Curved Mirrors
The focal length of a curved mirror with radius of curvature R is given by
2
Rf
The relation between object distance p image distance q and focal length f is
1 1 1
p q f
If the mirror is a diverging mirror f should be negative and if q should turn out to be
negative the image is virtual
Lateral Magnification
The lateral magnification in an optical system is defined by the ratio of the image size
to the object size
17
Image height
Object height
qM
p
for curved mirrors
As usual there are sign conventions here too but this formula just gives the
magnitude It is better to keep keep track of upright images versus inverted images by
means of ray diagrams rather than by memorizing sign conventions
Thin Lenses
The focal length of a thin lens is related to the radii of curvature of the two faces R1
and R2 of the lens by
1 2
1 1 11n
f R R
Note that this formula differs from Eq (3611) in the text by not having a minus sign
between the two R-terms We like this form better because for a simple converging
lens like a magnifying glass we just use positive values of R for both surfaces If one
of the faces is concave producing divergence use a negative value for R And if a
surface is flat use R = infin If the face is flat the radius is infinite
The relation between the image object and focal distances for a thin lens is the same
as that for a curved mirror
1 1 1
q p f
Use a negative focal length if the lens is diverging
The lateral magnification for a thin lens is the same as for a curved mirror
Image height
Object height
qM
p
Camera
The lens system in a camera projects a real image of an object onto the film (or CCD
array in a digital camera) The position of the image is adjusted to be on the film by
moving the lens into or out of the camera
18
Eye
The eye is like a camera in that a real image is formed on the retina Unlike a camera
the image position is adjusted by changing the focal length of the lens This is done by
the ciliary muscle which squeezes the lens changing its shape
Near Point The near point is the closest distance from the eye for which the lens can
focus an image on the retina It is usually 18-25 cm for young persons
Far Point The far point is the greatest distance from the eye for which the lens can
focus an image on the retina For a person with normal vision the far point is at
infinity
Nearsightedness A person is nearsighted if their far point is at some finite distance
less than infinity This condition can be corrected with a lens that takes an object at
infinity and produces a virtual image at the personrsquos far point
Farsightedness and Presbyopia A person is farsighted if their near point is too far
away for comfortable near work like reading or knitting This can be corrected by a
lens which takes an object at a normal near point distance of 18-25 cm and produces a
virtual image at the personrsquos natural near point Presbyopia involves a similar
problem which nearly all people experience as they age The ciliary muscle becomes
too weak and the lens becomes too stiff to allow the eye to provide for both near and
far vision The solution for this problem is either reading glasses or bifocal lenses
Reading classes are just weak magnifying glasses mounted on eyeglass frames
Bifocal lenses are split into upper and lower halves The lower half is a lens which
gives the proper correction for near work and the upper half is a different lens for
proper focusing at infinity
Angular Size
When an object is brought closer to the eye it appears to be larger because the image
on the retina is larger The size of this image is directly proportional to the objectrsquos
angular size which is the angle subtended by the object measured from the center of
the lens of the eye In optical instruments which are to be used with the eye the
angular size of the final image is whatrsquos important because it determines how large
the image will appear to the viewer
19
Simple Magnifier
A simple magnifier is a single converging lens or magnifying glass It takes an object
closer to the eye than a normal near point and produces a virtual image at or beyond
this near point The angular magnification is defined to be the ratio of the angular size
when viewed through the lens to the angular size of the object when viewed at the
normal near point (without aid of the lens)
Microscope
This instrument has two lenses (1) The objective is near the object being viewed and
produces a greatly magnified real image (2) The eyepiece is a simple magnifier which
the viewer uses to closely examine the image from (1)
Telescope
This instrument also has two lenses (1) The objective at the front of the telescope
takes light from a distant object and produces a real inverted image (which is rather
small) near its focal point (2) This small real image is then examined by the eyepiece
functioning as a simple magnifier to produce a virtual image with a larger angular
size
20
Serway Chapter 37
Two-Slit Interference
If light is incident on two closely spaced narrow slits a pattern of light and dark
stripes is produced beyond the slits The bright stripes or fringes are caused by
constructive interference of the two waves coming from the slits Constructive
interference occurs whenever two waves arrive at a location in phase with each other
This occurs when the distance x1 from slit 1 to a point P on the screen and the
distance x2 from slit 2 to point P differ by in integral number of wavelengths
1 2 where 0 1 2x x m m
where λ is the wavelength of the light When x1 and x2 are much larger than the slit
spacing d this condition reduces to
sind m
where θ is the angle between the direction of the incident light and the direction of the
light arriving at the screen
Thin Films
When light is partially reflected and partially transmitted by a thin film of transparent
material it is possible to have interference between the wave reflected from the front
of the film and light reflected from the back of the film (The colored reflections from
the thin film of oil on the water in a rain-soaked parking lot are an example of this
effect) It is difficult to write down formulas that will work in all cases so we will
just review the important principles here
1 If the two reflected waves are in phase with each other the film has enhanced
reflection (constructive interference) but if the two reflected waves are out of phase
with each other reflection is diminished (destructive interference) Phase shifts occur
due to reflection and due to the extra path length through the film of the wave
reflected from the back of the film
2 The phase change due to reflection is determined by the difference in index of
refraction between the two media involved in the reflection If the wave is incident
21
from a medium with a low index of refraction into a medium with a high index of
refraction a phase change of 180deg occurs and the reflected wave is inverted If
incident from high to low no phase shift occurs and the reflected wave is non-
inverted
3 The extra path length through the film of the wave reflected from the back of the
film is equal to 2t where t is the thickness of the film (The incident light is assumed
to be normal to the surface of the film) The number of wavelengths contained in the
extra path length is equal to 2tλn where λn = λn is the wavelength of the light in the
film
4 Rules for reflection from thin films
If one ray is inverted and the other is not then we have
12 constructive
2 nt m
2 destructivent m
If either both rays are inverted or both are non-inverted we have
2 constructivent m
12 destructive
2 nt m
22
Serway Chapter 38
Diffraction Grating
A diffraction grating is simply a fancy version of two-slit interference with the two
slits replaced by thousands of slits Just as in the two-slit case the bright fringes
occur at angles given by
sin md
but in this case d the distance between neighboring slits is made to be very small
The effect of having many slits instead of two is to make each bright fringe highly
localized with wide dark regions between neighboring maxima
Single Slit Diffraction
When light passes through an opening in an opaque screen an interference pattern is
produced beyond the opening To understand why we may replace the single opening
by many small coherent sources of light These many sources interfere with each
other producing a pattern known as a diffraction pattern (Note that many authors do
not distinguish between interference and diffraction treating them as interchangeable
terms) If the opening is a slit of width a then the diffraction pattern far from the slit
will have a bright central maximum with a succession of minima and weaker maxima
on either side The angle between the incident direction and the minima is given by
sin ma
where m = plusmn1 plusmn2
If the opening is circular with diameter D the angle between the incident direction
and the first minimum is given by
sin 122D
Optical Resolution and Rayleighrsquos Criterion
Two point sources can just be resolved (distinguished from each other) if the peak of
the diffraction image of the first source overlies the first minimum of the diffraction
image of the second source For circular holes of the kind usually encountered in
23
optical devices this condition is approximately satisfied when the angular separation
between the two sources as viewed from the optical instrument is greater than or equal
to the critical angle
min 122D
where λ is the wavelength of the light and where D is the diameter of the aperture in
the instrument
Polarization
We say that an electromagnetic wave is polarized if its electric field vector doesnrsquot
change direction in a random fashion The simplest kind of polarization is linear
polarization in which the electric field vector oscillates back and forth along the
same axis in space Polarized light can be produced from normal unpolarized light by
selective absorption (as in Polaroid sunglasses) by passing light through crystals that
have different indices of refraction for different polarizations (double refraction) by
scattering (the blue sky is polarized) and by reflection (glare) Polarization by
reflection occurs when light reflects from a shiny insulating (non-metallic) surface
The amount of polarization is greatest for reflection at Brewsterrsquos angle
2
1
tan P
n
n
where θP is the incidence angle of light from medium 1 onto medium 2 and where n1
and n2 are the indices of refraction for the media (In our everyday experience n1 = 1
since the light comes in through the air and n2 is the index of refraction of the shiny
insulating material producing the glare eg water glass plastic paint etc)
Malusrsquos Law
The intensity of transmitted polarized light through a perfect polarizer is related to the
incident intensity of polarized light by Malusrsquos law
2cosoI I
where Io is the intensity of the incident light and where θ is the angle between the
electric field vector in the incident wave and the transmission axis the polarizer
24
Serway Chapter 39
Principles of Relativity
All of the weirdness of relativity flows from two simple principles
(1) The laws of physics must be the same in all inertial (non-accelerating) reference
frames
(2) The speed of light in vacuum has the same value in all inertial reference frames
Note that (1) seems reasonable but (2) is very odd It says that if two space ships are
approaching each other at nearly the speed of light and a laser pulse is shot from ship
1 toward ship 2 then when the people on ship 2 measure the speed of the pulse as it
goes by it is moving at 3 times 108 ms the same as if the ships were stationary
Simultaneity
If observer 1 sees two events in her own frame as simultaneous at two different
locations a moving observer 2 will see these two events happening at different times
Relativistic gamma
The factor γ (gamma) appears regularly in the formulas of relativity
2
2
1
1 vc
where v is the relative speed between two inertial frames
Time Dilation
If observer 1 sees two events at the same location in space separated by time Δtp in his
own frame then observer 2 moving at speed v relative to observer 1 will see these
two events separated by a longer time Δt
pt t
Moving clocks run slow
Length Contraction
If observer 1 measures the length of an object along the x-direction at some instant of
time in her frame to be Lp then observer 2 moving in the x-direction at speed v will
measure the length L of the object to be shorter
pLL
25
Moving meter sticks are short
Velocity Addition
If a particle is moving at velocity vac relative to frame c and if frame c is moving at
speed vcb with respect to frame b (with both velocities directed along the same line)
what would be the velocity of the particle in frame b In classical physics the answer is
ab ac cbv v v
but if any of these velocities are an appreciable fraction of the speed of light the
answer changes
21ac cb
abac cb
v Vv
v v c
In using this equation make sure that vac has the same sign in the numerator and in the
denominator and that vbc also has the same sign in both places Note that this equation
is different (and in our opinion easier to understand) than the equations in the text
Momentum and Energy
The momentum and total energy of a particle moving at velocity v (its speed is v v )
are given by
22
2 2
22
1 1 vvc c
m mcm E mc
vp v
Rest Energy
When a particle is at rest (v = 0 and γ = 1) then E = mc2 is called the rest energy It
includes the potential energy of the particle so that if we change the potential energy
of a particle we change its rest mass
Kinetic Energy The kinetic energy is the difference between the total energy and the
rest energy
2 2K mc mc
For v ltlt c this reduces to K = mv22
General Relativity
Einsteinrsquos principle of equivalence in the theory of general relativity is In the vicinity
of a point a gravitational field is equivalent to an accelerated frame of reference in
26
the absence of gravitational effects For example there is no difference whatever in
the physics experienced by an astronaut on earth and one who is in interstellar space
far away from any stars or planets but who has just turned on her spaceshiprsquos engines
and is accelerating at 98 ms2
27
Serway Chapter 40
Photons
We now know that even though light behaves like a wave it is better described as
consisting of small packets of energy called photons The energy of a photon is related
to its frequency by
E hf
where h is Planckrsquos constant
346626 10 J sh
Photons also carry momentum given by
hf hp
c
Since the amount of energy in each photon is fixed the difference between dim light
and intense light (of the same wavelength) is that dim light consists of fewer photons
than intense light
Photoelectric Effect
In the photoelectric effect light shining onto a metal surface gives the electrons in the
metal enough energy to escape and be detected It requires a certain amount of energy
(called the work function typically = 3 ndash 5 eV) just to kick the electron out of the
metal so the light must deliver at least this much energy to an electron to produce the
effect It is observed that red light no matter how intense never produces electrons
But ultraviolet light even if quite dim will eject electrons from the metal Since red
light consists of 2 eV photons and ultraviolet light has photons with energies around
4-6 eV the photon idea explains the behavior of the photoelectric effect The
maximum energy that an ejected electron can have is
maxK hf
where f is the photon frequency The minimum frequency that light can have and cause
electrons to be ejected is called the cutoff energy and is found by setting Kmax = 0
cf h
28
Compton Effect
When high-frequency light interacts with free electrons the scattered light does not
have the same wavelength as the incident light contrary to what classical
electromagnetism would predict But the photon picture in which photons have
momentum and energy allows us to treat scattering as a collision between the photon
and the electron In this collision the electron and the scattered photon both have a
different momentum and energy than they did before And since p = hλ for a photon
if the momentum is different the wavelength will also be different
after before 1 cose
h
m c
where the angle θ is the angle between the incoming photon direction and the
direction of the scattered photon
Particles are Waves
Since photons behave like particles sometimes it is not surprising that elementary
particles can behave like waves sometimes The wavelength of a particle with
momentum p is given by
h
p
Wave-particle Duality
Both photons and elementary particles have a dual nature sometimes they behave like
particles and sometimes they behave like waves It is hard for us to comprehend the
nature of such an object by making mental pictures but experiments definitively show
that this is the case Since we have no direct experience with photons and elementary
particles (because their energies are so much smaller than the energies of the everyday
objects we encounter) it is perhaps not surprising that we have a hard time forming a
mental picture of how they behave
Electron Interference
Since an electron is both a particle and a wave just like a photon it should be able to
produce an interference pattern This is observed If an electron beam is shot at two
closely-spaced slits and if the electrons that pass through the slits are detected
downstream it is found that there are some locations where electrons are never detected
29
and others where lots of electrons are detected The pattern is exactly the same as the
one observed for light waves provided that we use the electron wavelength λ = hp in
place of the wavelength of light This pattern is observed even though each electron is
detected as a single dot on the screen Only after many such dots are collected does the
pattern emerge And if we try to understand how this effect could possibly work by
looking closely at each slit to see which one the electron came through the pattern
disappears the act of measurement destroys the interference This means that each
single electron somehow comes through both slits (as a wave would)
Uncertainty Principle
In classical physics we always imagine that the positions and momenta of moving
particles have definite values It might be hard to measure them but surely at each
instant of time a particle should be precisely located at some point in space and have a
similarly precise momentum This turns out not to be true Instead both position and
momentum are required to be uncertain with their uncertainties Δx and Δp satisfying
the Heisenberg uncertainty relation
2x p
So if the particle were known to be precisely at some particular location (so that Δx =
0) we couldnrsquot know anything about its momentum (Δp = infin) and if its momentum
were exactly known we couldnrsquot know its position
There is a similar relation involving the particlersquos energy E and the time interval Δt
over which this energy is measured
2E t
This means that energy is not actually conserved if we are considering very small time
intervals and this brief non-conservation of energy has been observed
30
Serway Chapter 41
Wave Function ψ and Probability
Quantum mechanics does not predict exactly what an electron or a photon will do Instead
it specifies the wave function or probability amplitude ψ of an electron or a photon
This wave function is a complex-valued function of space and time whose squared
magnitude is the probability density P for finding a particle at a particular place in
space at a certain time
2P
where is the complex conjugate of ψ
For example the wave function of an electron with perfectly specified momentum p
would have a wavelength given by
h
p
and its probability amplitude would be proportional to
2i x ipxe e
The corresponding probability density would then be
21ipx ip ipxP e e e
which means that the electron is equally probable to be anywhere along the x axis
This is in accord with the uncertainty principle since we specified the momentum
precisely we canrsquot have any idea about the position of the electron
As another example you have probably seen ldquofuzzy ballrdquo drawings of electron
orbitals in a chemistry book These fuzzy balls are meant to indicate the distribution
of the probability density 2
P in the orbital
Particle in a Box
A simple example in which we can calculate the wave function is the case of a particle of
mass m constrained to be inside a 1-dimensional box between x = 0 and x = L In this
case the general wave function is a linear superposition of wave functions ψn of the form
sin niE tn
n xx t A e
L
31
where A is a positive constant where n = 1 2 3 and where the energy associated
with each of the quantum states ψn is given by
22
28n
hE n
mL
This wave function is zero at x = 0 and x = L which means that the particle will never
be found at the walls of the box The wave function has maximum values in the
interior and at these places the particle is most likely to be found and it also has
places where it is zero and at these places the particle will also never be found as
expressed by the formula
2sinn x
PL
The particle in the box is interfering with itself producing a probability interference
pattern across the box just like the interference patterns we studied with light and
sound
32
Serway Chapter 42
Atoms
Many people picture an atom as a miniature solar system where electrons orbit around
a massive nucleus at the center This picture is misleading because in atoms the wave
nature of electrons dominates The electrons form 3-dimensional standing waves
(called orbitals) centered at the nucleus And if you ask what is it thatrsquos waving the
answer is ψ the probability amplitude (see Chapter 41)
Standing waves on a string can occur only for certain frequencies (the fundamental
and higher harmonics) Similarly atomic orbitals occur only for certain energies For
the hydrogen atom the energies of the orbitals have the a particularly simple form
2
1136 eVnE
n
where n = 1 2 3 is called the principal quantum number For other atoms the
determination of the orbital energies requires numerical calculation by computers
Atomic Spectra
If an electron is somehow given extra extra energy (we say that it is excited) so that it
occupies a higher orbital it will eventually ldquofallrdquo back down to a lower orbital Each
time an electron falls to a lower orbital it loses the difference in energy between the
two orbitals in the form of a photon Since the orbital energies are discrete so are the
energy differences and so are the wavelengths of the emitted light The entire set of
these discrete wavelengths is called the atomic spectrum and it is unique to each type
of atom For hydrogen the spectrum can be simply written as
H 2 2
1 1 1
f i
Rn n
where RH = 10973732 times 107 m-1 For other atoms the spectrum cannot be expressed as
a formula
Orbital Angular Momentum Electrons in atoms also have quantized values of
angular momentum The orbital quantum number ℓ specifies the value of this
quantized angular momentum through the formula
1L
33
If we want to know the value of the angular momentum along some direction in space
say the z direction the answer is not L but rather
zL m
where mℓ is another quantum number which runs from
1 1m
This quantum number is important when an atom sits in a magnetic field
Spin Angular Momentum It has been found experimentally that electrons and other
charged particles also carry internal angular momentum which we call spin
Electrons have an intrinsic spin angular momentum s along a specified axis that is
extremely quantized it can only take on 2 possible values
1
2zs s
We thus say that electrons have spin s = 12 or that electrons are rdquospin one-half parti-
cles Since s is an angular momentum it obeys the same rule as that for orbital angular
momentum namely that if its value along some axis is s then its total magnitude is
given by
31
2S s s
I know this seems weird but quantum mechanics is weird The only excuse for this
bizarre way of looking at the world is that it predicts what happens in experiments
Exclusion Principle The answer to the question of how many electrons (or any other
spin one-half particle) can be in one particular quantum state was discovered by
Wolfgang Pauli and is called the exclusion principle
ldquoNo two electrons can ever be in the same quantum state therefore no two electrons
in the same atom can have the same set of quantum numbersrdquo
This is the reason that we have atoms with different properties instead of every atom
simple having all of its electrons in the ground state All of the variety we see around
us in the world is the result of chemical differences and these differences would not
exist unless electrons obeyed this important principle The entire structure of the
periodic table (see pages 1377-1379 in Serway) is an expression of this principle
34
Serway Chapter 44
Nuclear Properties
The nucleus is composed of protons with charge +e and mass mp = 1007226 u and of
neutrons with zero charge and mass mn = 1008665 u where u is one atomic mass unit
-271 u=1660540 10 kg
The atomic number Z counts the number of protons in a nucleus while the neutron
number N counts the number of neutrons The mass number A is the sum of the two
A N Z
Protons and nuetrons have about the same mass and some times called baryons (heavy
ones) or nucleons (partcles in the nucleus) A is sometimes called the baryon
number The chemical elements are distinguished by Z ie hydrogen has Z = 1 iron
has Z = 26 uranium has Z = 92 etc But for a given Z there might be several nuclei
with different numbers of neutrons These nuclear siblings are called isotopes
These numbers are used to label nuclei according to the pattern
5626 ie FeA
Z X
denotes the isotope of iron with N = 56 ndash 26 = 30 neutrons
The nucleus is roughly spherical with a radius given approximately by
1 3 150 0where 12 10 mr r A r
Nuclear Stability
Because the positively charged protons electrically repel each other with an enormous
force at distances as small as 10-15 m there must be some really strong force that that
overcomes electrical repulsion to hold protons and neutrons together This force is
called with some lack of imagination the strong force It is a very short-range force
(it only acts over a distance of about 2 x 10-15 m) and attracts protons to protons
neutrons to neutrons and neutrons to protons But in spite of this strong nuclear force
the coulomb repulsion of the protons is still present so anything that might keep the
protons from being right next to each other would help keep the nucleus from
35
exploding This role is played by the neutrons and for nuclei with Z le 20 the stable
nuclei roughly have N = Z
For Z gt 20 the coulomb repulsion force is more powerful and more neutrons are
needed to dilute it up to about N = 15Z around Z = 80 For Z gt 83 no amount of
neutrons can help and these nuclei are unstable (radioactive) There is an ldquoisland of
(relative) stabilityrdquo around Z=90-92 (thorium and uranium respectively) These
elements have one or two isotopes with half-lifes of billions of years so there are
substantial amounts of such elements on Earth This fact makes it possible to have
practical fission devices
Radioactive Decay
There are three types of radioactive decay
Alpha decay the nucleus kicks out a helium nucleus (N = 2 Z = 2 A = 4) So if the
original nucleus is called X and the new nucleus is called Y then the decay would look
like this
4 42 2X Y+ HeA A
Z Z
Beta decay the nucleus either kicks out an electron (endash) or its positively-charged
antimatter twin the positron (e+) plus either an electron neutrino v or an electron
anti-neutrino v
1X Y+e electron decayA AZ Z v
1X Y+e positron decayA AZ Z v
A neutrino is a particle with no charge hardly any mass (much less than the electron
mass) and interacts so weakly with matter that most neutrinos upon encountering the
planet earth just pass right through it as if it werenrsquot there
Gamma decay the nucleons in the nucleus X are in an excited energy state X
(perhaps as a result of having undergone alpha or beta decay) and they drop down to a
lower energy state shedding the energy as a high frequency photon
X XA AZ Z
36
This process is exactly analogous to the way that the electrons in atoms emit photons
Decay Rate and Half Life
There is no way to predict exactly when an unstable or excited nucleus will decay but
there is an average rate at which this decay occurs called the decay constant λ The
meaning of this constant is that if there are a large number N of nuclei in a sample
then the number of decays per second that will be observed (called the decay rate R)
is R = λN In mathematical language
dMR N
dt
This simple differential equation has for its solution
0tN t N e
where N0 is the number of nuclei in the sample at time t = 0
The half-life is the time it takes for half of the nuclei in the sample to decay and is
related to the decay constant by
1 2
ln 2 0693T
Disintegration Energy
When a nucleus decays it is making a transition to an overall state of lower energy
which means according to Einsteinrsquos famous formula E = mc2 that the sum of the
masses after the decay must be less than the mass before with the lost mass appearing
as kinetic energy among the decay products For example in alpha decay this kinetic
energy called the disintegration energy Q is given by
2X YQ M M M c
37
Serway Chapter 45
Nuclear energy
There are two ways to extract energy from the nucleus fission and fusion For nuclei
with Z greater than 26 breaking the nucleus apart into pieces leads to a lower
2mc energy than the original nucleus so energy can be extracted by fission For
nuclei with Z less than 26 a lower energy is achieved by combining nuclei so energy
can be extracted by fusion
Fission
Since neutrons have no charge they are not repelled from nuclei as protons are for
this reason their behavior is the key to understanding how fission works
Fast neutrons mostly bounce off other nuclei with each collision slowing the neutron
down This elastic energy loss is most effective if the other nuclei have low mass (like
hydrogen) and these materials are called moderators because of their ability to slow
down fast neutrons
The reason that slowing neutrons is important is that slow neutrons are much more
likely to be absorbed by a nucleus which then leads to nuclear reactions of various
kinds For a few very large nuclei like uranium-235 and some plutonium isotopes
absorption of a slow neutron causes the nucleus to split into two large fragments plus
2 or 3 fast neutrons (energy is released in the process too) Thus one slow neutron
can produce energy plus 2 or 3 more neutrons which if moderated from fast to slow
can split 2 or 3 more nuclei leading to a runaway chain reaction If the reaction is
allowed to proceed unchecked it produces a large explosion If a neutron absorbing
material is added to the mix (like the cadmium in reactor control rods) it is possible to
keep the reaction under control and to extract the released energy as heat to drive
steam turbines and produce electricity
Fusion
Fusion involves mashing two nuclei together and since they are both charged and repel
each other this reaction is much harder to make go The nuclei must have enough energy to
overcome the coulomb repulsion which is why this reaction requires a high temperature
(hundreds of millions of degrees K) like that in the sun (which is burning hydrogen to
helium via fusion) or like that in the center of the fission explosion that is used to detonate
38
a hydrogen bomb
This reaction is of interest for power production in spite of this difficult temperature
requirement because of the abundance of fusion fuel on the planet There are about
012 g of deuterium ( 21H ) in every gallon of water on earth and it only costs about 4
cents to extract it The fusion energy available from this minuscule amount of
deuterium would run a 1000-Megawatt power plant for 10 seconds Water is so
abundant on earth that if fusion were to work we would have an essentially
inexhaustible source of energy
So why donrsquot we have fusion power plants Well the fuel is cheap but the match is
incredibly expensive The only way we know to control this difficult high-temperature
reaction is with large and expensive pieces of equipment involving either large
magnetic fields and complex high-power electromagnetic antennas or with gigantic
(football-fieldsized) laser facilities involving more than a hundred of the highest-
energy lasers ever built Power plants based on these current methods for controlling
fusion are unattractive to the fiscally-minded people who run the electric power
industry Hopefully better designs will be discovered as experiments continue
39
c actinium 68 Er erbium 101 Md mendelevium 104 Rf rutherfordium l aluminum 63 Eu europium 80 Hg mercury 62 Sm samarium m americium 100 Fm fermium 42 Mo molybdenum 21 Sc scandium b antimony 9 F fluorine 60 Nd neodymium 106 Sg seaborgium r argon 87 Fr francium 10 Ne neon 34 Se selenium s arsenic 64 Gd gadolinium 93 Np neptunium 14 Si silicon t astatine 31 Ga gallium 28 Ni nickel 47 Ag silver a barium 32 Ge germanium 41 Nb niobium 11 Na sodium k berkelium 79 Au gold 7 N nitrogen 38 Sr strontium e beryllium 72 Hf hafnium 102 No nobelium 16 S sulfur i bismuth 108 Hs hassium 76 Os osmium 73 Ta tantalum h bohrium 2 He helium 8 O oxygen 43 Tc technetium
boron 67 Ho holmium 46 Pd palladium 52 Te tellurium r bromine 1 H hydrogen 15 P phosphorus 65 Tb terbium d cadmium 49 In indium 78 Pt platinum 81 Tl thallium a calcium 53 I iodine 94 Pu plutonium 90 Th thorium f californium 77 Ir iridium 84 Po polonium 69 Tm thulium
carbon 26 Fe iron 19 K potassium 50 Sn tin e cerium 36 Kr krypton 59 Pr praseodymium 22 Ti titanium s cesium 57 La lanthanum 61 Pm promethium 74 W tungsten l chlorine 103 Lr lawrencium 91 Pa protactinium 92 U uranium r chromium 82 Pb lead 88 Ra radium 23 V vanadium o cobalt 3 Li lithium 86 Rn radon 54 Xe xenon u copper 71 Lu lutetium 75 Re rhenium 70 Yb ytterbium m curium 12 Mg magnesium 45 Rh rhodium 39 Y yttrium b dubnium 25 Mn manganese 37 Rb rubidium 30 Zn zinc y dysprosium 109 Mt meitnerium 44 Ru ruthenium 40 Zr zirconium s einsteinium
40
UNITS The SI units are given in parentheses Other commonly used units are given in terms of the SI units Symbols conform with the recommendations of the American National Standards Institute (ANSI) and the American Institute of Physics (AIP) Commonly used multiples of SI units
prefix symbol factor tera T 1012 giga G 109
mega M 106 kilo k 103
centi c 10-2 milli m 10-3
micro μ 10-6 nano n 10-9 pico p 10-12
femto f 10-15
length meter (m) angstrom 1 Ǻ = 10-10 m inch 1 in = 254 x 10-2 m foot 1 ft = 03048 m mile 1 mi = 1609 m
mass kilogram (kg) atomic mass unit 1 u = 1661 x 10-27 kgslug 1 slug = 1459 kg
force newton (N = kg bull ms2) dyne 1 dyn = 10-5 N pound 1 lb = 4448 N
pressure Pascal (Pa = kgm bull s2) atmosphere 1 atm = 1013 x 105 Pa poundssquare inch 1 psi = 6895 Pa cm of mercury 1 cm Hg = 1333 Pa bar 1 bar = 1000 x 105 Pa torr 1 torr = 1333 Pa
time second (s) minute 1 min = 60 s hour 1 h =3600 s
frequency hertz (Hz = s-1) radianssecond 1 rads = 12π Hz
energy joule (J = kg bull m2s2) erg 1 erg = 10-7 J electron volt 1 eV = 1602 x 10-19 J calorie 1 cal = 4187 J kilowatt-hour 1 kWh = 36 x 106 J British thermal unit 1 Btu = 1055 J
power watt (W = kg bull m2s3) horsepower 1 hp = 7457 W
charge coulomb (C = A bull s)
electric potential volt (V = kg bull m2 s3 bull A)
current ampere (A)
resistance ohm (Ω = kg bull m2s3 bull A2)
capacitance farad (F = s4 bull A2kg bull m2)
magnetic field tesla (T = kgs2 bull A) gauss 1 G = 10-4 T
magnetic flux weber (Wb = kg m2s2 bull A) maxwell 1 Mx = 10-8 Wb
magnetic inductance henry (H = kg bull m2s2 bullA2)
temperature kelvin (K) degrees Celsius 0degC = 27315 K
angle radian (rad) degree 1 deg = π180 rad revolution 1 rev = 2π rad
41
Some Physical Constantsa Quantity Symbol Valueb Atomic mass unit μ 1660 538 73 (13) x 10-27 kg
931494 013 (37) MeVc2 Avogadros number NA 6022 141 99 (47) x 1023 particlesmol Avogadrorsquos number 6022 x 1023 mol Bohr magneton
2B e
em 9274 008 99 (37) x 10-24 JT
Bohr magneton μB 927 x 10-24 JT Bohr radius
0
2
2e em e k
a 5291 772 083 (19) x 10-11 m
Boltzmanns constant B A
RNk 1380 650 3 (24) x 10-23 JK
Boltzmannrsquos constant kB 1380 x 10-23 JK Compton wavelength
C eh
m c 2426 310 215 (18) x 10-12 m
Deuteron mass md 3343 583 09 (26) x 10-27 kg 2013 553 212 71 (35) u
electron charge e 1602 x 10-19 C Electron mass me 9109 381 88 (72) x 10-31 kg
5485 799 110 (12) x 10-4 u 0510 998 902 (21) MeVc2
electron mass 911 x 10-31 kg Electron volt eV 1602 176 462 (63) x 10-19 J Elementary charge e 1602 176 462 (63) x 10-19 C Gas constant R 8314 472 (15) JKmol Gravitational constant G 6673 (10) x 10-11 Nm2kg2 Neutron mass mn 1674 927 16 (13) x 10-27 kg
1008 664 915 78 (55) u 939565 330 (38) MeVc2
neutron mass 1675 x 10-27 kg Nuclear magneton
2n p
em 5050 783 17 (20) x 10-27 JT
permeability constant μ0 1257 x 10-6 Hm permittivity constant ε0 8854 x 10-12 Fm Plancks constant h
2h
6626 068 76 (52) x 10-34 Js 1054 571 596 (82) x 10-34 Js
Planckrsquos constant h ħ
6626 x 10-34 Js 1055 x 10-34 Js
Proton mass mp 1672621 58 (13) x 10-27 kg 1007 276 466 88 (13) u 938271 998 (38) MeVc2
proton mass 1673 x 10-27 kg Rydberg constant RH 1097 373 156 854 9 (83) x 107 m-1 Speed of light in vacuum c 2997 92458 x 108 ms (exact) speed of light c 300 x 108 ms a
These constants are the values recommended in 1998 by CODATA based on a least-squares adjustment of data from different measurements For a more complete list see P J Mohr and B N Taylor Rev Mod Phys 72351 2000 b The numbers in parentheses for the values above represent the uncertainties of the last two digits
NOTE The ones in red are the ones from Appendix 13 The ones in yellow are the ones that are from the Serway book table but that were already on the Appendix 13 table
42
INDEX
Absolute zero 9
Adiabatic 17
Adiabatic exponent 18
Adiabatic process 17
Alpha decay 46
Amplitude 4
Angle of reflection 25
Angular frequency 3
Angular magnification 30
Angular momentum atomic 43
Angular size 29
Archimedes Principle 1
Atomic spectra 43
Atoms 43
Avogadrorsquos number 10
Beats 8
Bernoullirsquos Equation 2
Beta decay 46
Bifocals 29
Brewsterrsquos angle 34
British Thermal Unit Btu 11
Buoyancy 2
calorie 11
Calorie 11
Camera 28
Carnot cycle 21
Carnot efficiency 22
Celsius scale 9
Ciliary muscle 29
Coefficient of performance 21
Compton effect 39
Constructive interference 7
Continuity equation of 2
Contraction length 35
Convection 14
Curved mirrors 27
Decay constant 46
Decay rate radioactive 46
Decibel scale 5
Degrees of freedom 16
Density 1
Destructive interference 7
Diffraction single slit 33
Diffraction grating 33
Dilation time 35
Disintegration energy 47
Dispersion of light 26
Doppler effect 6
Double slit interference 31
Efficiency 20
Electron interference 39
Emissivity 15
Energy nuclear decay 47
Energy relativistic 36
43
Engineering work 20
Entropy 22
Entropy ideal gas 24
Equipartition of energy 18
Equivalence principle 36
Exclusion principle 44
Expansion thermal 9
Expansion coefficient linear 9
Eye 29
Fahrenheit scale 9
Far point 29
Farsightedness 29
First Law of Thermodynamics 12
Fission 48
Flux volume 2
Free expansion 23
Fringes 31
Fusion 48
Fusion heat of 11
Gamma relativistic 35
Gamma decay 46
General relativity 36
Half-life 46
Heat 11
Heat capacity 11
Heat conduction 13
Heat engine 20
Heat of fusion 11
Heat of vaporization 11
Heat pump 20
Heats of transformation 11
Hydrostatics 1
Ideal Gas Law 10
Images real and virtual 27
Intensity sound 5
Interference 7
Interference two-slit 31
Internal energy 12
Internal energy degrees of freedom 16
Irreversible process 19 23
Isotopes 45
Joule 11
Kelvin scale 9
Kinetic energy relativity 36
Kinetic theory 16
Latent heat 11
Length contraction 35
Linear expansion coefficient 9
Linear polarization 34
Linear superposition 7
Longitudinal wave 3
Loudness 5
Magnification lateral 27
Magnifying glass 30
Malusrsquos law 34
Microscope 30
Momentum relativistic 36
Muscle ciliary 29
44
Musical instruments 8
Musical scale 8
Near point 29
Nearsightedness 29
Nonlinear 7
Nuclear energy 48
Nuclear properties 45
Nuclear stability 45
Octave 8
Optical resolution 33
Orbital quantum number ℓ 43
Orbitals 43
Particle in a Box 41
Particles are waves 39
Pascalrsquos Principle 1
Period 3
Photoelectric effect 38
Photons 38
Pitch 8
Polarization 34
Power sound 5
Presbyopia 29
Pressure 1
Principal quantum number n 43
Principle of equivalence 36
Principle of linear superposition 7
Probability amplitude ψ 41
Processes thermodynamic 13
R-value 14
Radiation thermal 14
Radioactive decay 46
Radioactive decay rate 46
Ray tracing 27
Rayleighrsquos criterion 33
Reading glasses 29
Real image 27
Refraction 25
Refrigerator 20
Relativistic gamma 35
Relativity principles 35
Resolved for light sources 33
Rest energy 36
Reversible process 19 22
Rope wave speed 4
Second Law of Thermodynamics 19
Shock waves 6
Simple magnifier 30
Simultaneity 35
Single slit diffraction 33
Snellrsquos law 25
Sound speed 5
Specific heat 11
Spin Angular momentum 44
Standing waves 7
Stefanrsquos law 14
Strong force 45
Telescope 30
Temperature 9
45
Temperature Scales 9
Thermal conductivity 14
Thermal energy 12
Thermal expansion 9
Thin film interference 31
Thin lenses 28
Time dilation 35
Tone musical 8
Total internal reflection 25
Transverse wave 3
Traveling Waves 3
Two-slit interference 31
Uncertainty principle 40
Vaporization heat of 11
Velocity addition relativity 36
Virtual image 27
Volume flux 2
Wave function 41
Wave Function ψ and Probability 41
Wave speed 3
Wave-particle duality 39
Wavelength 3
Wavenumber 3
Wien Displacement Law 16
Work 11
10
where Li is the initial length of a rod of the material and ΔL is the change in its length due to a
small temperature change ΔT The coefficient of volume expansion is defined similarly
1
i
V
V T
where Vi is the initial volume of a piece of material and where ΔV is the change in its volume due
to a small temperature change ΔT
Avogadrorsquos Number (N or NA)
One mole of any substance corresponds to 6022 times 1023 molecules The atomic mass given on the
periodic table (p A32-33 in the text) is the mass in grams of one mole of atoms For example
the atomic mass of oxygen is 15999 gmol The mass of a single atom is given by the same
number in units of atomic mass units (u) For example the mass of an oxygen atom is 15999 u
(on the average)
Ideal Gas Law (an example of an equation of state)
When the molecules of a gas are sufficiently inert and widely separated that interactions between
them are negligible we say that it is an ideal gas The pressure P volume V and temperature T
(in kelvins) of such a gas are State Variables and are related by the ideal gas law
Bor PV=NkPV nRT T
where n is the number of moles of the gas where R is the gas constant
8314 Jmol KR
where N is the number of molecules and where kB is Boltzmannrsquos constant
231380 10 J KBk
It works well for air at atmosphere pressure and even better for partial vaccuums The relative
ease of measuring pressure and the linear relationship between pressure and temperature (if V
and n are held fixed) makes an ideal gas an ideal thermometer Thermometers can be based on
properties of solids or liquids but the behavior of these materials with temperature is more
complicated
11
Serway Chapter 20
Heat
Heat is energy that flows between a system and its environment because of a tempera-
ture difference between them The units of heat are Joules as expected for an energy
Unfortunately there are several competing units of energy They are related by
1 cal 4186 J 1 Cal 4186 J 1 Btu 1054 J
Heat Capacity
There is often a simple linear relation between the heat that flows in or out of part of a
system and the temperature change that results from this energy transfer When this
linear relation holds it is convenient to define the heat capacity C and the specific
heat c as follows
For the entire object the heat Q it absorbs is given by Q = C(Tf - Ti)
For a particular material the specific heat is defined by c = Cm which is the heat
capacity per unit mass so that
Q = mc(Tf ndash Ti)
Note C has units of energy (J or Cal)(Kelvin kg)
Heats of Transformation or Latent Heat Q = plusmn mL
When a substance changes phase from solid to liquid or from liquid to gas it absorbs
heat without a change in temperature The latent heat or heat of transformation is
usually given per unit mass of the substance For example for water the heat of fusion
(melting) is L = 333 kJkg while the heat of vaporization (boiling) is L = 2260 kJkg
Note that heat for boiling is considerably bigger than melting for water You have to
be careful with signs heat is given off (negative) if you go down in temperature and
condense steam
Work
In general the small amount of work done on a system as a force Fon is exerted on it
through a vector displacement dx is given by
on xdW d F
12
But if the displacement is done very slowly (as we always assume in thermodynamics)
then the force exerted on the system and the force exerted by the system are in
balance so the force exerted by the system is ndash Fon In thermodynamics it is more
convenient to talk about the force exerted by the system so we change the above
formula for the work done on the system to
xdW d F
where F is the force exerted by the system This has confused students for more than a
century now but this is the way your book and many other books do it so you are
stuck You will need to memorize the minus sign in this definition of the work to be
able to use your textbook
There are many chances to get signs wrong in this and the next two chapters (Mosiah
2 )
When an external agent changes the volume of a gas at pressure P by a small amount
dV the (small amount) of work done on the system is given by
dW PdV
Notice that this minus sign is just what we need to make dW be positive if the external
agent compresses the gas for then dV is negative If on the other hand the external
agent gives way allowing the gas to expand against it then dV is positive and we say
that the work done on the gas is negative
The work done on the system (eg by the gas in a cylinder) in a thermodynamic
process is the area under the curve in a PV diagram It is positive for compressions
and negative for expansions If the volume of gas remains constant in a process then
no work is done by the gas
Cyclic processes are important For cyclic processes represented by PV diagrams the
magnitude of the net work during one cycle is simply the area enclosed by the cycle
on the diagram Be careful to keep track of signs when you are calculating that
enclosed area In cyclic process Q = ndashW for a cycle Put a PV diagram in your notes
1
Path A-B B-C C-D D-A A to A net
Q
W
ΔU
ΔS
Internal Energy
The energy stored in a substance is called its internal energy Eint This energy may be
stored as random kinetic energy or as potential energy in each molecule (stretched
chemical bonds electrons in excited states etc) For ideal gases all states with the
same temperature will have the same Eint
First Law of Thermodynamics
The change ΔEint in the internal energy of a system is given by
intE Q W
where Q is the heat absorbed by the system and where W is the work done on the
system Hence if a system absorbs heat (and if Wge0) the internal energy increases
Likewise if the system does work (W on the system is negative) and if Qge0 the
internal energy decreases Potential Pitfall Many times people talk about work done
by the system It is the minus of W on the system Donrsquot get tripped up
Processes
Adiabatic process No heat is exchanged Q = 0 requiring that the system be insulated
from the environment A process may be approximately adiabatic if it happens so
rapidly that heat does not have time to enter or leave the system Work + or ndash is done
and ΔEint = W
Adiabatic free expansion A gas is allowed to expand into a vacuum without pushing
on anythingndashit just rushes into the vacuum In this process Q = W = 0 so the internal
energy and temperature does not change (Note the difference between an adiabatic
process and a free expansion is that NO work is done in the adiabatic free expansion)
Isobaric process The pressure is held fixed ΔP = 0 For example usually the
pressure increases when a gas is heated but if it were allowed to expand during the
2
heating process in just the right way its pressure could remain fixed In isobaric
processes both Q and W are non-zero (Work is eacy = ndashP(Vf ndash Vi)
Isovolumetric process The volume is held fixed ΔV = 0 The work done by the gas is
then zero and so we have ΔEint = Q
Isothermal process The temperature is held fixed ΔT = 0 There is no change in Eint
so Q = ndashW for isothermal processes For an ideal gas PV = nRT so PV = constant in
an isothermal process The work done on the gas is then given by
lnf
i
Vi
Vf
VW PdV nRT
V
Heat Conduction
The quantity P is defined to be the rate at which heat flows through an object and is a
power having units of watts It is analogous to electric current which is the rate at
which charge flows through an object If the flow of heat through a slab of length L
and cross-sectional area A is steady in time then P is given by the equation
h cT TdQkA
dt L
P =
where Th and Tc are the (hot and cold) temperatures of the two ends of the slab The
heat flows of course because of this temperature difference The quantity k is called
the thermal conductivity and is a constant that is characteristic of the material It is
analogous to the electrical conductivity h cT TL is sometimes called the temperature
gradient and is written dTd dTdx
R-Values
It is common to have the heat-conducting properties of materials described by their R-
values especially for insulating materials like fiberglass batting The connection
between k and R is R = Lk where L is the material thickness In this country R-values
always have units of 2ft F hourBtu
Polyurethane foam 6 (1 in thick) Air space 1 (35 in thick) Fiber glass batting 11
(35 in thick) Wood about 1 (1 in thick) Glass 09 (0125 in thick)
The heat flow rate through a slab of area A is given by
3
h cT TA
R
P
in units of Btuhour Note that A must be in square feet and the temperatures must be
in degrees Fahrenheit
Convection
Convection is the transfer of thermal energy by flow of material For instance a home
furnace doesnrsquot heat a house by waiting for the heat from the burner to slowly conduct
throughout the house instead it quickly pumps warm air to all of the rooms
Generally convection is a much faster way to transfer heat than conduction
Radiation
Electromagnetic radiation can also transfer heat When you warm yourself near a
campfire which has burned itself down into a bed of glowing embers you are
receiving radiant heat from the infrared portion of the electromagnetic spectrum The
rate at which an object emits radiant heat is given by Stefanrsquos law
4AeTP
where P is the radiated power in watts σ is a constant
8 2 45696 10 W m K
A is the surface area of the object in m2 and T is the temperature in kelvins The
constant e is called the emissivity and it varies from substance to substance A perfect
absorber (think black velvet) has e = 1 while a perfect reflector (think mirror) has e =
0 Hence black objects radiate very well while shiny ones do not Also an object that
is hotter than its surroundings radiates more energy than it absorbs whereas an object
that is cooler than its surroundings absorbs more energy than it radiates
Terminology
Transfer variables vs state variables
Energy transfer by heat as well as work done depends on the initial final and
intermediate states of the system They are transfer variables But their sum (Q + W =
4
Eint) is a state variable
Figure 205
5
Serway Chapter 21
Kinetic Theory
The ideal gas law works for all atoms and molecules at low pressure It is rather
amazing that it does Kinetic theory explains why The properties of an ideal gas can
be understood by thinking of it as N rapidly moving particles of mass m As these
particles collide with the container walls momentum is imparted to the walls which
we call the force of gas pressure In this picture the pressure is related to the average
of the square of the particle velocity 2v by
22 1( )
3 2
NP mv
V
Using the ideal gas law we obtain the average translational kinetic energy per
molecule
21 3
2 2 Bmv k T
The rms speed is then given by
2rms
3 3Bk T RTv v
m M
where M is the molecular mass in kgmol
Degrees of Freedom
Roughly speaking a degree of freedom is a way in which a molecule can store energy
For instance since there are three different directions in space along which a molecule
can move there are three degrees of freedom for the translational kinetic energy
There are also three different axes of rotation about which a polyatomic molecule can
spin so we say there are three degrees of freedom for the rotational kinetic energy
There are even degrees of freedom associated with the various ways in which a
molecule can vibrate and with the different energy levels in which the electrons of
the molecule can exist
Internal Energy and Degrees of Freedom The internal energy of an ideal gas made
up of molecules with J degrees of freedom is given by
int 2 2 B
J JE nRT Nk T
6
Heat Capacities of an Ideal Gas The heat capacity of a gas is described by means of
molar heat capacities CV and CP These are the heat capacities per mole and the
subscript V on CV means that the volume is being held constant while for CP the
pressure is held constant For example to raise the temperature of n moles of a gas
whose pressure is held constant by 10 K we would have to supply an amount of heat Q
= nCP (10) K
Molar Specific Heat of an Ideal Gas at Constant Volume
VQ nC T
3monatomic
2VC R
5diatomic
2VC R
5polyatomic
2VC
Real gases deviate from these formulas because in addition to the translational and ro-
tational degrees of freedom they also have vibrational and electronic degrees of
freedom These are unimportant at low temperatures due to quantum mechanical
effects but become increasingly important at higher temperatures The rough rule is
No of degrees of freedom
2VC R
Molar Specific Heat of an Ideal Gas at Constant Pressure
PQ nC T
P VC C R
The internal energy of an ideal gas depends only on the temperature
int VE nC T
Adiabatic Processes in an Ideal Gas
7
An adiabatic process is one in which no heat is exchanged between the system and the
environment When an ideal gas expands or contracts adiabatically not only does its
pressure change as expected from the ideal gas law but its temperature changes as
well Under these conditions the final pressure Pf can be computed from the initial
pressure Pi and from the final and initial volumes Vf and Vi by
or constantf f i iP V PV PV
where γ = CPCV The quantity γ is called the adiabatic exponent Note that this
doesnrsquot mean that the ideal gas law no longer holds it does and in fact it can be
combined with the adiabatic law for pressure given above to obtain the adiabatic law
for temperatures
1 constantTV
Compressions in sound waves are adiabatic because they happen too rapidly for any
appreciable amount of heat to flow This is why the adiabatic exponent appears in the
formula for the speed of sound in an ideal gas
RTv
M
Note that v depends only on T and not on P Because it depends only on the
temperature the speed of sound is the same in Provo as at sea level in spite of the
lower pressure here due to the difference in elevation
Equipartition of Energy
Every kind of molecule has a certain number of degrees of freedom which are
independent ways in which it can store energy Each such degree of freedom has
associated with it ndash on average ndash an energy of 12 Bk T per molecule (or 1
2 RT per mole)
(Note since a molecule has so many possible degrees of freedom it would seem that
there should be a lot of 12 sBk T to spread around But because energy is quantized
some of these degrees of freedom are not ldquoactiverdquo until the temperature becomes high
enough that 12 Bk T is as big as the lowest quantum of energy
8
Serway Chapter 22
Second Law of Thermodynamics
There are several equivalent forms of this important law
Kelvin It is not possible to change heat completely into work with no other change
taking place Or in other words there are no perfect heat engines
Clausius It is not possible for heat to flow from one body to another body at a higher
temperature with no other change taking place Or in other words there are no
perfect refrigerators
Entropy In any thermodynamic process that proceeds from one equilibrium state to
another the entropy of the system + environment either remains unchanged or
increases The total entropy never decreases This law is a bit of an oddity among the
laws of physics because it is not absolute Things are forbidden by the second law not
because it is impossible for them to happen but because it is extremely unlikely for
them to happen (See below for more information about entropy)
Reversible and Irreversible Processes
A reversible process is one which occurs so slowly that it is in thermal equilibrium (or
very nearly so) at all times A hallmark of such processes is that a motion picture of
them looks perfectly normal whether run forward or backward Imagine for instance
the slow expansion of a gas at constant temperature in a cylinder whose volume is being
increased by a slowly moving piston Run the movie backwards and what do you see
You see the slow compression of a gas at constant temperature which looks perfectly
normal
An irreversible process is one which occurs in such a way that thermal equilibrium is
not maintained throughout the process The mark of this kind of process is that a motion
picture of it looks very odd when run backward Imagine the sudden expansion of a gas
into a previously evacuated chamber because a hole was punched in the wall between a
pressurized chamber and the evacuated one Run the movie backward and what do you
see You see the gas in the soon-to-be-evacuated chamber gather itself together and
stream through a tiny hole into a chamber in which there is already plenty of gas If you
have ever seen this happen get in touch with the support group for those who have
witnessed the spontaneous combustion of large mammals by calling 1-800-PYROCOW
9
Heat Engines
Because of the vexing difference in sign between work done on and system and work done
by a system we will invent a new work variable Weng Heat engines do work and so the net
W for these engines is negative But in engineering applications hidden minus signs are
regarded as evil so for heat engines we donrsquot talk about W instead we talk about its
magnitude engW W So for heat engines the first law is
int engE Q W
But for heat pumps and refrigerators work is done on the system so we use the usual work
W when we talk about these systems
A heat engine is a machine that absorbs heat converts part of it to work and exhausts the
rest The heat must be absorbed at high temperature and exhausted at low temperature If the
absorbed heat is Qh the exhausted heat is Qc and the work done by the engine is Weng then
eng h cW Q Q
and the efficiency of the engine is defined to be
eng
h
We
Q
A perfect engine would convert the heat hQ completely into work Weng giving an effi-
ciency of e = 1 Energy conservation alone allows a perfect engine but the second law
requires e lt 1
Refrigerators and Heat Pumps
A refrigerator is a machine that absorbs heat at low temperature and exhausts it at high
temperature the ldquobackwardsrdquo heat transfer being driven by the work done on the machine
by some source of power A heat pump is a machine that either works like a refrigerator
keeping a place cold by transferring heat from this cold place to a higher temperature
environment (cooling mode like an air conditioner) or it functions as a heater
transferring heat into a warm place from a cooler one (heating mode like a window unit
that heats a house by extracting thermal energy from the cold outdoors) The coefficient
of performance of a refrigerator or of a heat pump in cooling mode is defined to be
COP cooling modecQ
W
10
For a heat pump in heating mode the coefficient of performance is
COP heating modehQ
W
Note that we donrsquot have to use the engineering work here because in these systems
work is done on the system and W is naturally positive
A perfect refrigerator would take in heat Qc and exhaust the same amount of heat without
doing any work giving an infinite coefficient of performance Energy conservation alone
allows a perfect refrigerator but the second law requires COP lt infin A perfect heat pump
in heating mode would transfer Qh into the house without doing any work and so would
also have an infinite coefficient of performance The second law forbids this too A good
coefficient of performance for a real device would be around 5 or 6
Carnot Cycle
The most efficient of all possible engines is one that uses the Carnot cycle This cycle
employs an ideal gas has no friction and operates very slowly so that the gas can be in
thermal equilibrium at all parts of the cycle This means of course that it canrsquot
possibly be built and even if it could be built it would not run fast enough to be useful
Nevertheless this cycle is very important because it gives an upper bound on the
efficiency of real engines There cannot possibly be an engine that is more efficient
than one based on the Carnot cycle This cycle consists of the following four steps
1 The ideal gas absorbs heat Qh at constant temperature Th while the gas increases its
volume The reason that heat is absorbed is that expansion tends to cool the gas but
thermal contact with the environment at Th keeps the temperature high by heat
conduction into the ideal gas
2 The ideal gas further increases its volume by an adiabatic expansion This expansion
causes the gas to cool so at the end of this part of the cycle the gas is at temperature Tc
3 The gas exhausts heat Qc at constant temperature Tc while the gas decreases in
volume The reason that heat is exhausted is that compression tends to heat the gas
but thermal contact with the environment at Tc keeps the temperature low by heat
conduction out of the ideal gas
4 The gas is adiabatically compressed back to its original volume (the volume it started
with in step 1) This compression heats the gas from Tc up to Th
11
The efficiency of a Carnot engine is given by the very simple formula
1 cC
h
Te
T
where the temperatures must in be Kelvin No real engine can be more efficient than
this
The coefficient of performance of a Carnot refrigerator or heat pump in cooling mode
is given by
COP cooling mode cC
h c
T
T T
and no real refrigerator can have a coefficient of performance greater than this
The coefficient of performance of a Carnot heat pump in heating mode is
COP heating mode hC
h c
T
T T
So why donrsquot we just use these wonderful Carnot engines and have perfect efficiency
To make the reversible steps in the cycle really reversible they would have to occur
infinitely slowly So the price you pay for making a perfect engine is that it takes
forever to get it to do any work
Entropy
The entropy of a system is defined in terms of its molecular makeup and measures
roughly the disorder of the system If the system is packed into a very small volume
then it is quite ordered and the entropy will be low If it occupies a large volume the
entropy is high (To see what this has to do with disorder note that socks in a drawer
occupy a small volume while socks on the bed in the corner by the door and
hanging from the chandelier occupy a large volume) If the system is very cold then
the molecules hardly move and may even reach out to each other and form a crystal
This is a highly ordered state and therefore has low entropy If the system is very hot
with rapidly speeding molecules crashing into the container walls and bouncing off
each other things are disordered and the entropy is high
It is possible to calculate the entropy of a system in terms of its macroscopic thermody-
namic properties ie pressure volume temperature number of moles etc The key to this
calculation is the concept of a reversible process A reversible process is one that is
12
carried out without friction and so slowly that the process can be reversed at any stage by
making an infinitesimal change in the environment of the system The slow expansion of the
gas at Th in the Carnot cycle with tiny amounts of heat being transferred to the environment
is reversible If we turned around and began slowly to compress the gas would just slowly
exhaust heat to the environment in the exact reverse way that it absorbed it during expansion
Most processes however are irreversible For example if a gas-filled box were suddenly
increased in size so that the particles were free to wander into the void created by the sudden
expansion then the gas would eventually fill the new volume uniformly at the same
temperature as before the expansion (The temperature is unchanged in this imaginary
process because the kinetic energy of the molecules would be unaffected by such an
instantaneous expansion of the container walls) This imaginary but highly thought-
stimulating process is called a free expansion and it is impossible to reverse it During the
expansion we didnrsquot push on any of the molecules so reversing this process would mean
making them go back into their original volume without pushing on them they simply will
not cooperate to this extent Another way to see that just pushing them back where they came
from does not reverse the free expansion is to think about what would happen if we just
compressed either adiabatically or isothermally An adiabatic compression back to the
original volume would heat the gas above its original temperature and an isothermal
compression would require that heat be exhausted to the environment But the free expansion
involved neither temperature changes nor heat exchanges so neither of these two processes
is the reverse of the free expansion It is simply impossible to reverse this rapid expansion in
a way that takes us back to the initial state
It is possible to calculate the change of entropy for both reversible and irreversible processes
Letrsquos consider a reversible process first In a reversible process the entropy change is given
by the formula
dQS
T
where dQ is the amount of heat added to the system during a small step of the process
The total energy change during the process may then simply be calculated by integration
f f
f i i i
dQS S S dS
T
13
(Just as in the case of energy we are mainly interested in differences rather than in
absolute magnitudes)
This integration method does not work for irreversible processes For instance in the
rapid free expansion discussed above no heat is added to the system but its disorder
obviously goes up We need to find some other way to calculate the entropy The key
is the fact that the entropy of a system depends only on its current state and not at all
on how it arrived there So to calculate the entropy change in an irreversible process
first find out what the initial and final conditions of the process are Then invent a
reversible process that takes the system from the initial state to the final state Since
the entropy depends only on the state of system and not on the process the entropy
change for the reversible process is the same as that for the irreversible process
Entropy is thus like pressure or temperature if the pressure changes from Pl to to P2
then the pressure difference is P2 - Pl regardless of how the pressure change was
made This seems obvious for pressure but not for entropy only because you arent
comfortable with entropy yet
Entropy of an Ideal Gas
For n moles of an ideal gas the difference in entropy between a state with temperature
T and volume V and some standard state with temperature To and volume Vo is given
by
ln lno o Vo o
T VS T V S T V nC nR
T V
Entropy in a Phase Change
Calculating entropy change in a phase change like melting or boiling is easy It is
QT Where Q is in the latent heat for example the flows in to cause the melting and
T is the temperature of the phase change There is NO integral to do
14
Serway Chapter 35
Angle of Reflection
If plane waves are incident on a reflecting surface with the propagation direction of
the waves making angle θ1 with the normal direction to the surface then the reflected
angle θ1prime relative to the surface normal is simply
1 1
ie the incident angles and reflected angles are the same
Refraction
If plane waves traveling through medium 1 are incident on a plane interface between
medium 1 and medium 2 then the angle of incidence of the incoming wave θ1 and
the angle of refraction of the transmitted wave θ2 are related by Snellrsquos law
2 2
1 1
sin
sin
v
v
where v1 and v2 are the wave speeds in medium 1 and medium 2 The angle of
incidence and the angle of refraction are both measured between the wave propagation
direction and the normal to the interface In terms of indices of refraction in the case
of light waves Snellrsquos law takes the more familiar form
1 1 2 2sin sinn n
where ni = cvi
Total Internal Reflection
If a wave is incident from a medium of low wave speed into a medium of high wave
speed the law of refraction requires that the angle of refraction be greater than the
angle of incidence If the angle of refraction is required to be greater than 90deg then no
refracted wave can exist and total internal reflection occurs The critical incident
angle θc beyond which total internal reflection occurs is given by
2
1
sin c
n
n
15
Dispersion of Light
In addition to the speed of light varying from material to material it also varies with
wavelength within each material This means that the index of refraction is generally a
function of wavelength
c
nv
Since the wave speed is not constant such a medium is dispersive meaning in this
context that refraction actually disperses white light into its various colors because
Snellrsquos law gives a different angle for each wavelength In most materials the
variation with wavelength is quite small but this small effect is responsible for some
of the most spectacular color effects we ever see including rainbows a flashing
crystal chandelier and the colored fire of a diamond solitaire by candlelight
16
Serway Chapter 36
Real and Virtual Images
When light rays are focused at a certain plane producing an image if a sheet of white
paper is placed there we call the image a real image The images produced by film
projectors and overhead projectors are examples of real images
When light rays appear to come from a certain location but no image is produced
when a screen is placed there we say that there is a virtual image at that location For
instance when you look in a mirror it appears that someone is behind the mirror but
a screen placed back there in the dark would show nothing Your image in the mirror
is a virtual image
Ray Tracing
There are lots of rules about how to find the images in optical systems but the best
way to keep things straight is to learn how to draw the principal rays for curved
mirrors and lenses The rays for convex and concave mirrors are shown in Fig 3615
The rays for converging and diverging thin lenses are shown in Fig 3627 You should
memorize the principal rays and know how to use them to locate images
Curved Mirrors
The focal length of a curved mirror with radius of curvature R is given by
2
Rf
The relation between object distance p image distance q and focal length f is
1 1 1
p q f
If the mirror is a diverging mirror f should be negative and if q should turn out to be
negative the image is virtual
Lateral Magnification
The lateral magnification in an optical system is defined by the ratio of the image size
to the object size
17
Image height
Object height
qM
p
for curved mirrors
As usual there are sign conventions here too but this formula just gives the
magnitude It is better to keep keep track of upright images versus inverted images by
means of ray diagrams rather than by memorizing sign conventions
Thin Lenses
The focal length of a thin lens is related to the radii of curvature of the two faces R1
and R2 of the lens by
1 2
1 1 11n
f R R
Note that this formula differs from Eq (3611) in the text by not having a minus sign
between the two R-terms We like this form better because for a simple converging
lens like a magnifying glass we just use positive values of R for both surfaces If one
of the faces is concave producing divergence use a negative value for R And if a
surface is flat use R = infin If the face is flat the radius is infinite
The relation between the image object and focal distances for a thin lens is the same
as that for a curved mirror
1 1 1
q p f
Use a negative focal length if the lens is diverging
The lateral magnification for a thin lens is the same as for a curved mirror
Image height
Object height
qM
p
Camera
The lens system in a camera projects a real image of an object onto the film (or CCD
array in a digital camera) The position of the image is adjusted to be on the film by
moving the lens into or out of the camera
18
Eye
The eye is like a camera in that a real image is formed on the retina Unlike a camera
the image position is adjusted by changing the focal length of the lens This is done by
the ciliary muscle which squeezes the lens changing its shape
Near Point The near point is the closest distance from the eye for which the lens can
focus an image on the retina It is usually 18-25 cm for young persons
Far Point The far point is the greatest distance from the eye for which the lens can
focus an image on the retina For a person with normal vision the far point is at
infinity
Nearsightedness A person is nearsighted if their far point is at some finite distance
less than infinity This condition can be corrected with a lens that takes an object at
infinity and produces a virtual image at the personrsquos far point
Farsightedness and Presbyopia A person is farsighted if their near point is too far
away for comfortable near work like reading or knitting This can be corrected by a
lens which takes an object at a normal near point distance of 18-25 cm and produces a
virtual image at the personrsquos natural near point Presbyopia involves a similar
problem which nearly all people experience as they age The ciliary muscle becomes
too weak and the lens becomes too stiff to allow the eye to provide for both near and
far vision The solution for this problem is either reading glasses or bifocal lenses
Reading classes are just weak magnifying glasses mounted on eyeglass frames
Bifocal lenses are split into upper and lower halves The lower half is a lens which
gives the proper correction for near work and the upper half is a different lens for
proper focusing at infinity
Angular Size
When an object is brought closer to the eye it appears to be larger because the image
on the retina is larger The size of this image is directly proportional to the objectrsquos
angular size which is the angle subtended by the object measured from the center of
the lens of the eye In optical instruments which are to be used with the eye the
angular size of the final image is whatrsquos important because it determines how large
the image will appear to the viewer
19
Simple Magnifier
A simple magnifier is a single converging lens or magnifying glass It takes an object
closer to the eye than a normal near point and produces a virtual image at or beyond
this near point The angular magnification is defined to be the ratio of the angular size
when viewed through the lens to the angular size of the object when viewed at the
normal near point (without aid of the lens)
Microscope
This instrument has two lenses (1) The objective is near the object being viewed and
produces a greatly magnified real image (2) The eyepiece is a simple magnifier which
the viewer uses to closely examine the image from (1)
Telescope
This instrument also has two lenses (1) The objective at the front of the telescope
takes light from a distant object and produces a real inverted image (which is rather
small) near its focal point (2) This small real image is then examined by the eyepiece
functioning as a simple magnifier to produce a virtual image with a larger angular
size
20
Serway Chapter 37
Two-Slit Interference
If light is incident on two closely spaced narrow slits a pattern of light and dark
stripes is produced beyond the slits The bright stripes or fringes are caused by
constructive interference of the two waves coming from the slits Constructive
interference occurs whenever two waves arrive at a location in phase with each other
This occurs when the distance x1 from slit 1 to a point P on the screen and the
distance x2 from slit 2 to point P differ by in integral number of wavelengths
1 2 where 0 1 2x x m m
where λ is the wavelength of the light When x1 and x2 are much larger than the slit
spacing d this condition reduces to
sind m
where θ is the angle between the direction of the incident light and the direction of the
light arriving at the screen
Thin Films
When light is partially reflected and partially transmitted by a thin film of transparent
material it is possible to have interference between the wave reflected from the front
of the film and light reflected from the back of the film (The colored reflections from
the thin film of oil on the water in a rain-soaked parking lot are an example of this
effect) It is difficult to write down formulas that will work in all cases so we will
just review the important principles here
1 If the two reflected waves are in phase with each other the film has enhanced
reflection (constructive interference) but if the two reflected waves are out of phase
with each other reflection is diminished (destructive interference) Phase shifts occur
due to reflection and due to the extra path length through the film of the wave
reflected from the back of the film
2 The phase change due to reflection is determined by the difference in index of
refraction between the two media involved in the reflection If the wave is incident
21
from a medium with a low index of refraction into a medium with a high index of
refraction a phase change of 180deg occurs and the reflected wave is inverted If
incident from high to low no phase shift occurs and the reflected wave is non-
inverted
3 The extra path length through the film of the wave reflected from the back of the
film is equal to 2t where t is the thickness of the film (The incident light is assumed
to be normal to the surface of the film) The number of wavelengths contained in the
extra path length is equal to 2tλn where λn = λn is the wavelength of the light in the
film
4 Rules for reflection from thin films
If one ray is inverted and the other is not then we have
12 constructive
2 nt m
2 destructivent m
If either both rays are inverted or both are non-inverted we have
2 constructivent m
12 destructive
2 nt m
22
Serway Chapter 38
Diffraction Grating
A diffraction grating is simply a fancy version of two-slit interference with the two
slits replaced by thousands of slits Just as in the two-slit case the bright fringes
occur at angles given by
sin md
but in this case d the distance between neighboring slits is made to be very small
The effect of having many slits instead of two is to make each bright fringe highly
localized with wide dark regions between neighboring maxima
Single Slit Diffraction
When light passes through an opening in an opaque screen an interference pattern is
produced beyond the opening To understand why we may replace the single opening
by many small coherent sources of light These many sources interfere with each
other producing a pattern known as a diffraction pattern (Note that many authors do
not distinguish between interference and diffraction treating them as interchangeable
terms) If the opening is a slit of width a then the diffraction pattern far from the slit
will have a bright central maximum with a succession of minima and weaker maxima
on either side The angle between the incident direction and the minima is given by
sin ma
where m = plusmn1 plusmn2
If the opening is circular with diameter D the angle between the incident direction
and the first minimum is given by
sin 122D
Optical Resolution and Rayleighrsquos Criterion
Two point sources can just be resolved (distinguished from each other) if the peak of
the diffraction image of the first source overlies the first minimum of the diffraction
image of the second source For circular holes of the kind usually encountered in
23
optical devices this condition is approximately satisfied when the angular separation
between the two sources as viewed from the optical instrument is greater than or equal
to the critical angle
min 122D
where λ is the wavelength of the light and where D is the diameter of the aperture in
the instrument
Polarization
We say that an electromagnetic wave is polarized if its electric field vector doesnrsquot
change direction in a random fashion The simplest kind of polarization is linear
polarization in which the electric field vector oscillates back and forth along the
same axis in space Polarized light can be produced from normal unpolarized light by
selective absorption (as in Polaroid sunglasses) by passing light through crystals that
have different indices of refraction for different polarizations (double refraction) by
scattering (the blue sky is polarized) and by reflection (glare) Polarization by
reflection occurs when light reflects from a shiny insulating (non-metallic) surface
The amount of polarization is greatest for reflection at Brewsterrsquos angle
2
1
tan P
n
n
where θP is the incidence angle of light from medium 1 onto medium 2 and where n1
and n2 are the indices of refraction for the media (In our everyday experience n1 = 1
since the light comes in through the air and n2 is the index of refraction of the shiny
insulating material producing the glare eg water glass plastic paint etc)
Malusrsquos Law
The intensity of transmitted polarized light through a perfect polarizer is related to the
incident intensity of polarized light by Malusrsquos law
2cosoI I
where Io is the intensity of the incident light and where θ is the angle between the
electric field vector in the incident wave and the transmission axis the polarizer
24
Serway Chapter 39
Principles of Relativity
All of the weirdness of relativity flows from two simple principles
(1) The laws of physics must be the same in all inertial (non-accelerating) reference
frames
(2) The speed of light in vacuum has the same value in all inertial reference frames
Note that (1) seems reasonable but (2) is very odd It says that if two space ships are
approaching each other at nearly the speed of light and a laser pulse is shot from ship
1 toward ship 2 then when the people on ship 2 measure the speed of the pulse as it
goes by it is moving at 3 times 108 ms the same as if the ships were stationary
Simultaneity
If observer 1 sees two events in her own frame as simultaneous at two different
locations a moving observer 2 will see these two events happening at different times
Relativistic gamma
The factor γ (gamma) appears regularly in the formulas of relativity
2
2
1
1 vc
where v is the relative speed between two inertial frames
Time Dilation
If observer 1 sees two events at the same location in space separated by time Δtp in his
own frame then observer 2 moving at speed v relative to observer 1 will see these
two events separated by a longer time Δt
pt t
Moving clocks run slow
Length Contraction
If observer 1 measures the length of an object along the x-direction at some instant of
time in her frame to be Lp then observer 2 moving in the x-direction at speed v will
measure the length L of the object to be shorter
pLL
25
Moving meter sticks are short
Velocity Addition
If a particle is moving at velocity vac relative to frame c and if frame c is moving at
speed vcb with respect to frame b (with both velocities directed along the same line)
what would be the velocity of the particle in frame b In classical physics the answer is
ab ac cbv v v
but if any of these velocities are an appreciable fraction of the speed of light the
answer changes
21ac cb
abac cb
v Vv
v v c
In using this equation make sure that vac has the same sign in the numerator and in the
denominator and that vbc also has the same sign in both places Note that this equation
is different (and in our opinion easier to understand) than the equations in the text
Momentum and Energy
The momentum and total energy of a particle moving at velocity v (its speed is v v )
are given by
22
2 2
22
1 1 vvc c
m mcm E mc
vp v
Rest Energy
When a particle is at rest (v = 0 and γ = 1) then E = mc2 is called the rest energy It
includes the potential energy of the particle so that if we change the potential energy
of a particle we change its rest mass
Kinetic Energy The kinetic energy is the difference between the total energy and the
rest energy
2 2K mc mc
For v ltlt c this reduces to K = mv22
General Relativity
Einsteinrsquos principle of equivalence in the theory of general relativity is In the vicinity
of a point a gravitational field is equivalent to an accelerated frame of reference in
26
the absence of gravitational effects For example there is no difference whatever in
the physics experienced by an astronaut on earth and one who is in interstellar space
far away from any stars or planets but who has just turned on her spaceshiprsquos engines
and is accelerating at 98 ms2
27
Serway Chapter 40
Photons
We now know that even though light behaves like a wave it is better described as
consisting of small packets of energy called photons The energy of a photon is related
to its frequency by
E hf
where h is Planckrsquos constant
346626 10 J sh
Photons also carry momentum given by
hf hp
c
Since the amount of energy in each photon is fixed the difference between dim light
and intense light (of the same wavelength) is that dim light consists of fewer photons
than intense light
Photoelectric Effect
In the photoelectric effect light shining onto a metal surface gives the electrons in the
metal enough energy to escape and be detected It requires a certain amount of energy
(called the work function typically = 3 ndash 5 eV) just to kick the electron out of the
metal so the light must deliver at least this much energy to an electron to produce the
effect It is observed that red light no matter how intense never produces electrons
But ultraviolet light even if quite dim will eject electrons from the metal Since red
light consists of 2 eV photons and ultraviolet light has photons with energies around
4-6 eV the photon idea explains the behavior of the photoelectric effect The
maximum energy that an ejected electron can have is
maxK hf
where f is the photon frequency The minimum frequency that light can have and cause
electrons to be ejected is called the cutoff energy and is found by setting Kmax = 0
cf h
28
Compton Effect
When high-frequency light interacts with free electrons the scattered light does not
have the same wavelength as the incident light contrary to what classical
electromagnetism would predict But the photon picture in which photons have
momentum and energy allows us to treat scattering as a collision between the photon
and the electron In this collision the electron and the scattered photon both have a
different momentum and energy than they did before And since p = hλ for a photon
if the momentum is different the wavelength will also be different
after before 1 cose
h
m c
where the angle θ is the angle between the incoming photon direction and the
direction of the scattered photon
Particles are Waves
Since photons behave like particles sometimes it is not surprising that elementary
particles can behave like waves sometimes The wavelength of a particle with
momentum p is given by
h
p
Wave-particle Duality
Both photons and elementary particles have a dual nature sometimes they behave like
particles and sometimes they behave like waves It is hard for us to comprehend the
nature of such an object by making mental pictures but experiments definitively show
that this is the case Since we have no direct experience with photons and elementary
particles (because their energies are so much smaller than the energies of the everyday
objects we encounter) it is perhaps not surprising that we have a hard time forming a
mental picture of how they behave
Electron Interference
Since an electron is both a particle and a wave just like a photon it should be able to
produce an interference pattern This is observed If an electron beam is shot at two
closely-spaced slits and if the electrons that pass through the slits are detected
downstream it is found that there are some locations where electrons are never detected
29
and others where lots of electrons are detected The pattern is exactly the same as the
one observed for light waves provided that we use the electron wavelength λ = hp in
place of the wavelength of light This pattern is observed even though each electron is
detected as a single dot on the screen Only after many such dots are collected does the
pattern emerge And if we try to understand how this effect could possibly work by
looking closely at each slit to see which one the electron came through the pattern
disappears the act of measurement destroys the interference This means that each
single electron somehow comes through both slits (as a wave would)
Uncertainty Principle
In classical physics we always imagine that the positions and momenta of moving
particles have definite values It might be hard to measure them but surely at each
instant of time a particle should be precisely located at some point in space and have a
similarly precise momentum This turns out not to be true Instead both position and
momentum are required to be uncertain with their uncertainties Δx and Δp satisfying
the Heisenberg uncertainty relation
2x p
So if the particle were known to be precisely at some particular location (so that Δx =
0) we couldnrsquot know anything about its momentum (Δp = infin) and if its momentum
were exactly known we couldnrsquot know its position
There is a similar relation involving the particlersquos energy E and the time interval Δt
over which this energy is measured
2E t
This means that energy is not actually conserved if we are considering very small time
intervals and this brief non-conservation of energy has been observed
30
Serway Chapter 41
Wave Function ψ and Probability
Quantum mechanics does not predict exactly what an electron or a photon will do Instead
it specifies the wave function or probability amplitude ψ of an electron or a photon
This wave function is a complex-valued function of space and time whose squared
magnitude is the probability density P for finding a particle at a particular place in
space at a certain time
2P
where is the complex conjugate of ψ
For example the wave function of an electron with perfectly specified momentum p
would have a wavelength given by
h
p
and its probability amplitude would be proportional to
2i x ipxe e
The corresponding probability density would then be
21ipx ip ipxP e e e
which means that the electron is equally probable to be anywhere along the x axis
This is in accord with the uncertainty principle since we specified the momentum
precisely we canrsquot have any idea about the position of the electron
As another example you have probably seen ldquofuzzy ballrdquo drawings of electron
orbitals in a chemistry book These fuzzy balls are meant to indicate the distribution
of the probability density 2
P in the orbital
Particle in a Box
A simple example in which we can calculate the wave function is the case of a particle of
mass m constrained to be inside a 1-dimensional box between x = 0 and x = L In this
case the general wave function is a linear superposition of wave functions ψn of the form
sin niE tn
n xx t A e
L
31
where A is a positive constant where n = 1 2 3 and where the energy associated
with each of the quantum states ψn is given by
22
28n
hE n
mL
This wave function is zero at x = 0 and x = L which means that the particle will never
be found at the walls of the box The wave function has maximum values in the
interior and at these places the particle is most likely to be found and it also has
places where it is zero and at these places the particle will also never be found as
expressed by the formula
2sinn x
PL
The particle in the box is interfering with itself producing a probability interference
pattern across the box just like the interference patterns we studied with light and
sound
32
Serway Chapter 42
Atoms
Many people picture an atom as a miniature solar system where electrons orbit around
a massive nucleus at the center This picture is misleading because in atoms the wave
nature of electrons dominates The electrons form 3-dimensional standing waves
(called orbitals) centered at the nucleus And if you ask what is it thatrsquos waving the
answer is ψ the probability amplitude (see Chapter 41)
Standing waves on a string can occur only for certain frequencies (the fundamental
and higher harmonics) Similarly atomic orbitals occur only for certain energies For
the hydrogen atom the energies of the orbitals have the a particularly simple form
2
1136 eVnE
n
where n = 1 2 3 is called the principal quantum number For other atoms the
determination of the orbital energies requires numerical calculation by computers
Atomic Spectra
If an electron is somehow given extra extra energy (we say that it is excited) so that it
occupies a higher orbital it will eventually ldquofallrdquo back down to a lower orbital Each
time an electron falls to a lower orbital it loses the difference in energy between the
two orbitals in the form of a photon Since the orbital energies are discrete so are the
energy differences and so are the wavelengths of the emitted light The entire set of
these discrete wavelengths is called the atomic spectrum and it is unique to each type
of atom For hydrogen the spectrum can be simply written as
H 2 2
1 1 1
f i
Rn n
where RH = 10973732 times 107 m-1 For other atoms the spectrum cannot be expressed as
a formula
Orbital Angular Momentum Electrons in atoms also have quantized values of
angular momentum The orbital quantum number ℓ specifies the value of this
quantized angular momentum through the formula
1L
33
If we want to know the value of the angular momentum along some direction in space
say the z direction the answer is not L but rather
zL m
where mℓ is another quantum number which runs from
1 1m
This quantum number is important when an atom sits in a magnetic field
Spin Angular Momentum It has been found experimentally that electrons and other
charged particles also carry internal angular momentum which we call spin
Electrons have an intrinsic spin angular momentum s along a specified axis that is
extremely quantized it can only take on 2 possible values
1
2zs s
We thus say that electrons have spin s = 12 or that electrons are rdquospin one-half parti-
cles Since s is an angular momentum it obeys the same rule as that for orbital angular
momentum namely that if its value along some axis is s then its total magnitude is
given by
31
2S s s
I know this seems weird but quantum mechanics is weird The only excuse for this
bizarre way of looking at the world is that it predicts what happens in experiments
Exclusion Principle The answer to the question of how many electrons (or any other
spin one-half particle) can be in one particular quantum state was discovered by
Wolfgang Pauli and is called the exclusion principle
ldquoNo two electrons can ever be in the same quantum state therefore no two electrons
in the same atom can have the same set of quantum numbersrdquo
This is the reason that we have atoms with different properties instead of every atom
simple having all of its electrons in the ground state All of the variety we see around
us in the world is the result of chemical differences and these differences would not
exist unless electrons obeyed this important principle The entire structure of the
periodic table (see pages 1377-1379 in Serway) is an expression of this principle
34
Serway Chapter 44
Nuclear Properties
The nucleus is composed of protons with charge +e and mass mp = 1007226 u and of
neutrons with zero charge and mass mn = 1008665 u where u is one atomic mass unit
-271 u=1660540 10 kg
The atomic number Z counts the number of protons in a nucleus while the neutron
number N counts the number of neutrons The mass number A is the sum of the two
A N Z
Protons and nuetrons have about the same mass and some times called baryons (heavy
ones) or nucleons (partcles in the nucleus) A is sometimes called the baryon
number The chemical elements are distinguished by Z ie hydrogen has Z = 1 iron
has Z = 26 uranium has Z = 92 etc But for a given Z there might be several nuclei
with different numbers of neutrons These nuclear siblings are called isotopes
These numbers are used to label nuclei according to the pattern
5626 ie FeA
Z X
denotes the isotope of iron with N = 56 ndash 26 = 30 neutrons
The nucleus is roughly spherical with a radius given approximately by
1 3 150 0where 12 10 mr r A r
Nuclear Stability
Because the positively charged protons electrically repel each other with an enormous
force at distances as small as 10-15 m there must be some really strong force that that
overcomes electrical repulsion to hold protons and neutrons together This force is
called with some lack of imagination the strong force It is a very short-range force
(it only acts over a distance of about 2 x 10-15 m) and attracts protons to protons
neutrons to neutrons and neutrons to protons But in spite of this strong nuclear force
the coulomb repulsion of the protons is still present so anything that might keep the
protons from being right next to each other would help keep the nucleus from
35
exploding This role is played by the neutrons and for nuclei with Z le 20 the stable
nuclei roughly have N = Z
For Z gt 20 the coulomb repulsion force is more powerful and more neutrons are
needed to dilute it up to about N = 15Z around Z = 80 For Z gt 83 no amount of
neutrons can help and these nuclei are unstable (radioactive) There is an ldquoisland of
(relative) stabilityrdquo around Z=90-92 (thorium and uranium respectively) These
elements have one or two isotopes with half-lifes of billions of years so there are
substantial amounts of such elements on Earth This fact makes it possible to have
practical fission devices
Radioactive Decay
There are three types of radioactive decay
Alpha decay the nucleus kicks out a helium nucleus (N = 2 Z = 2 A = 4) So if the
original nucleus is called X and the new nucleus is called Y then the decay would look
like this
4 42 2X Y+ HeA A
Z Z
Beta decay the nucleus either kicks out an electron (endash) or its positively-charged
antimatter twin the positron (e+) plus either an electron neutrino v or an electron
anti-neutrino v
1X Y+e electron decayA AZ Z v
1X Y+e positron decayA AZ Z v
A neutrino is a particle with no charge hardly any mass (much less than the electron
mass) and interacts so weakly with matter that most neutrinos upon encountering the
planet earth just pass right through it as if it werenrsquot there
Gamma decay the nucleons in the nucleus X are in an excited energy state X
(perhaps as a result of having undergone alpha or beta decay) and they drop down to a
lower energy state shedding the energy as a high frequency photon
X XA AZ Z
36
This process is exactly analogous to the way that the electrons in atoms emit photons
Decay Rate and Half Life
There is no way to predict exactly when an unstable or excited nucleus will decay but
there is an average rate at which this decay occurs called the decay constant λ The
meaning of this constant is that if there are a large number N of nuclei in a sample
then the number of decays per second that will be observed (called the decay rate R)
is R = λN In mathematical language
dMR N
dt
This simple differential equation has for its solution
0tN t N e
where N0 is the number of nuclei in the sample at time t = 0
The half-life is the time it takes for half of the nuclei in the sample to decay and is
related to the decay constant by
1 2
ln 2 0693T
Disintegration Energy
When a nucleus decays it is making a transition to an overall state of lower energy
which means according to Einsteinrsquos famous formula E = mc2 that the sum of the
masses after the decay must be less than the mass before with the lost mass appearing
as kinetic energy among the decay products For example in alpha decay this kinetic
energy called the disintegration energy Q is given by
2X YQ M M M c
37
Serway Chapter 45
Nuclear energy
There are two ways to extract energy from the nucleus fission and fusion For nuclei
with Z greater than 26 breaking the nucleus apart into pieces leads to a lower
2mc energy than the original nucleus so energy can be extracted by fission For
nuclei with Z less than 26 a lower energy is achieved by combining nuclei so energy
can be extracted by fusion
Fission
Since neutrons have no charge they are not repelled from nuclei as protons are for
this reason their behavior is the key to understanding how fission works
Fast neutrons mostly bounce off other nuclei with each collision slowing the neutron
down This elastic energy loss is most effective if the other nuclei have low mass (like
hydrogen) and these materials are called moderators because of their ability to slow
down fast neutrons
The reason that slowing neutrons is important is that slow neutrons are much more
likely to be absorbed by a nucleus which then leads to nuclear reactions of various
kinds For a few very large nuclei like uranium-235 and some plutonium isotopes
absorption of a slow neutron causes the nucleus to split into two large fragments plus
2 or 3 fast neutrons (energy is released in the process too) Thus one slow neutron
can produce energy plus 2 or 3 more neutrons which if moderated from fast to slow
can split 2 or 3 more nuclei leading to a runaway chain reaction If the reaction is
allowed to proceed unchecked it produces a large explosion If a neutron absorbing
material is added to the mix (like the cadmium in reactor control rods) it is possible to
keep the reaction under control and to extract the released energy as heat to drive
steam turbines and produce electricity
Fusion
Fusion involves mashing two nuclei together and since they are both charged and repel
each other this reaction is much harder to make go The nuclei must have enough energy to
overcome the coulomb repulsion which is why this reaction requires a high temperature
(hundreds of millions of degrees K) like that in the sun (which is burning hydrogen to
helium via fusion) or like that in the center of the fission explosion that is used to detonate
38
a hydrogen bomb
This reaction is of interest for power production in spite of this difficult temperature
requirement because of the abundance of fusion fuel on the planet There are about
012 g of deuterium ( 21H ) in every gallon of water on earth and it only costs about 4
cents to extract it The fusion energy available from this minuscule amount of
deuterium would run a 1000-Megawatt power plant for 10 seconds Water is so
abundant on earth that if fusion were to work we would have an essentially
inexhaustible source of energy
So why donrsquot we have fusion power plants Well the fuel is cheap but the match is
incredibly expensive The only way we know to control this difficult high-temperature
reaction is with large and expensive pieces of equipment involving either large
magnetic fields and complex high-power electromagnetic antennas or with gigantic
(football-fieldsized) laser facilities involving more than a hundred of the highest-
energy lasers ever built Power plants based on these current methods for controlling
fusion are unattractive to the fiscally-minded people who run the electric power
industry Hopefully better designs will be discovered as experiments continue
39
c actinium 68 Er erbium 101 Md mendelevium 104 Rf rutherfordium l aluminum 63 Eu europium 80 Hg mercury 62 Sm samarium m americium 100 Fm fermium 42 Mo molybdenum 21 Sc scandium b antimony 9 F fluorine 60 Nd neodymium 106 Sg seaborgium r argon 87 Fr francium 10 Ne neon 34 Se selenium s arsenic 64 Gd gadolinium 93 Np neptunium 14 Si silicon t astatine 31 Ga gallium 28 Ni nickel 47 Ag silver a barium 32 Ge germanium 41 Nb niobium 11 Na sodium k berkelium 79 Au gold 7 N nitrogen 38 Sr strontium e beryllium 72 Hf hafnium 102 No nobelium 16 S sulfur i bismuth 108 Hs hassium 76 Os osmium 73 Ta tantalum h bohrium 2 He helium 8 O oxygen 43 Tc technetium
boron 67 Ho holmium 46 Pd palladium 52 Te tellurium r bromine 1 H hydrogen 15 P phosphorus 65 Tb terbium d cadmium 49 In indium 78 Pt platinum 81 Tl thallium a calcium 53 I iodine 94 Pu plutonium 90 Th thorium f californium 77 Ir iridium 84 Po polonium 69 Tm thulium
carbon 26 Fe iron 19 K potassium 50 Sn tin e cerium 36 Kr krypton 59 Pr praseodymium 22 Ti titanium s cesium 57 La lanthanum 61 Pm promethium 74 W tungsten l chlorine 103 Lr lawrencium 91 Pa protactinium 92 U uranium r chromium 82 Pb lead 88 Ra radium 23 V vanadium o cobalt 3 Li lithium 86 Rn radon 54 Xe xenon u copper 71 Lu lutetium 75 Re rhenium 70 Yb ytterbium m curium 12 Mg magnesium 45 Rh rhodium 39 Y yttrium b dubnium 25 Mn manganese 37 Rb rubidium 30 Zn zinc y dysprosium 109 Mt meitnerium 44 Ru ruthenium 40 Zr zirconium s einsteinium
40
UNITS The SI units are given in parentheses Other commonly used units are given in terms of the SI units Symbols conform with the recommendations of the American National Standards Institute (ANSI) and the American Institute of Physics (AIP) Commonly used multiples of SI units
prefix symbol factor tera T 1012 giga G 109
mega M 106 kilo k 103
centi c 10-2 milli m 10-3
micro μ 10-6 nano n 10-9 pico p 10-12
femto f 10-15
length meter (m) angstrom 1 Ǻ = 10-10 m inch 1 in = 254 x 10-2 m foot 1 ft = 03048 m mile 1 mi = 1609 m
mass kilogram (kg) atomic mass unit 1 u = 1661 x 10-27 kgslug 1 slug = 1459 kg
force newton (N = kg bull ms2) dyne 1 dyn = 10-5 N pound 1 lb = 4448 N
pressure Pascal (Pa = kgm bull s2) atmosphere 1 atm = 1013 x 105 Pa poundssquare inch 1 psi = 6895 Pa cm of mercury 1 cm Hg = 1333 Pa bar 1 bar = 1000 x 105 Pa torr 1 torr = 1333 Pa
time second (s) minute 1 min = 60 s hour 1 h =3600 s
frequency hertz (Hz = s-1) radianssecond 1 rads = 12π Hz
energy joule (J = kg bull m2s2) erg 1 erg = 10-7 J electron volt 1 eV = 1602 x 10-19 J calorie 1 cal = 4187 J kilowatt-hour 1 kWh = 36 x 106 J British thermal unit 1 Btu = 1055 J
power watt (W = kg bull m2s3) horsepower 1 hp = 7457 W
charge coulomb (C = A bull s)
electric potential volt (V = kg bull m2 s3 bull A)
current ampere (A)
resistance ohm (Ω = kg bull m2s3 bull A2)
capacitance farad (F = s4 bull A2kg bull m2)
magnetic field tesla (T = kgs2 bull A) gauss 1 G = 10-4 T
magnetic flux weber (Wb = kg m2s2 bull A) maxwell 1 Mx = 10-8 Wb
magnetic inductance henry (H = kg bull m2s2 bullA2)
temperature kelvin (K) degrees Celsius 0degC = 27315 K
angle radian (rad) degree 1 deg = π180 rad revolution 1 rev = 2π rad
41
Some Physical Constantsa Quantity Symbol Valueb Atomic mass unit μ 1660 538 73 (13) x 10-27 kg
931494 013 (37) MeVc2 Avogadros number NA 6022 141 99 (47) x 1023 particlesmol Avogadrorsquos number 6022 x 1023 mol Bohr magneton
2B e
em 9274 008 99 (37) x 10-24 JT
Bohr magneton μB 927 x 10-24 JT Bohr radius
0
2
2e em e k
a 5291 772 083 (19) x 10-11 m
Boltzmanns constant B A
RNk 1380 650 3 (24) x 10-23 JK
Boltzmannrsquos constant kB 1380 x 10-23 JK Compton wavelength
C eh
m c 2426 310 215 (18) x 10-12 m
Deuteron mass md 3343 583 09 (26) x 10-27 kg 2013 553 212 71 (35) u
electron charge e 1602 x 10-19 C Electron mass me 9109 381 88 (72) x 10-31 kg
5485 799 110 (12) x 10-4 u 0510 998 902 (21) MeVc2
electron mass 911 x 10-31 kg Electron volt eV 1602 176 462 (63) x 10-19 J Elementary charge e 1602 176 462 (63) x 10-19 C Gas constant R 8314 472 (15) JKmol Gravitational constant G 6673 (10) x 10-11 Nm2kg2 Neutron mass mn 1674 927 16 (13) x 10-27 kg
1008 664 915 78 (55) u 939565 330 (38) MeVc2
neutron mass 1675 x 10-27 kg Nuclear magneton
2n p
em 5050 783 17 (20) x 10-27 JT
permeability constant μ0 1257 x 10-6 Hm permittivity constant ε0 8854 x 10-12 Fm Plancks constant h
2h
6626 068 76 (52) x 10-34 Js 1054 571 596 (82) x 10-34 Js
Planckrsquos constant h ħ
6626 x 10-34 Js 1055 x 10-34 Js
Proton mass mp 1672621 58 (13) x 10-27 kg 1007 276 466 88 (13) u 938271 998 (38) MeVc2
proton mass 1673 x 10-27 kg Rydberg constant RH 1097 373 156 854 9 (83) x 107 m-1 Speed of light in vacuum c 2997 92458 x 108 ms (exact) speed of light c 300 x 108 ms a
These constants are the values recommended in 1998 by CODATA based on a least-squares adjustment of data from different measurements For a more complete list see P J Mohr and B N Taylor Rev Mod Phys 72351 2000 b The numbers in parentheses for the values above represent the uncertainties of the last two digits
NOTE The ones in red are the ones from Appendix 13 The ones in yellow are the ones that are from the Serway book table but that were already on the Appendix 13 table
42
INDEX
Absolute zero 9
Adiabatic 17
Adiabatic exponent 18
Adiabatic process 17
Alpha decay 46
Amplitude 4
Angle of reflection 25
Angular frequency 3
Angular magnification 30
Angular momentum atomic 43
Angular size 29
Archimedes Principle 1
Atomic spectra 43
Atoms 43
Avogadrorsquos number 10
Beats 8
Bernoullirsquos Equation 2
Beta decay 46
Bifocals 29
Brewsterrsquos angle 34
British Thermal Unit Btu 11
Buoyancy 2
calorie 11
Calorie 11
Camera 28
Carnot cycle 21
Carnot efficiency 22
Celsius scale 9
Ciliary muscle 29
Coefficient of performance 21
Compton effect 39
Constructive interference 7
Continuity equation of 2
Contraction length 35
Convection 14
Curved mirrors 27
Decay constant 46
Decay rate radioactive 46
Decibel scale 5
Degrees of freedom 16
Density 1
Destructive interference 7
Diffraction single slit 33
Diffraction grating 33
Dilation time 35
Disintegration energy 47
Dispersion of light 26
Doppler effect 6
Double slit interference 31
Efficiency 20
Electron interference 39
Emissivity 15
Energy nuclear decay 47
Energy relativistic 36
43
Engineering work 20
Entropy 22
Entropy ideal gas 24
Equipartition of energy 18
Equivalence principle 36
Exclusion principle 44
Expansion thermal 9
Expansion coefficient linear 9
Eye 29
Fahrenheit scale 9
Far point 29
Farsightedness 29
First Law of Thermodynamics 12
Fission 48
Flux volume 2
Free expansion 23
Fringes 31
Fusion 48
Fusion heat of 11
Gamma relativistic 35
Gamma decay 46
General relativity 36
Half-life 46
Heat 11
Heat capacity 11
Heat conduction 13
Heat engine 20
Heat of fusion 11
Heat of vaporization 11
Heat pump 20
Heats of transformation 11
Hydrostatics 1
Ideal Gas Law 10
Images real and virtual 27
Intensity sound 5
Interference 7
Interference two-slit 31
Internal energy 12
Internal energy degrees of freedom 16
Irreversible process 19 23
Isotopes 45
Joule 11
Kelvin scale 9
Kinetic energy relativity 36
Kinetic theory 16
Latent heat 11
Length contraction 35
Linear expansion coefficient 9
Linear polarization 34
Linear superposition 7
Longitudinal wave 3
Loudness 5
Magnification lateral 27
Magnifying glass 30
Malusrsquos law 34
Microscope 30
Momentum relativistic 36
Muscle ciliary 29
44
Musical instruments 8
Musical scale 8
Near point 29
Nearsightedness 29
Nonlinear 7
Nuclear energy 48
Nuclear properties 45
Nuclear stability 45
Octave 8
Optical resolution 33
Orbital quantum number ℓ 43
Orbitals 43
Particle in a Box 41
Particles are waves 39
Pascalrsquos Principle 1
Period 3
Photoelectric effect 38
Photons 38
Pitch 8
Polarization 34
Power sound 5
Presbyopia 29
Pressure 1
Principal quantum number n 43
Principle of equivalence 36
Principle of linear superposition 7
Probability amplitude ψ 41
Processes thermodynamic 13
R-value 14
Radiation thermal 14
Radioactive decay 46
Radioactive decay rate 46
Ray tracing 27
Rayleighrsquos criterion 33
Reading glasses 29
Real image 27
Refraction 25
Refrigerator 20
Relativistic gamma 35
Relativity principles 35
Resolved for light sources 33
Rest energy 36
Reversible process 19 22
Rope wave speed 4
Second Law of Thermodynamics 19
Shock waves 6
Simple magnifier 30
Simultaneity 35
Single slit diffraction 33
Snellrsquos law 25
Sound speed 5
Specific heat 11
Spin Angular momentum 44
Standing waves 7
Stefanrsquos law 14
Strong force 45
Telescope 30
Temperature 9
45
Temperature Scales 9
Thermal conductivity 14
Thermal energy 12
Thermal expansion 9
Thin film interference 31
Thin lenses 28
Time dilation 35
Tone musical 8
Total internal reflection 25
Transverse wave 3
Traveling Waves 3
Two-slit interference 31
Uncertainty principle 40
Vaporization heat of 11
Velocity addition relativity 36
Virtual image 27
Volume flux 2
Wave function 41
Wave Function ψ and Probability 41
Wave speed 3
Wave-particle duality 39
Wavelength 3
Wavenumber 3
Wien Displacement Law 16
Work 11
11
Serway Chapter 20
Heat
Heat is energy that flows between a system and its environment because of a tempera-
ture difference between them The units of heat are Joules as expected for an energy
Unfortunately there are several competing units of energy They are related by
1 cal 4186 J 1 Cal 4186 J 1 Btu 1054 J
Heat Capacity
There is often a simple linear relation between the heat that flows in or out of part of a
system and the temperature change that results from this energy transfer When this
linear relation holds it is convenient to define the heat capacity C and the specific
heat c as follows
For the entire object the heat Q it absorbs is given by Q = C(Tf - Ti)
For a particular material the specific heat is defined by c = Cm which is the heat
capacity per unit mass so that
Q = mc(Tf ndash Ti)
Note C has units of energy (J or Cal)(Kelvin kg)
Heats of Transformation or Latent Heat Q = plusmn mL
When a substance changes phase from solid to liquid or from liquid to gas it absorbs
heat without a change in temperature The latent heat or heat of transformation is
usually given per unit mass of the substance For example for water the heat of fusion
(melting) is L = 333 kJkg while the heat of vaporization (boiling) is L = 2260 kJkg
Note that heat for boiling is considerably bigger than melting for water You have to
be careful with signs heat is given off (negative) if you go down in temperature and
condense steam
Work
In general the small amount of work done on a system as a force Fon is exerted on it
through a vector displacement dx is given by
on xdW d F
12
But if the displacement is done very slowly (as we always assume in thermodynamics)
then the force exerted on the system and the force exerted by the system are in
balance so the force exerted by the system is ndash Fon In thermodynamics it is more
convenient to talk about the force exerted by the system so we change the above
formula for the work done on the system to
xdW d F
where F is the force exerted by the system This has confused students for more than a
century now but this is the way your book and many other books do it so you are
stuck You will need to memorize the minus sign in this definition of the work to be
able to use your textbook
There are many chances to get signs wrong in this and the next two chapters (Mosiah
2 )
When an external agent changes the volume of a gas at pressure P by a small amount
dV the (small amount) of work done on the system is given by
dW PdV
Notice that this minus sign is just what we need to make dW be positive if the external
agent compresses the gas for then dV is negative If on the other hand the external
agent gives way allowing the gas to expand against it then dV is positive and we say
that the work done on the gas is negative
The work done on the system (eg by the gas in a cylinder) in a thermodynamic
process is the area under the curve in a PV diagram It is positive for compressions
and negative for expansions If the volume of gas remains constant in a process then
no work is done by the gas
Cyclic processes are important For cyclic processes represented by PV diagrams the
magnitude of the net work during one cycle is simply the area enclosed by the cycle
on the diagram Be careful to keep track of signs when you are calculating that
enclosed area In cyclic process Q = ndashW for a cycle Put a PV diagram in your notes
1
Path A-B B-C C-D D-A A to A net
Q
W
ΔU
ΔS
Internal Energy
The energy stored in a substance is called its internal energy Eint This energy may be
stored as random kinetic energy or as potential energy in each molecule (stretched
chemical bonds electrons in excited states etc) For ideal gases all states with the
same temperature will have the same Eint
First Law of Thermodynamics
The change ΔEint in the internal energy of a system is given by
intE Q W
where Q is the heat absorbed by the system and where W is the work done on the
system Hence if a system absorbs heat (and if Wge0) the internal energy increases
Likewise if the system does work (W on the system is negative) and if Qge0 the
internal energy decreases Potential Pitfall Many times people talk about work done
by the system It is the minus of W on the system Donrsquot get tripped up
Processes
Adiabatic process No heat is exchanged Q = 0 requiring that the system be insulated
from the environment A process may be approximately adiabatic if it happens so
rapidly that heat does not have time to enter or leave the system Work + or ndash is done
and ΔEint = W
Adiabatic free expansion A gas is allowed to expand into a vacuum without pushing
on anythingndashit just rushes into the vacuum In this process Q = W = 0 so the internal
energy and temperature does not change (Note the difference between an adiabatic
process and a free expansion is that NO work is done in the adiabatic free expansion)
Isobaric process The pressure is held fixed ΔP = 0 For example usually the
pressure increases when a gas is heated but if it were allowed to expand during the
2
heating process in just the right way its pressure could remain fixed In isobaric
processes both Q and W are non-zero (Work is eacy = ndashP(Vf ndash Vi)
Isovolumetric process The volume is held fixed ΔV = 0 The work done by the gas is
then zero and so we have ΔEint = Q
Isothermal process The temperature is held fixed ΔT = 0 There is no change in Eint
so Q = ndashW for isothermal processes For an ideal gas PV = nRT so PV = constant in
an isothermal process The work done on the gas is then given by
lnf
i
Vi
Vf
VW PdV nRT
V
Heat Conduction
The quantity P is defined to be the rate at which heat flows through an object and is a
power having units of watts It is analogous to electric current which is the rate at
which charge flows through an object If the flow of heat through a slab of length L
and cross-sectional area A is steady in time then P is given by the equation
h cT TdQkA
dt L
P =
where Th and Tc are the (hot and cold) temperatures of the two ends of the slab The
heat flows of course because of this temperature difference The quantity k is called
the thermal conductivity and is a constant that is characteristic of the material It is
analogous to the electrical conductivity h cT TL is sometimes called the temperature
gradient and is written dTd dTdx
R-Values
It is common to have the heat-conducting properties of materials described by their R-
values especially for insulating materials like fiberglass batting The connection
between k and R is R = Lk where L is the material thickness In this country R-values
always have units of 2ft F hourBtu
Polyurethane foam 6 (1 in thick) Air space 1 (35 in thick) Fiber glass batting 11
(35 in thick) Wood about 1 (1 in thick) Glass 09 (0125 in thick)
The heat flow rate through a slab of area A is given by
3
h cT TA
R
P
in units of Btuhour Note that A must be in square feet and the temperatures must be
in degrees Fahrenheit
Convection
Convection is the transfer of thermal energy by flow of material For instance a home
furnace doesnrsquot heat a house by waiting for the heat from the burner to slowly conduct
throughout the house instead it quickly pumps warm air to all of the rooms
Generally convection is a much faster way to transfer heat than conduction
Radiation
Electromagnetic radiation can also transfer heat When you warm yourself near a
campfire which has burned itself down into a bed of glowing embers you are
receiving radiant heat from the infrared portion of the electromagnetic spectrum The
rate at which an object emits radiant heat is given by Stefanrsquos law
4AeTP
where P is the radiated power in watts σ is a constant
8 2 45696 10 W m K
A is the surface area of the object in m2 and T is the temperature in kelvins The
constant e is called the emissivity and it varies from substance to substance A perfect
absorber (think black velvet) has e = 1 while a perfect reflector (think mirror) has e =
0 Hence black objects radiate very well while shiny ones do not Also an object that
is hotter than its surroundings radiates more energy than it absorbs whereas an object
that is cooler than its surroundings absorbs more energy than it radiates
Terminology
Transfer variables vs state variables
Energy transfer by heat as well as work done depends on the initial final and
intermediate states of the system They are transfer variables But their sum (Q + W =
4
Eint) is a state variable
Figure 205
5
Serway Chapter 21
Kinetic Theory
The ideal gas law works for all atoms and molecules at low pressure It is rather
amazing that it does Kinetic theory explains why The properties of an ideal gas can
be understood by thinking of it as N rapidly moving particles of mass m As these
particles collide with the container walls momentum is imparted to the walls which
we call the force of gas pressure In this picture the pressure is related to the average
of the square of the particle velocity 2v by
22 1( )
3 2
NP mv
V
Using the ideal gas law we obtain the average translational kinetic energy per
molecule
21 3
2 2 Bmv k T
The rms speed is then given by
2rms
3 3Bk T RTv v
m M
where M is the molecular mass in kgmol
Degrees of Freedom
Roughly speaking a degree of freedom is a way in which a molecule can store energy
For instance since there are three different directions in space along which a molecule
can move there are three degrees of freedom for the translational kinetic energy
There are also three different axes of rotation about which a polyatomic molecule can
spin so we say there are three degrees of freedom for the rotational kinetic energy
There are even degrees of freedom associated with the various ways in which a
molecule can vibrate and with the different energy levels in which the electrons of
the molecule can exist
Internal Energy and Degrees of Freedom The internal energy of an ideal gas made
up of molecules with J degrees of freedom is given by
int 2 2 B
J JE nRT Nk T
6
Heat Capacities of an Ideal Gas The heat capacity of a gas is described by means of
molar heat capacities CV and CP These are the heat capacities per mole and the
subscript V on CV means that the volume is being held constant while for CP the
pressure is held constant For example to raise the temperature of n moles of a gas
whose pressure is held constant by 10 K we would have to supply an amount of heat Q
= nCP (10) K
Molar Specific Heat of an Ideal Gas at Constant Volume
VQ nC T
3monatomic
2VC R
5diatomic
2VC R
5polyatomic
2VC
Real gases deviate from these formulas because in addition to the translational and ro-
tational degrees of freedom they also have vibrational and electronic degrees of
freedom These are unimportant at low temperatures due to quantum mechanical
effects but become increasingly important at higher temperatures The rough rule is
No of degrees of freedom
2VC R
Molar Specific Heat of an Ideal Gas at Constant Pressure
PQ nC T
P VC C R
The internal energy of an ideal gas depends only on the temperature
int VE nC T
Adiabatic Processes in an Ideal Gas
7
An adiabatic process is one in which no heat is exchanged between the system and the
environment When an ideal gas expands or contracts adiabatically not only does its
pressure change as expected from the ideal gas law but its temperature changes as
well Under these conditions the final pressure Pf can be computed from the initial
pressure Pi and from the final and initial volumes Vf and Vi by
or constantf f i iP V PV PV
where γ = CPCV The quantity γ is called the adiabatic exponent Note that this
doesnrsquot mean that the ideal gas law no longer holds it does and in fact it can be
combined with the adiabatic law for pressure given above to obtain the adiabatic law
for temperatures
1 constantTV
Compressions in sound waves are adiabatic because they happen too rapidly for any
appreciable amount of heat to flow This is why the adiabatic exponent appears in the
formula for the speed of sound in an ideal gas
RTv
M
Note that v depends only on T and not on P Because it depends only on the
temperature the speed of sound is the same in Provo as at sea level in spite of the
lower pressure here due to the difference in elevation
Equipartition of Energy
Every kind of molecule has a certain number of degrees of freedom which are
independent ways in which it can store energy Each such degree of freedom has
associated with it ndash on average ndash an energy of 12 Bk T per molecule (or 1
2 RT per mole)
(Note since a molecule has so many possible degrees of freedom it would seem that
there should be a lot of 12 sBk T to spread around But because energy is quantized
some of these degrees of freedom are not ldquoactiverdquo until the temperature becomes high
enough that 12 Bk T is as big as the lowest quantum of energy
8
Serway Chapter 22
Second Law of Thermodynamics
There are several equivalent forms of this important law
Kelvin It is not possible to change heat completely into work with no other change
taking place Or in other words there are no perfect heat engines
Clausius It is not possible for heat to flow from one body to another body at a higher
temperature with no other change taking place Or in other words there are no
perfect refrigerators
Entropy In any thermodynamic process that proceeds from one equilibrium state to
another the entropy of the system + environment either remains unchanged or
increases The total entropy never decreases This law is a bit of an oddity among the
laws of physics because it is not absolute Things are forbidden by the second law not
because it is impossible for them to happen but because it is extremely unlikely for
them to happen (See below for more information about entropy)
Reversible and Irreversible Processes
A reversible process is one which occurs so slowly that it is in thermal equilibrium (or
very nearly so) at all times A hallmark of such processes is that a motion picture of
them looks perfectly normal whether run forward or backward Imagine for instance
the slow expansion of a gas at constant temperature in a cylinder whose volume is being
increased by a slowly moving piston Run the movie backwards and what do you see
You see the slow compression of a gas at constant temperature which looks perfectly
normal
An irreversible process is one which occurs in such a way that thermal equilibrium is
not maintained throughout the process The mark of this kind of process is that a motion
picture of it looks very odd when run backward Imagine the sudden expansion of a gas
into a previously evacuated chamber because a hole was punched in the wall between a
pressurized chamber and the evacuated one Run the movie backward and what do you
see You see the gas in the soon-to-be-evacuated chamber gather itself together and
stream through a tiny hole into a chamber in which there is already plenty of gas If you
have ever seen this happen get in touch with the support group for those who have
witnessed the spontaneous combustion of large mammals by calling 1-800-PYROCOW
9
Heat Engines
Because of the vexing difference in sign between work done on and system and work done
by a system we will invent a new work variable Weng Heat engines do work and so the net
W for these engines is negative But in engineering applications hidden minus signs are
regarded as evil so for heat engines we donrsquot talk about W instead we talk about its
magnitude engW W So for heat engines the first law is
int engE Q W
But for heat pumps and refrigerators work is done on the system so we use the usual work
W when we talk about these systems
A heat engine is a machine that absorbs heat converts part of it to work and exhausts the
rest The heat must be absorbed at high temperature and exhausted at low temperature If the
absorbed heat is Qh the exhausted heat is Qc and the work done by the engine is Weng then
eng h cW Q Q
and the efficiency of the engine is defined to be
eng
h
We
Q
A perfect engine would convert the heat hQ completely into work Weng giving an effi-
ciency of e = 1 Energy conservation alone allows a perfect engine but the second law
requires e lt 1
Refrigerators and Heat Pumps
A refrigerator is a machine that absorbs heat at low temperature and exhausts it at high
temperature the ldquobackwardsrdquo heat transfer being driven by the work done on the machine
by some source of power A heat pump is a machine that either works like a refrigerator
keeping a place cold by transferring heat from this cold place to a higher temperature
environment (cooling mode like an air conditioner) or it functions as a heater
transferring heat into a warm place from a cooler one (heating mode like a window unit
that heats a house by extracting thermal energy from the cold outdoors) The coefficient
of performance of a refrigerator or of a heat pump in cooling mode is defined to be
COP cooling modecQ
W
10
For a heat pump in heating mode the coefficient of performance is
COP heating modehQ
W
Note that we donrsquot have to use the engineering work here because in these systems
work is done on the system and W is naturally positive
A perfect refrigerator would take in heat Qc and exhaust the same amount of heat without
doing any work giving an infinite coefficient of performance Energy conservation alone
allows a perfect refrigerator but the second law requires COP lt infin A perfect heat pump
in heating mode would transfer Qh into the house without doing any work and so would
also have an infinite coefficient of performance The second law forbids this too A good
coefficient of performance for a real device would be around 5 or 6
Carnot Cycle
The most efficient of all possible engines is one that uses the Carnot cycle This cycle
employs an ideal gas has no friction and operates very slowly so that the gas can be in
thermal equilibrium at all parts of the cycle This means of course that it canrsquot
possibly be built and even if it could be built it would not run fast enough to be useful
Nevertheless this cycle is very important because it gives an upper bound on the
efficiency of real engines There cannot possibly be an engine that is more efficient
than one based on the Carnot cycle This cycle consists of the following four steps
1 The ideal gas absorbs heat Qh at constant temperature Th while the gas increases its
volume The reason that heat is absorbed is that expansion tends to cool the gas but
thermal contact with the environment at Th keeps the temperature high by heat
conduction into the ideal gas
2 The ideal gas further increases its volume by an adiabatic expansion This expansion
causes the gas to cool so at the end of this part of the cycle the gas is at temperature Tc
3 The gas exhausts heat Qc at constant temperature Tc while the gas decreases in
volume The reason that heat is exhausted is that compression tends to heat the gas
but thermal contact with the environment at Tc keeps the temperature low by heat
conduction out of the ideal gas
4 The gas is adiabatically compressed back to its original volume (the volume it started
with in step 1) This compression heats the gas from Tc up to Th
11
The efficiency of a Carnot engine is given by the very simple formula
1 cC
h
Te
T
where the temperatures must in be Kelvin No real engine can be more efficient than
this
The coefficient of performance of a Carnot refrigerator or heat pump in cooling mode
is given by
COP cooling mode cC
h c
T
T T
and no real refrigerator can have a coefficient of performance greater than this
The coefficient of performance of a Carnot heat pump in heating mode is
COP heating mode hC
h c
T
T T
So why donrsquot we just use these wonderful Carnot engines and have perfect efficiency
To make the reversible steps in the cycle really reversible they would have to occur
infinitely slowly So the price you pay for making a perfect engine is that it takes
forever to get it to do any work
Entropy
The entropy of a system is defined in terms of its molecular makeup and measures
roughly the disorder of the system If the system is packed into a very small volume
then it is quite ordered and the entropy will be low If it occupies a large volume the
entropy is high (To see what this has to do with disorder note that socks in a drawer
occupy a small volume while socks on the bed in the corner by the door and
hanging from the chandelier occupy a large volume) If the system is very cold then
the molecules hardly move and may even reach out to each other and form a crystal
This is a highly ordered state and therefore has low entropy If the system is very hot
with rapidly speeding molecules crashing into the container walls and bouncing off
each other things are disordered and the entropy is high
It is possible to calculate the entropy of a system in terms of its macroscopic thermody-
namic properties ie pressure volume temperature number of moles etc The key to this
calculation is the concept of a reversible process A reversible process is one that is
12
carried out without friction and so slowly that the process can be reversed at any stage by
making an infinitesimal change in the environment of the system The slow expansion of the
gas at Th in the Carnot cycle with tiny amounts of heat being transferred to the environment
is reversible If we turned around and began slowly to compress the gas would just slowly
exhaust heat to the environment in the exact reverse way that it absorbed it during expansion
Most processes however are irreversible For example if a gas-filled box were suddenly
increased in size so that the particles were free to wander into the void created by the sudden
expansion then the gas would eventually fill the new volume uniformly at the same
temperature as before the expansion (The temperature is unchanged in this imaginary
process because the kinetic energy of the molecules would be unaffected by such an
instantaneous expansion of the container walls) This imaginary but highly thought-
stimulating process is called a free expansion and it is impossible to reverse it During the
expansion we didnrsquot push on any of the molecules so reversing this process would mean
making them go back into their original volume without pushing on them they simply will
not cooperate to this extent Another way to see that just pushing them back where they came
from does not reverse the free expansion is to think about what would happen if we just
compressed either adiabatically or isothermally An adiabatic compression back to the
original volume would heat the gas above its original temperature and an isothermal
compression would require that heat be exhausted to the environment But the free expansion
involved neither temperature changes nor heat exchanges so neither of these two processes
is the reverse of the free expansion It is simply impossible to reverse this rapid expansion in
a way that takes us back to the initial state
It is possible to calculate the change of entropy for both reversible and irreversible processes
Letrsquos consider a reversible process first In a reversible process the entropy change is given
by the formula
dQS
T
where dQ is the amount of heat added to the system during a small step of the process
The total energy change during the process may then simply be calculated by integration
f f
f i i i
dQS S S dS
T
13
(Just as in the case of energy we are mainly interested in differences rather than in
absolute magnitudes)
This integration method does not work for irreversible processes For instance in the
rapid free expansion discussed above no heat is added to the system but its disorder
obviously goes up We need to find some other way to calculate the entropy The key
is the fact that the entropy of a system depends only on its current state and not at all
on how it arrived there So to calculate the entropy change in an irreversible process
first find out what the initial and final conditions of the process are Then invent a
reversible process that takes the system from the initial state to the final state Since
the entropy depends only on the state of system and not on the process the entropy
change for the reversible process is the same as that for the irreversible process
Entropy is thus like pressure or temperature if the pressure changes from Pl to to P2
then the pressure difference is P2 - Pl regardless of how the pressure change was
made This seems obvious for pressure but not for entropy only because you arent
comfortable with entropy yet
Entropy of an Ideal Gas
For n moles of an ideal gas the difference in entropy between a state with temperature
T and volume V and some standard state with temperature To and volume Vo is given
by
ln lno o Vo o
T VS T V S T V nC nR
T V
Entropy in a Phase Change
Calculating entropy change in a phase change like melting or boiling is easy It is
QT Where Q is in the latent heat for example the flows in to cause the melting and
T is the temperature of the phase change There is NO integral to do
14
Serway Chapter 35
Angle of Reflection
If plane waves are incident on a reflecting surface with the propagation direction of
the waves making angle θ1 with the normal direction to the surface then the reflected
angle θ1prime relative to the surface normal is simply
1 1
ie the incident angles and reflected angles are the same
Refraction
If plane waves traveling through medium 1 are incident on a plane interface between
medium 1 and medium 2 then the angle of incidence of the incoming wave θ1 and
the angle of refraction of the transmitted wave θ2 are related by Snellrsquos law
2 2
1 1
sin
sin
v
v
where v1 and v2 are the wave speeds in medium 1 and medium 2 The angle of
incidence and the angle of refraction are both measured between the wave propagation
direction and the normal to the interface In terms of indices of refraction in the case
of light waves Snellrsquos law takes the more familiar form
1 1 2 2sin sinn n
where ni = cvi
Total Internal Reflection
If a wave is incident from a medium of low wave speed into a medium of high wave
speed the law of refraction requires that the angle of refraction be greater than the
angle of incidence If the angle of refraction is required to be greater than 90deg then no
refracted wave can exist and total internal reflection occurs The critical incident
angle θc beyond which total internal reflection occurs is given by
2
1
sin c
n
n
15
Dispersion of Light
In addition to the speed of light varying from material to material it also varies with
wavelength within each material This means that the index of refraction is generally a
function of wavelength
c
nv
Since the wave speed is not constant such a medium is dispersive meaning in this
context that refraction actually disperses white light into its various colors because
Snellrsquos law gives a different angle for each wavelength In most materials the
variation with wavelength is quite small but this small effect is responsible for some
of the most spectacular color effects we ever see including rainbows a flashing
crystal chandelier and the colored fire of a diamond solitaire by candlelight
16
Serway Chapter 36
Real and Virtual Images
When light rays are focused at a certain plane producing an image if a sheet of white
paper is placed there we call the image a real image The images produced by film
projectors and overhead projectors are examples of real images
When light rays appear to come from a certain location but no image is produced
when a screen is placed there we say that there is a virtual image at that location For
instance when you look in a mirror it appears that someone is behind the mirror but
a screen placed back there in the dark would show nothing Your image in the mirror
is a virtual image
Ray Tracing
There are lots of rules about how to find the images in optical systems but the best
way to keep things straight is to learn how to draw the principal rays for curved
mirrors and lenses The rays for convex and concave mirrors are shown in Fig 3615
The rays for converging and diverging thin lenses are shown in Fig 3627 You should
memorize the principal rays and know how to use them to locate images
Curved Mirrors
The focal length of a curved mirror with radius of curvature R is given by
2
Rf
The relation between object distance p image distance q and focal length f is
1 1 1
p q f
If the mirror is a diverging mirror f should be negative and if q should turn out to be
negative the image is virtual
Lateral Magnification
The lateral magnification in an optical system is defined by the ratio of the image size
to the object size
17
Image height
Object height
qM
p
for curved mirrors
As usual there are sign conventions here too but this formula just gives the
magnitude It is better to keep keep track of upright images versus inverted images by
means of ray diagrams rather than by memorizing sign conventions
Thin Lenses
The focal length of a thin lens is related to the radii of curvature of the two faces R1
and R2 of the lens by
1 2
1 1 11n
f R R
Note that this formula differs from Eq (3611) in the text by not having a minus sign
between the two R-terms We like this form better because for a simple converging
lens like a magnifying glass we just use positive values of R for both surfaces If one
of the faces is concave producing divergence use a negative value for R And if a
surface is flat use R = infin If the face is flat the radius is infinite
The relation between the image object and focal distances for a thin lens is the same
as that for a curved mirror
1 1 1
q p f
Use a negative focal length if the lens is diverging
The lateral magnification for a thin lens is the same as for a curved mirror
Image height
Object height
qM
p
Camera
The lens system in a camera projects a real image of an object onto the film (or CCD
array in a digital camera) The position of the image is adjusted to be on the film by
moving the lens into or out of the camera
18
Eye
The eye is like a camera in that a real image is formed on the retina Unlike a camera
the image position is adjusted by changing the focal length of the lens This is done by
the ciliary muscle which squeezes the lens changing its shape
Near Point The near point is the closest distance from the eye for which the lens can
focus an image on the retina It is usually 18-25 cm for young persons
Far Point The far point is the greatest distance from the eye for which the lens can
focus an image on the retina For a person with normal vision the far point is at
infinity
Nearsightedness A person is nearsighted if their far point is at some finite distance
less than infinity This condition can be corrected with a lens that takes an object at
infinity and produces a virtual image at the personrsquos far point
Farsightedness and Presbyopia A person is farsighted if their near point is too far
away for comfortable near work like reading or knitting This can be corrected by a
lens which takes an object at a normal near point distance of 18-25 cm and produces a
virtual image at the personrsquos natural near point Presbyopia involves a similar
problem which nearly all people experience as they age The ciliary muscle becomes
too weak and the lens becomes too stiff to allow the eye to provide for both near and
far vision The solution for this problem is either reading glasses or bifocal lenses
Reading classes are just weak magnifying glasses mounted on eyeglass frames
Bifocal lenses are split into upper and lower halves The lower half is a lens which
gives the proper correction for near work and the upper half is a different lens for
proper focusing at infinity
Angular Size
When an object is brought closer to the eye it appears to be larger because the image
on the retina is larger The size of this image is directly proportional to the objectrsquos
angular size which is the angle subtended by the object measured from the center of
the lens of the eye In optical instruments which are to be used with the eye the
angular size of the final image is whatrsquos important because it determines how large
the image will appear to the viewer
19
Simple Magnifier
A simple magnifier is a single converging lens or magnifying glass It takes an object
closer to the eye than a normal near point and produces a virtual image at or beyond
this near point The angular magnification is defined to be the ratio of the angular size
when viewed through the lens to the angular size of the object when viewed at the
normal near point (without aid of the lens)
Microscope
This instrument has two lenses (1) The objective is near the object being viewed and
produces a greatly magnified real image (2) The eyepiece is a simple magnifier which
the viewer uses to closely examine the image from (1)
Telescope
This instrument also has two lenses (1) The objective at the front of the telescope
takes light from a distant object and produces a real inverted image (which is rather
small) near its focal point (2) This small real image is then examined by the eyepiece
functioning as a simple magnifier to produce a virtual image with a larger angular
size
20
Serway Chapter 37
Two-Slit Interference
If light is incident on two closely spaced narrow slits a pattern of light and dark
stripes is produced beyond the slits The bright stripes or fringes are caused by
constructive interference of the two waves coming from the slits Constructive
interference occurs whenever two waves arrive at a location in phase with each other
This occurs when the distance x1 from slit 1 to a point P on the screen and the
distance x2 from slit 2 to point P differ by in integral number of wavelengths
1 2 where 0 1 2x x m m
where λ is the wavelength of the light When x1 and x2 are much larger than the slit
spacing d this condition reduces to
sind m
where θ is the angle between the direction of the incident light and the direction of the
light arriving at the screen
Thin Films
When light is partially reflected and partially transmitted by a thin film of transparent
material it is possible to have interference between the wave reflected from the front
of the film and light reflected from the back of the film (The colored reflections from
the thin film of oil on the water in a rain-soaked parking lot are an example of this
effect) It is difficult to write down formulas that will work in all cases so we will
just review the important principles here
1 If the two reflected waves are in phase with each other the film has enhanced
reflection (constructive interference) but if the two reflected waves are out of phase
with each other reflection is diminished (destructive interference) Phase shifts occur
due to reflection and due to the extra path length through the film of the wave
reflected from the back of the film
2 The phase change due to reflection is determined by the difference in index of
refraction between the two media involved in the reflection If the wave is incident
21
from a medium with a low index of refraction into a medium with a high index of
refraction a phase change of 180deg occurs and the reflected wave is inverted If
incident from high to low no phase shift occurs and the reflected wave is non-
inverted
3 The extra path length through the film of the wave reflected from the back of the
film is equal to 2t where t is the thickness of the film (The incident light is assumed
to be normal to the surface of the film) The number of wavelengths contained in the
extra path length is equal to 2tλn where λn = λn is the wavelength of the light in the
film
4 Rules for reflection from thin films
If one ray is inverted and the other is not then we have
12 constructive
2 nt m
2 destructivent m
If either both rays are inverted or both are non-inverted we have
2 constructivent m
12 destructive
2 nt m
22
Serway Chapter 38
Diffraction Grating
A diffraction grating is simply a fancy version of two-slit interference with the two
slits replaced by thousands of slits Just as in the two-slit case the bright fringes
occur at angles given by
sin md
but in this case d the distance between neighboring slits is made to be very small
The effect of having many slits instead of two is to make each bright fringe highly
localized with wide dark regions between neighboring maxima
Single Slit Diffraction
When light passes through an opening in an opaque screen an interference pattern is
produced beyond the opening To understand why we may replace the single opening
by many small coherent sources of light These many sources interfere with each
other producing a pattern known as a diffraction pattern (Note that many authors do
not distinguish between interference and diffraction treating them as interchangeable
terms) If the opening is a slit of width a then the diffraction pattern far from the slit
will have a bright central maximum with a succession of minima and weaker maxima
on either side The angle between the incident direction and the minima is given by
sin ma
where m = plusmn1 plusmn2
If the opening is circular with diameter D the angle between the incident direction
and the first minimum is given by
sin 122D
Optical Resolution and Rayleighrsquos Criterion
Two point sources can just be resolved (distinguished from each other) if the peak of
the diffraction image of the first source overlies the first minimum of the diffraction
image of the second source For circular holes of the kind usually encountered in
23
optical devices this condition is approximately satisfied when the angular separation
between the two sources as viewed from the optical instrument is greater than or equal
to the critical angle
min 122D
where λ is the wavelength of the light and where D is the diameter of the aperture in
the instrument
Polarization
We say that an electromagnetic wave is polarized if its electric field vector doesnrsquot
change direction in a random fashion The simplest kind of polarization is linear
polarization in which the electric field vector oscillates back and forth along the
same axis in space Polarized light can be produced from normal unpolarized light by
selective absorption (as in Polaroid sunglasses) by passing light through crystals that
have different indices of refraction for different polarizations (double refraction) by
scattering (the blue sky is polarized) and by reflection (glare) Polarization by
reflection occurs when light reflects from a shiny insulating (non-metallic) surface
The amount of polarization is greatest for reflection at Brewsterrsquos angle
2
1
tan P
n
n
where θP is the incidence angle of light from medium 1 onto medium 2 and where n1
and n2 are the indices of refraction for the media (In our everyday experience n1 = 1
since the light comes in through the air and n2 is the index of refraction of the shiny
insulating material producing the glare eg water glass plastic paint etc)
Malusrsquos Law
The intensity of transmitted polarized light through a perfect polarizer is related to the
incident intensity of polarized light by Malusrsquos law
2cosoI I
where Io is the intensity of the incident light and where θ is the angle between the
electric field vector in the incident wave and the transmission axis the polarizer
24
Serway Chapter 39
Principles of Relativity
All of the weirdness of relativity flows from two simple principles
(1) The laws of physics must be the same in all inertial (non-accelerating) reference
frames
(2) The speed of light in vacuum has the same value in all inertial reference frames
Note that (1) seems reasonable but (2) is very odd It says that if two space ships are
approaching each other at nearly the speed of light and a laser pulse is shot from ship
1 toward ship 2 then when the people on ship 2 measure the speed of the pulse as it
goes by it is moving at 3 times 108 ms the same as if the ships were stationary
Simultaneity
If observer 1 sees two events in her own frame as simultaneous at two different
locations a moving observer 2 will see these two events happening at different times
Relativistic gamma
The factor γ (gamma) appears regularly in the formulas of relativity
2
2
1
1 vc
where v is the relative speed between two inertial frames
Time Dilation
If observer 1 sees two events at the same location in space separated by time Δtp in his
own frame then observer 2 moving at speed v relative to observer 1 will see these
two events separated by a longer time Δt
pt t
Moving clocks run slow
Length Contraction
If observer 1 measures the length of an object along the x-direction at some instant of
time in her frame to be Lp then observer 2 moving in the x-direction at speed v will
measure the length L of the object to be shorter
pLL
25
Moving meter sticks are short
Velocity Addition
If a particle is moving at velocity vac relative to frame c and if frame c is moving at
speed vcb with respect to frame b (with both velocities directed along the same line)
what would be the velocity of the particle in frame b In classical physics the answer is
ab ac cbv v v
but if any of these velocities are an appreciable fraction of the speed of light the
answer changes
21ac cb
abac cb
v Vv
v v c
In using this equation make sure that vac has the same sign in the numerator and in the
denominator and that vbc also has the same sign in both places Note that this equation
is different (and in our opinion easier to understand) than the equations in the text
Momentum and Energy
The momentum and total energy of a particle moving at velocity v (its speed is v v )
are given by
22
2 2
22
1 1 vvc c
m mcm E mc
vp v
Rest Energy
When a particle is at rest (v = 0 and γ = 1) then E = mc2 is called the rest energy It
includes the potential energy of the particle so that if we change the potential energy
of a particle we change its rest mass
Kinetic Energy The kinetic energy is the difference between the total energy and the
rest energy
2 2K mc mc
For v ltlt c this reduces to K = mv22
General Relativity
Einsteinrsquos principle of equivalence in the theory of general relativity is In the vicinity
of a point a gravitational field is equivalent to an accelerated frame of reference in
26
the absence of gravitational effects For example there is no difference whatever in
the physics experienced by an astronaut on earth and one who is in interstellar space
far away from any stars or planets but who has just turned on her spaceshiprsquos engines
and is accelerating at 98 ms2
27
Serway Chapter 40
Photons
We now know that even though light behaves like a wave it is better described as
consisting of small packets of energy called photons The energy of a photon is related
to its frequency by
E hf
where h is Planckrsquos constant
346626 10 J sh
Photons also carry momentum given by
hf hp
c
Since the amount of energy in each photon is fixed the difference between dim light
and intense light (of the same wavelength) is that dim light consists of fewer photons
than intense light
Photoelectric Effect
In the photoelectric effect light shining onto a metal surface gives the electrons in the
metal enough energy to escape and be detected It requires a certain amount of energy
(called the work function typically = 3 ndash 5 eV) just to kick the electron out of the
metal so the light must deliver at least this much energy to an electron to produce the
effect It is observed that red light no matter how intense never produces electrons
But ultraviolet light even if quite dim will eject electrons from the metal Since red
light consists of 2 eV photons and ultraviolet light has photons with energies around
4-6 eV the photon idea explains the behavior of the photoelectric effect The
maximum energy that an ejected electron can have is
maxK hf
where f is the photon frequency The minimum frequency that light can have and cause
electrons to be ejected is called the cutoff energy and is found by setting Kmax = 0
cf h
28
Compton Effect
When high-frequency light interacts with free electrons the scattered light does not
have the same wavelength as the incident light contrary to what classical
electromagnetism would predict But the photon picture in which photons have
momentum and energy allows us to treat scattering as a collision between the photon
and the electron In this collision the electron and the scattered photon both have a
different momentum and energy than they did before And since p = hλ for a photon
if the momentum is different the wavelength will also be different
after before 1 cose
h
m c
where the angle θ is the angle between the incoming photon direction and the
direction of the scattered photon
Particles are Waves
Since photons behave like particles sometimes it is not surprising that elementary
particles can behave like waves sometimes The wavelength of a particle with
momentum p is given by
h
p
Wave-particle Duality
Both photons and elementary particles have a dual nature sometimes they behave like
particles and sometimes they behave like waves It is hard for us to comprehend the
nature of such an object by making mental pictures but experiments definitively show
that this is the case Since we have no direct experience with photons and elementary
particles (because their energies are so much smaller than the energies of the everyday
objects we encounter) it is perhaps not surprising that we have a hard time forming a
mental picture of how they behave
Electron Interference
Since an electron is both a particle and a wave just like a photon it should be able to
produce an interference pattern This is observed If an electron beam is shot at two
closely-spaced slits and if the electrons that pass through the slits are detected
downstream it is found that there are some locations where electrons are never detected
29
and others where lots of electrons are detected The pattern is exactly the same as the
one observed for light waves provided that we use the electron wavelength λ = hp in
place of the wavelength of light This pattern is observed even though each electron is
detected as a single dot on the screen Only after many such dots are collected does the
pattern emerge And if we try to understand how this effect could possibly work by
looking closely at each slit to see which one the electron came through the pattern
disappears the act of measurement destroys the interference This means that each
single electron somehow comes through both slits (as a wave would)
Uncertainty Principle
In classical physics we always imagine that the positions and momenta of moving
particles have definite values It might be hard to measure them but surely at each
instant of time a particle should be precisely located at some point in space and have a
similarly precise momentum This turns out not to be true Instead both position and
momentum are required to be uncertain with their uncertainties Δx and Δp satisfying
the Heisenberg uncertainty relation
2x p
So if the particle were known to be precisely at some particular location (so that Δx =
0) we couldnrsquot know anything about its momentum (Δp = infin) and if its momentum
were exactly known we couldnrsquot know its position
There is a similar relation involving the particlersquos energy E and the time interval Δt
over which this energy is measured
2E t
This means that energy is not actually conserved if we are considering very small time
intervals and this brief non-conservation of energy has been observed
30
Serway Chapter 41
Wave Function ψ and Probability
Quantum mechanics does not predict exactly what an electron or a photon will do Instead
it specifies the wave function or probability amplitude ψ of an electron or a photon
This wave function is a complex-valued function of space and time whose squared
magnitude is the probability density P for finding a particle at a particular place in
space at a certain time
2P
where is the complex conjugate of ψ
For example the wave function of an electron with perfectly specified momentum p
would have a wavelength given by
h
p
and its probability amplitude would be proportional to
2i x ipxe e
The corresponding probability density would then be
21ipx ip ipxP e e e
which means that the electron is equally probable to be anywhere along the x axis
This is in accord with the uncertainty principle since we specified the momentum
precisely we canrsquot have any idea about the position of the electron
As another example you have probably seen ldquofuzzy ballrdquo drawings of electron
orbitals in a chemistry book These fuzzy balls are meant to indicate the distribution
of the probability density 2
P in the orbital
Particle in a Box
A simple example in which we can calculate the wave function is the case of a particle of
mass m constrained to be inside a 1-dimensional box between x = 0 and x = L In this
case the general wave function is a linear superposition of wave functions ψn of the form
sin niE tn
n xx t A e
L
31
where A is a positive constant where n = 1 2 3 and where the energy associated
with each of the quantum states ψn is given by
22
28n
hE n
mL
This wave function is zero at x = 0 and x = L which means that the particle will never
be found at the walls of the box The wave function has maximum values in the
interior and at these places the particle is most likely to be found and it also has
places where it is zero and at these places the particle will also never be found as
expressed by the formula
2sinn x
PL
The particle in the box is interfering with itself producing a probability interference
pattern across the box just like the interference patterns we studied with light and
sound
32
Serway Chapter 42
Atoms
Many people picture an atom as a miniature solar system where electrons orbit around
a massive nucleus at the center This picture is misleading because in atoms the wave
nature of electrons dominates The electrons form 3-dimensional standing waves
(called orbitals) centered at the nucleus And if you ask what is it thatrsquos waving the
answer is ψ the probability amplitude (see Chapter 41)
Standing waves on a string can occur only for certain frequencies (the fundamental
and higher harmonics) Similarly atomic orbitals occur only for certain energies For
the hydrogen atom the energies of the orbitals have the a particularly simple form
2
1136 eVnE
n
where n = 1 2 3 is called the principal quantum number For other atoms the
determination of the orbital energies requires numerical calculation by computers
Atomic Spectra
If an electron is somehow given extra extra energy (we say that it is excited) so that it
occupies a higher orbital it will eventually ldquofallrdquo back down to a lower orbital Each
time an electron falls to a lower orbital it loses the difference in energy between the
two orbitals in the form of a photon Since the orbital energies are discrete so are the
energy differences and so are the wavelengths of the emitted light The entire set of
these discrete wavelengths is called the atomic spectrum and it is unique to each type
of atom For hydrogen the spectrum can be simply written as
H 2 2
1 1 1
f i
Rn n
where RH = 10973732 times 107 m-1 For other atoms the spectrum cannot be expressed as
a formula
Orbital Angular Momentum Electrons in atoms also have quantized values of
angular momentum The orbital quantum number ℓ specifies the value of this
quantized angular momentum through the formula
1L
33
If we want to know the value of the angular momentum along some direction in space
say the z direction the answer is not L but rather
zL m
where mℓ is another quantum number which runs from
1 1m
This quantum number is important when an atom sits in a magnetic field
Spin Angular Momentum It has been found experimentally that electrons and other
charged particles also carry internal angular momentum which we call spin
Electrons have an intrinsic spin angular momentum s along a specified axis that is
extremely quantized it can only take on 2 possible values
1
2zs s
We thus say that electrons have spin s = 12 or that electrons are rdquospin one-half parti-
cles Since s is an angular momentum it obeys the same rule as that for orbital angular
momentum namely that if its value along some axis is s then its total magnitude is
given by
31
2S s s
I know this seems weird but quantum mechanics is weird The only excuse for this
bizarre way of looking at the world is that it predicts what happens in experiments
Exclusion Principle The answer to the question of how many electrons (or any other
spin one-half particle) can be in one particular quantum state was discovered by
Wolfgang Pauli and is called the exclusion principle
ldquoNo two electrons can ever be in the same quantum state therefore no two electrons
in the same atom can have the same set of quantum numbersrdquo
This is the reason that we have atoms with different properties instead of every atom
simple having all of its electrons in the ground state All of the variety we see around
us in the world is the result of chemical differences and these differences would not
exist unless electrons obeyed this important principle The entire structure of the
periodic table (see pages 1377-1379 in Serway) is an expression of this principle
34
Serway Chapter 44
Nuclear Properties
The nucleus is composed of protons with charge +e and mass mp = 1007226 u and of
neutrons with zero charge and mass mn = 1008665 u where u is one atomic mass unit
-271 u=1660540 10 kg
The atomic number Z counts the number of protons in a nucleus while the neutron
number N counts the number of neutrons The mass number A is the sum of the two
A N Z
Protons and nuetrons have about the same mass and some times called baryons (heavy
ones) or nucleons (partcles in the nucleus) A is sometimes called the baryon
number The chemical elements are distinguished by Z ie hydrogen has Z = 1 iron
has Z = 26 uranium has Z = 92 etc But for a given Z there might be several nuclei
with different numbers of neutrons These nuclear siblings are called isotopes
These numbers are used to label nuclei according to the pattern
5626 ie FeA
Z X
denotes the isotope of iron with N = 56 ndash 26 = 30 neutrons
The nucleus is roughly spherical with a radius given approximately by
1 3 150 0where 12 10 mr r A r
Nuclear Stability
Because the positively charged protons electrically repel each other with an enormous
force at distances as small as 10-15 m there must be some really strong force that that
overcomes electrical repulsion to hold protons and neutrons together This force is
called with some lack of imagination the strong force It is a very short-range force
(it only acts over a distance of about 2 x 10-15 m) and attracts protons to protons
neutrons to neutrons and neutrons to protons But in spite of this strong nuclear force
the coulomb repulsion of the protons is still present so anything that might keep the
protons from being right next to each other would help keep the nucleus from
35
exploding This role is played by the neutrons and for nuclei with Z le 20 the stable
nuclei roughly have N = Z
For Z gt 20 the coulomb repulsion force is more powerful and more neutrons are
needed to dilute it up to about N = 15Z around Z = 80 For Z gt 83 no amount of
neutrons can help and these nuclei are unstable (radioactive) There is an ldquoisland of
(relative) stabilityrdquo around Z=90-92 (thorium and uranium respectively) These
elements have one or two isotopes with half-lifes of billions of years so there are
substantial amounts of such elements on Earth This fact makes it possible to have
practical fission devices
Radioactive Decay
There are three types of radioactive decay
Alpha decay the nucleus kicks out a helium nucleus (N = 2 Z = 2 A = 4) So if the
original nucleus is called X and the new nucleus is called Y then the decay would look
like this
4 42 2X Y+ HeA A
Z Z
Beta decay the nucleus either kicks out an electron (endash) or its positively-charged
antimatter twin the positron (e+) plus either an electron neutrino v or an electron
anti-neutrino v
1X Y+e electron decayA AZ Z v
1X Y+e positron decayA AZ Z v
A neutrino is a particle with no charge hardly any mass (much less than the electron
mass) and interacts so weakly with matter that most neutrinos upon encountering the
planet earth just pass right through it as if it werenrsquot there
Gamma decay the nucleons in the nucleus X are in an excited energy state X
(perhaps as a result of having undergone alpha or beta decay) and they drop down to a
lower energy state shedding the energy as a high frequency photon
X XA AZ Z
36
This process is exactly analogous to the way that the electrons in atoms emit photons
Decay Rate and Half Life
There is no way to predict exactly when an unstable or excited nucleus will decay but
there is an average rate at which this decay occurs called the decay constant λ The
meaning of this constant is that if there are a large number N of nuclei in a sample
then the number of decays per second that will be observed (called the decay rate R)
is R = λN In mathematical language
dMR N
dt
This simple differential equation has for its solution
0tN t N e
where N0 is the number of nuclei in the sample at time t = 0
The half-life is the time it takes for half of the nuclei in the sample to decay and is
related to the decay constant by
1 2
ln 2 0693T
Disintegration Energy
When a nucleus decays it is making a transition to an overall state of lower energy
which means according to Einsteinrsquos famous formula E = mc2 that the sum of the
masses after the decay must be less than the mass before with the lost mass appearing
as kinetic energy among the decay products For example in alpha decay this kinetic
energy called the disintegration energy Q is given by
2X YQ M M M c
37
Serway Chapter 45
Nuclear energy
There are two ways to extract energy from the nucleus fission and fusion For nuclei
with Z greater than 26 breaking the nucleus apart into pieces leads to a lower
2mc energy than the original nucleus so energy can be extracted by fission For
nuclei with Z less than 26 a lower energy is achieved by combining nuclei so energy
can be extracted by fusion
Fission
Since neutrons have no charge they are not repelled from nuclei as protons are for
this reason their behavior is the key to understanding how fission works
Fast neutrons mostly bounce off other nuclei with each collision slowing the neutron
down This elastic energy loss is most effective if the other nuclei have low mass (like
hydrogen) and these materials are called moderators because of their ability to slow
down fast neutrons
The reason that slowing neutrons is important is that slow neutrons are much more
likely to be absorbed by a nucleus which then leads to nuclear reactions of various
kinds For a few very large nuclei like uranium-235 and some plutonium isotopes
absorption of a slow neutron causes the nucleus to split into two large fragments plus
2 or 3 fast neutrons (energy is released in the process too) Thus one slow neutron
can produce energy plus 2 or 3 more neutrons which if moderated from fast to slow
can split 2 or 3 more nuclei leading to a runaway chain reaction If the reaction is
allowed to proceed unchecked it produces a large explosion If a neutron absorbing
material is added to the mix (like the cadmium in reactor control rods) it is possible to
keep the reaction under control and to extract the released energy as heat to drive
steam turbines and produce electricity
Fusion
Fusion involves mashing two nuclei together and since they are both charged and repel
each other this reaction is much harder to make go The nuclei must have enough energy to
overcome the coulomb repulsion which is why this reaction requires a high temperature
(hundreds of millions of degrees K) like that in the sun (which is burning hydrogen to
helium via fusion) or like that in the center of the fission explosion that is used to detonate
38
a hydrogen bomb
This reaction is of interest for power production in spite of this difficult temperature
requirement because of the abundance of fusion fuel on the planet There are about
012 g of deuterium ( 21H ) in every gallon of water on earth and it only costs about 4
cents to extract it The fusion energy available from this minuscule amount of
deuterium would run a 1000-Megawatt power plant for 10 seconds Water is so
abundant on earth that if fusion were to work we would have an essentially
inexhaustible source of energy
So why donrsquot we have fusion power plants Well the fuel is cheap but the match is
incredibly expensive The only way we know to control this difficult high-temperature
reaction is with large and expensive pieces of equipment involving either large
magnetic fields and complex high-power electromagnetic antennas or with gigantic
(football-fieldsized) laser facilities involving more than a hundred of the highest-
energy lasers ever built Power plants based on these current methods for controlling
fusion are unattractive to the fiscally-minded people who run the electric power
industry Hopefully better designs will be discovered as experiments continue
39
c actinium 68 Er erbium 101 Md mendelevium 104 Rf rutherfordium l aluminum 63 Eu europium 80 Hg mercury 62 Sm samarium m americium 100 Fm fermium 42 Mo molybdenum 21 Sc scandium b antimony 9 F fluorine 60 Nd neodymium 106 Sg seaborgium r argon 87 Fr francium 10 Ne neon 34 Se selenium s arsenic 64 Gd gadolinium 93 Np neptunium 14 Si silicon t astatine 31 Ga gallium 28 Ni nickel 47 Ag silver a barium 32 Ge germanium 41 Nb niobium 11 Na sodium k berkelium 79 Au gold 7 N nitrogen 38 Sr strontium e beryllium 72 Hf hafnium 102 No nobelium 16 S sulfur i bismuth 108 Hs hassium 76 Os osmium 73 Ta tantalum h bohrium 2 He helium 8 O oxygen 43 Tc technetium
boron 67 Ho holmium 46 Pd palladium 52 Te tellurium r bromine 1 H hydrogen 15 P phosphorus 65 Tb terbium d cadmium 49 In indium 78 Pt platinum 81 Tl thallium a calcium 53 I iodine 94 Pu plutonium 90 Th thorium f californium 77 Ir iridium 84 Po polonium 69 Tm thulium
carbon 26 Fe iron 19 K potassium 50 Sn tin e cerium 36 Kr krypton 59 Pr praseodymium 22 Ti titanium s cesium 57 La lanthanum 61 Pm promethium 74 W tungsten l chlorine 103 Lr lawrencium 91 Pa protactinium 92 U uranium r chromium 82 Pb lead 88 Ra radium 23 V vanadium o cobalt 3 Li lithium 86 Rn radon 54 Xe xenon u copper 71 Lu lutetium 75 Re rhenium 70 Yb ytterbium m curium 12 Mg magnesium 45 Rh rhodium 39 Y yttrium b dubnium 25 Mn manganese 37 Rb rubidium 30 Zn zinc y dysprosium 109 Mt meitnerium 44 Ru ruthenium 40 Zr zirconium s einsteinium
40
UNITS The SI units are given in parentheses Other commonly used units are given in terms of the SI units Symbols conform with the recommendations of the American National Standards Institute (ANSI) and the American Institute of Physics (AIP) Commonly used multiples of SI units
prefix symbol factor tera T 1012 giga G 109
mega M 106 kilo k 103
centi c 10-2 milli m 10-3
micro μ 10-6 nano n 10-9 pico p 10-12
femto f 10-15
length meter (m) angstrom 1 Ǻ = 10-10 m inch 1 in = 254 x 10-2 m foot 1 ft = 03048 m mile 1 mi = 1609 m
mass kilogram (kg) atomic mass unit 1 u = 1661 x 10-27 kgslug 1 slug = 1459 kg
force newton (N = kg bull ms2) dyne 1 dyn = 10-5 N pound 1 lb = 4448 N
pressure Pascal (Pa = kgm bull s2) atmosphere 1 atm = 1013 x 105 Pa poundssquare inch 1 psi = 6895 Pa cm of mercury 1 cm Hg = 1333 Pa bar 1 bar = 1000 x 105 Pa torr 1 torr = 1333 Pa
time second (s) minute 1 min = 60 s hour 1 h =3600 s
frequency hertz (Hz = s-1) radianssecond 1 rads = 12π Hz
energy joule (J = kg bull m2s2) erg 1 erg = 10-7 J electron volt 1 eV = 1602 x 10-19 J calorie 1 cal = 4187 J kilowatt-hour 1 kWh = 36 x 106 J British thermal unit 1 Btu = 1055 J
power watt (W = kg bull m2s3) horsepower 1 hp = 7457 W
charge coulomb (C = A bull s)
electric potential volt (V = kg bull m2 s3 bull A)
current ampere (A)
resistance ohm (Ω = kg bull m2s3 bull A2)
capacitance farad (F = s4 bull A2kg bull m2)
magnetic field tesla (T = kgs2 bull A) gauss 1 G = 10-4 T
magnetic flux weber (Wb = kg m2s2 bull A) maxwell 1 Mx = 10-8 Wb
magnetic inductance henry (H = kg bull m2s2 bullA2)
temperature kelvin (K) degrees Celsius 0degC = 27315 K
angle radian (rad) degree 1 deg = π180 rad revolution 1 rev = 2π rad
41
Some Physical Constantsa Quantity Symbol Valueb Atomic mass unit μ 1660 538 73 (13) x 10-27 kg
931494 013 (37) MeVc2 Avogadros number NA 6022 141 99 (47) x 1023 particlesmol Avogadrorsquos number 6022 x 1023 mol Bohr magneton
2B e
em 9274 008 99 (37) x 10-24 JT
Bohr magneton μB 927 x 10-24 JT Bohr radius
0
2
2e em e k
a 5291 772 083 (19) x 10-11 m
Boltzmanns constant B A
RNk 1380 650 3 (24) x 10-23 JK
Boltzmannrsquos constant kB 1380 x 10-23 JK Compton wavelength
C eh
m c 2426 310 215 (18) x 10-12 m
Deuteron mass md 3343 583 09 (26) x 10-27 kg 2013 553 212 71 (35) u
electron charge e 1602 x 10-19 C Electron mass me 9109 381 88 (72) x 10-31 kg
5485 799 110 (12) x 10-4 u 0510 998 902 (21) MeVc2
electron mass 911 x 10-31 kg Electron volt eV 1602 176 462 (63) x 10-19 J Elementary charge e 1602 176 462 (63) x 10-19 C Gas constant R 8314 472 (15) JKmol Gravitational constant G 6673 (10) x 10-11 Nm2kg2 Neutron mass mn 1674 927 16 (13) x 10-27 kg
1008 664 915 78 (55) u 939565 330 (38) MeVc2
neutron mass 1675 x 10-27 kg Nuclear magneton
2n p
em 5050 783 17 (20) x 10-27 JT
permeability constant μ0 1257 x 10-6 Hm permittivity constant ε0 8854 x 10-12 Fm Plancks constant h
2h
6626 068 76 (52) x 10-34 Js 1054 571 596 (82) x 10-34 Js
Planckrsquos constant h ħ
6626 x 10-34 Js 1055 x 10-34 Js
Proton mass mp 1672621 58 (13) x 10-27 kg 1007 276 466 88 (13) u 938271 998 (38) MeVc2
proton mass 1673 x 10-27 kg Rydberg constant RH 1097 373 156 854 9 (83) x 107 m-1 Speed of light in vacuum c 2997 92458 x 108 ms (exact) speed of light c 300 x 108 ms a
These constants are the values recommended in 1998 by CODATA based on a least-squares adjustment of data from different measurements For a more complete list see P J Mohr and B N Taylor Rev Mod Phys 72351 2000 b The numbers in parentheses for the values above represent the uncertainties of the last two digits
NOTE The ones in red are the ones from Appendix 13 The ones in yellow are the ones that are from the Serway book table but that were already on the Appendix 13 table
42
INDEX
Absolute zero 9
Adiabatic 17
Adiabatic exponent 18
Adiabatic process 17
Alpha decay 46
Amplitude 4
Angle of reflection 25
Angular frequency 3
Angular magnification 30
Angular momentum atomic 43
Angular size 29
Archimedes Principle 1
Atomic spectra 43
Atoms 43
Avogadrorsquos number 10
Beats 8
Bernoullirsquos Equation 2
Beta decay 46
Bifocals 29
Brewsterrsquos angle 34
British Thermal Unit Btu 11
Buoyancy 2
calorie 11
Calorie 11
Camera 28
Carnot cycle 21
Carnot efficiency 22
Celsius scale 9
Ciliary muscle 29
Coefficient of performance 21
Compton effect 39
Constructive interference 7
Continuity equation of 2
Contraction length 35
Convection 14
Curved mirrors 27
Decay constant 46
Decay rate radioactive 46
Decibel scale 5
Degrees of freedom 16
Density 1
Destructive interference 7
Diffraction single slit 33
Diffraction grating 33
Dilation time 35
Disintegration energy 47
Dispersion of light 26
Doppler effect 6
Double slit interference 31
Efficiency 20
Electron interference 39
Emissivity 15
Energy nuclear decay 47
Energy relativistic 36
43
Engineering work 20
Entropy 22
Entropy ideal gas 24
Equipartition of energy 18
Equivalence principle 36
Exclusion principle 44
Expansion thermal 9
Expansion coefficient linear 9
Eye 29
Fahrenheit scale 9
Far point 29
Farsightedness 29
First Law of Thermodynamics 12
Fission 48
Flux volume 2
Free expansion 23
Fringes 31
Fusion 48
Fusion heat of 11
Gamma relativistic 35
Gamma decay 46
General relativity 36
Half-life 46
Heat 11
Heat capacity 11
Heat conduction 13
Heat engine 20
Heat of fusion 11
Heat of vaporization 11
Heat pump 20
Heats of transformation 11
Hydrostatics 1
Ideal Gas Law 10
Images real and virtual 27
Intensity sound 5
Interference 7
Interference two-slit 31
Internal energy 12
Internal energy degrees of freedom 16
Irreversible process 19 23
Isotopes 45
Joule 11
Kelvin scale 9
Kinetic energy relativity 36
Kinetic theory 16
Latent heat 11
Length contraction 35
Linear expansion coefficient 9
Linear polarization 34
Linear superposition 7
Longitudinal wave 3
Loudness 5
Magnification lateral 27
Magnifying glass 30
Malusrsquos law 34
Microscope 30
Momentum relativistic 36
Muscle ciliary 29
44
Musical instruments 8
Musical scale 8
Near point 29
Nearsightedness 29
Nonlinear 7
Nuclear energy 48
Nuclear properties 45
Nuclear stability 45
Octave 8
Optical resolution 33
Orbital quantum number ℓ 43
Orbitals 43
Particle in a Box 41
Particles are waves 39
Pascalrsquos Principle 1
Period 3
Photoelectric effect 38
Photons 38
Pitch 8
Polarization 34
Power sound 5
Presbyopia 29
Pressure 1
Principal quantum number n 43
Principle of equivalence 36
Principle of linear superposition 7
Probability amplitude ψ 41
Processes thermodynamic 13
R-value 14
Radiation thermal 14
Radioactive decay 46
Radioactive decay rate 46
Ray tracing 27
Rayleighrsquos criterion 33
Reading glasses 29
Real image 27
Refraction 25
Refrigerator 20
Relativistic gamma 35
Relativity principles 35
Resolved for light sources 33
Rest energy 36
Reversible process 19 22
Rope wave speed 4
Second Law of Thermodynamics 19
Shock waves 6
Simple magnifier 30
Simultaneity 35
Single slit diffraction 33
Snellrsquos law 25
Sound speed 5
Specific heat 11
Spin Angular momentum 44
Standing waves 7
Stefanrsquos law 14
Strong force 45
Telescope 30
Temperature 9
45
Temperature Scales 9
Thermal conductivity 14
Thermal energy 12
Thermal expansion 9
Thin film interference 31
Thin lenses 28
Time dilation 35
Tone musical 8
Total internal reflection 25
Transverse wave 3
Traveling Waves 3
Two-slit interference 31
Uncertainty principle 40
Vaporization heat of 11
Velocity addition relativity 36
Virtual image 27
Volume flux 2
Wave function 41
Wave Function ψ and Probability 41
Wave speed 3
Wave-particle duality 39
Wavelength 3
Wavenumber 3
Wien Displacement Law 16
Work 11
12
But if the displacement is done very slowly (as we always assume in thermodynamics)
then the force exerted on the system and the force exerted by the system are in
balance so the force exerted by the system is ndash Fon In thermodynamics it is more
convenient to talk about the force exerted by the system so we change the above
formula for the work done on the system to
xdW d F
where F is the force exerted by the system This has confused students for more than a
century now but this is the way your book and many other books do it so you are
stuck You will need to memorize the minus sign in this definition of the work to be
able to use your textbook
There are many chances to get signs wrong in this and the next two chapters (Mosiah
2 )
When an external agent changes the volume of a gas at pressure P by a small amount
dV the (small amount) of work done on the system is given by
dW PdV
Notice that this minus sign is just what we need to make dW be positive if the external
agent compresses the gas for then dV is negative If on the other hand the external
agent gives way allowing the gas to expand against it then dV is positive and we say
that the work done on the gas is negative
The work done on the system (eg by the gas in a cylinder) in a thermodynamic
process is the area under the curve in a PV diagram It is positive for compressions
and negative for expansions If the volume of gas remains constant in a process then
no work is done by the gas
Cyclic processes are important For cyclic processes represented by PV diagrams the
magnitude of the net work during one cycle is simply the area enclosed by the cycle
on the diagram Be careful to keep track of signs when you are calculating that
enclosed area In cyclic process Q = ndashW for a cycle Put a PV diagram in your notes
1
Path A-B B-C C-D D-A A to A net
Q
W
ΔU
ΔS
Internal Energy
The energy stored in a substance is called its internal energy Eint This energy may be
stored as random kinetic energy or as potential energy in each molecule (stretched
chemical bonds electrons in excited states etc) For ideal gases all states with the
same temperature will have the same Eint
First Law of Thermodynamics
The change ΔEint in the internal energy of a system is given by
intE Q W
where Q is the heat absorbed by the system and where W is the work done on the
system Hence if a system absorbs heat (and if Wge0) the internal energy increases
Likewise if the system does work (W on the system is negative) and if Qge0 the
internal energy decreases Potential Pitfall Many times people talk about work done
by the system It is the minus of W on the system Donrsquot get tripped up
Processes
Adiabatic process No heat is exchanged Q = 0 requiring that the system be insulated
from the environment A process may be approximately adiabatic if it happens so
rapidly that heat does not have time to enter or leave the system Work + or ndash is done
and ΔEint = W
Adiabatic free expansion A gas is allowed to expand into a vacuum without pushing
on anythingndashit just rushes into the vacuum In this process Q = W = 0 so the internal
energy and temperature does not change (Note the difference between an adiabatic
process and a free expansion is that NO work is done in the adiabatic free expansion)
Isobaric process The pressure is held fixed ΔP = 0 For example usually the
pressure increases when a gas is heated but if it were allowed to expand during the
2
heating process in just the right way its pressure could remain fixed In isobaric
processes both Q and W are non-zero (Work is eacy = ndashP(Vf ndash Vi)
Isovolumetric process The volume is held fixed ΔV = 0 The work done by the gas is
then zero and so we have ΔEint = Q
Isothermal process The temperature is held fixed ΔT = 0 There is no change in Eint
so Q = ndashW for isothermal processes For an ideal gas PV = nRT so PV = constant in
an isothermal process The work done on the gas is then given by
lnf
i
Vi
Vf
VW PdV nRT
V
Heat Conduction
The quantity P is defined to be the rate at which heat flows through an object and is a
power having units of watts It is analogous to electric current which is the rate at
which charge flows through an object If the flow of heat through a slab of length L
and cross-sectional area A is steady in time then P is given by the equation
h cT TdQkA
dt L
P =
where Th and Tc are the (hot and cold) temperatures of the two ends of the slab The
heat flows of course because of this temperature difference The quantity k is called
the thermal conductivity and is a constant that is characteristic of the material It is
analogous to the electrical conductivity h cT TL is sometimes called the temperature
gradient and is written dTd dTdx
R-Values
It is common to have the heat-conducting properties of materials described by their R-
values especially for insulating materials like fiberglass batting The connection
between k and R is R = Lk where L is the material thickness In this country R-values
always have units of 2ft F hourBtu
Polyurethane foam 6 (1 in thick) Air space 1 (35 in thick) Fiber glass batting 11
(35 in thick) Wood about 1 (1 in thick) Glass 09 (0125 in thick)
The heat flow rate through a slab of area A is given by
3
h cT TA
R
P
in units of Btuhour Note that A must be in square feet and the temperatures must be
in degrees Fahrenheit
Convection
Convection is the transfer of thermal energy by flow of material For instance a home
furnace doesnrsquot heat a house by waiting for the heat from the burner to slowly conduct
throughout the house instead it quickly pumps warm air to all of the rooms
Generally convection is a much faster way to transfer heat than conduction
Radiation
Electromagnetic radiation can also transfer heat When you warm yourself near a
campfire which has burned itself down into a bed of glowing embers you are
receiving radiant heat from the infrared portion of the electromagnetic spectrum The
rate at which an object emits radiant heat is given by Stefanrsquos law
4AeTP
where P is the radiated power in watts σ is a constant
8 2 45696 10 W m K
A is the surface area of the object in m2 and T is the temperature in kelvins The
constant e is called the emissivity and it varies from substance to substance A perfect
absorber (think black velvet) has e = 1 while a perfect reflector (think mirror) has e =
0 Hence black objects radiate very well while shiny ones do not Also an object that
is hotter than its surroundings radiates more energy than it absorbs whereas an object
that is cooler than its surroundings absorbs more energy than it radiates
Terminology
Transfer variables vs state variables
Energy transfer by heat as well as work done depends on the initial final and
intermediate states of the system They are transfer variables But their sum (Q + W =
4
Eint) is a state variable
Figure 205
5
Serway Chapter 21
Kinetic Theory
The ideal gas law works for all atoms and molecules at low pressure It is rather
amazing that it does Kinetic theory explains why The properties of an ideal gas can
be understood by thinking of it as N rapidly moving particles of mass m As these
particles collide with the container walls momentum is imparted to the walls which
we call the force of gas pressure In this picture the pressure is related to the average
of the square of the particle velocity 2v by
22 1( )
3 2
NP mv
V
Using the ideal gas law we obtain the average translational kinetic energy per
molecule
21 3
2 2 Bmv k T
The rms speed is then given by
2rms
3 3Bk T RTv v
m M
where M is the molecular mass in kgmol
Degrees of Freedom
Roughly speaking a degree of freedom is a way in which a molecule can store energy
For instance since there are three different directions in space along which a molecule
can move there are three degrees of freedom for the translational kinetic energy
There are also three different axes of rotation about which a polyatomic molecule can
spin so we say there are three degrees of freedom for the rotational kinetic energy
There are even degrees of freedom associated with the various ways in which a
molecule can vibrate and with the different energy levels in which the electrons of
the molecule can exist
Internal Energy and Degrees of Freedom The internal energy of an ideal gas made
up of molecules with J degrees of freedom is given by
int 2 2 B
J JE nRT Nk T
6
Heat Capacities of an Ideal Gas The heat capacity of a gas is described by means of
molar heat capacities CV and CP These are the heat capacities per mole and the
subscript V on CV means that the volume is being held constant while for CP the
pressure is held constant For example to raise the temperature of n moles of a gas
whose pressure is held constant by 10 K we would have to supply an amount of heat Q
= nCP (10) K
Molar Specific Heat of an Ideal Gas at Constant Volume
VQ nC T
3monatomic
2VC R
5diatomic
2VC R
5polyatomic
2VC
Real gases deviate from these formulas because in addition to the translational and ro-
tational degrees of freedom they also have vibrational and electronic degrees of
freedom These are unimportant at low temperatures due to quantum mechanical
effects but become increasingly important at higher temperatures The rough rule is
No of degrees of freedom
2VC R
Molar Specific Heat of an Ideal Gas at Constant Pressure
PQ nC T
P VC C R
The internal energy of an ideal gas depends only on the temperature
int VE nC T
Adiabatic Processes in an Ideal Gas
7
An adiabatic process is one in which no heat is exchanged between the system and the
environment When an ideal gas expands or contracts adiabatically not only does its
pressure change as expected from the ideal gas law but its temperature changes as
well Under these conditions the final pressure Pf can be computed from the initial
pressure Pi and from the final and initial volumes Vf and Vi by
or constantf f i iP V PV PV
where γ = CPCV The quantity γ is called the adiabatic exponent Note that this
doesnrsquot mean that the ideal gas law no longer holds it does and in fact it can be
combined with the adiabatic law for pressure given above to obtain the adiabatic law
for temperatures
1 constantTV
Compressions in sound waves are adiabatic because they happen too rapidly for any
appreciable amount of heat to flow This is why the adiabatic exponent appears in the
formula for the speed of sound in an ideal gas
RTv
M
Note that v depends only on T and not on P Because it depends only on the
temperature the speed of sound is the same in Provo as at sea level in spite of the
lower pressure here due to the difference in elevation
Equipartition of Energy
Every kind of molecule has a certain number of degrees of freedom which are
independent ways in which it can store energy Each such degree of freedom has
associated with it ndash on average ndash an energy of 12 Bk T per molecule (or 1
2 RT per mole)
(Note since a molecule has so many possible degrees of freedom it would seem that
there should be a lot of 12 sBk T to spread around But because energy is quantized
some of these degrees of freedom are not ldquoactiverdquo until the temperature becomes high
enough that 12 Bk T is as big as the lowest quantum of energy
8
Serway Chapter 22
Second Law of Thermodynamics
There are several equivalent forms of this important law
Kelvin It is not possible to change heat completely into work with no other change
taking place Or in other words there are no perfect heat engines
Clausius It is not possible for heat to flow from one body to another body at a higher
temperature with no other change taking place Or in other words there are no
perfect refrigerators
Entropy In any thermodynamic process that proceeds from one equilibrium state to
another the entropy of the system + environment either remains unchanged or
increases The total entropy never decreases This law is a bit of an oddity among the
laws of physics because it is not absolute Things are forbidden by the second law not
because it is impossible for them to happen but because it is extremely unlikely for
them to happen (See below for more information about entropy)
Reversible and Irreversible Processes
A reversible process is one which occurs so slowly that it is in thermal equilibrium (or
very nearly so) at all times A hallmark of such processes is that a motion picture of
them looks perfectly normal whether run forward or backward Imagine for instance
the slow expansion of a gas at constant temperature in a cylinder whose volume is being
increased by a slowly moving piston Run the movie backwards and what do you see
You see the slow compression of a gas at constant temperature which looks perfectly
normal
An irreversible process is one which occurs in such a way that thermal equilibrium is
not maintained throughout the process The mark of this kind of process is that a motion
picture of it looks very odd when run backward Imagine the sudden expansion of a gas
into a previously evacuated chamber because a hole was punched in the wall between a
pressurized chamber and the evacuated one Run the movie backward and what do you
see You see the gas in the soon-to-be-evacuated chamber gather itself together and
stream through a tiny hole into a chamber in which there is already plenty of gas If you
have ever seen this happen get in touch with the support group for those who have
witnessed the spontaneous combustion of large mammals by calling 1-800-PYROCOW
9
Heat Engines
Because of the vexing difference in sign between work done on and system and work done
by a system we will invent a new work variable Weng Heat engines do work and so the net
W for these engines is negative But in engineering applications hidden minus signs are
regarded as evil so for heat engines we donrsquot talk about W instead we talk about its
magnitude engW W So for heat engines the first law is
int engE Q W
But for heat pumps and refrigerators work is done on the system so we use the usual work
W when we talk about these systems
A heat engine is a machine that absorbs heat converts part of it to work and exhausts the
rest The heat must be absorbed at high temperature and exhausted at low temperature If the
absorbed heat is Qh the exhausted heat is Qc and the work done by the engine is Weng then
eng h cW Q Q
and the efficiency of the engine is defined to be
eng
h
We
Q
A perfect engine would convert the heat hQ completely into work Weng giving an effi-
ciency of e = 1 Energy conservation alone allows a perfect engine but the second law
requires e lt 1
Refrigerators and Heat Pumps
A refrigerator is a machine that absorbs heat at low temperature and exhausts it at high
temperature the ldquobackwardsrdquo heat transfer being driven by the work done on the machine
by some source of power A heat pump is a machine that either works like a refrigerator
keeping a place cold by transferring heat from this cold place to a higher temperature
environment (cooling mode like an air conditioner) or it functions as a heater
transferring heat into a warm place from a cooler one (heating mode like a window unit
that heats a house by extracting thermal energy from the cold outdoors) The coefficient
of performance of a refrigerator or of a heat pump in cooling mode is defined to be
COP cooling modecQ
W
10
For a heat pump in heating mode the coefficient of performance is
COP heating modehQ
W
Note that we donrsquot have to use the engineering work here because in these systems
work is done on the system and W is naturally positive
A perfect refrigerator would take in heat Qc and exhaust the same amount of heat without
doing any work giving an infinite coefficient of performance Energy conservation alone
allows a perfect refrigerator but the second law requires COP lt infin A perfect heat pump
in heating mode would transfer Qh into the house without doing any work and so would
also have an infinite coefficient of performance The second law forbids this too A good
coefficient of performance for a real device would be around 5 or 6
Carnot Cycle
The most efficient of all possible engines is one that uses the Carnot cycle This cycle
employs an ideal gas has no friction and operates very slowly so that the gas can be in
thermal equilibrium at all parts of the cycle This means of course that it canrsquot
possibly be built and even if it could be built it would not run fast enough to be useful
Nevertheless this cycle is very important because it gives an upper bound on the
efficiency of real engines There cannot possibly be an engine that is more efficient
than one based on the Carnot cycle This cycle consists of the following four steps
1 The ideal gas absorbs heat Qh at constant temperature Th while the gas increases its
volume The reason that heat is absorbed is that expansion tends to cool the gas but
thermal contact with the environment at Th keeps the temperature high by heat
conduction into the ideal gas
2 The ideal gas further increases its volume by an adiabatic expansion This expansion
causes the gas to cool so at the end of this part of the cycle the gas is at temperature Tc
3 The gas exhausts heat Qc at constant temperature Tc while the gas decreases in
volume The reason that heat is exhausted is that compression tends to heat the gas
but thermal contact with the environment at Tc keeps the temperature low by heat
conduction out of the ideal gas
4 The gas is adiabatically compressed back to its original volume (the volume it started
with in step 1) This compression heats the gas from Tc up to Th
11
The efficiency of a Carnot engine is given by the very simple formula
1 cC
h
Te
T
where the temperatures must in be Kelvin No real engine can be more efficient than
this
The coefficient of performance of a Carnot refrigerator or heat pump in cooling mode
is given by
COP cooling mode cC
h c
T
T T
and no real refrigerator can have a coefficient of performance greater than this
The coefficient of performance of a Carnot heat pump in heating mode is
COP heating mode hC
h c
T
T T
So why donrsquot we just use these wonderful Carnot engines and have perfect efficiency
To make the reversible steps in the cycle really reversible they would have to occur
infinitely slowly So the price you pay for making a perfect engine is that it takes
forever to get it to do any work
Entropy
The entropy of a system is defined in terms of its molecular makeup and measures
roughly the disorder of the system If the system is packed into a very small volume
then it is quite ordered and the entropy will be low If it occupies a large volume the
entropy is high (To see what this has to do with disorder note that socks in a drawer
occupy a small volume while socks on the bed in the corner by the door and
hanging from the chandelier occupy a large volume) If the system is very cold then
the molecules hardly move and may even reach out to each other and form a crystal
This is a highly ordered state and therefore has low entropy If the system is very hot
with rapidly speeding molecules crashing into the container walls and bouncing off
each other things are disordered and the entropy is high
It is possible to calculate the entropy of a system in terms of its macroscopic thermody-
namic properties ie pressure volume temperature number of moles etc The key to this
calculation is the concept of a reversible process A reversible process is one that is
12
carried out without friction and so slowly that the process can be reversed at any stage by
making an infinitesimal change in the environment of the system The slow expansion of the
gas at Th in the Carnot cycle with tiny amounts of heat being transferred to the environment
is reversible If we turned around and began slowly to compress the gas would just slowly
exhaust heat to the environment in the exact reverse way that it absorbed it during expansion
Most processes however are irreversible For example if a gas-filled box were suddenly
increased in size so that the particles were free to wander into the void created by the sudden
expansion then the gas would eventually fill the new volume uniformly at the same
temperature as before the expansion (The temperature is unchanged in this imaginary
process because the kinetic energy of the molecules would be unaffected by such an
instantaneous expansion of the container walls) This imaginary but highly thought-
stimulating process is called a free expansion and it is impossible to reverse it During the
expansion we didnrsquot push on any of the molecules so reversing this process would mean
making them go back into their original volume without pushing on them they simply will
not cooperate to this extent Another way to see that just pushing them back where they came
from does not reverse the free expansion is to think about what would happen if we just
compressed either adiabatically or isothermally An adiabatic compression back to the
original volume would heat the gas above its original temperature and an isothermal
compression would require that heat be exhausted to the environment But the free expansion
involved neither temperature changes nor heat exchanges so neither of these two processes
is the reverse of the free expansion It is simply impossible to reverse this rapid expansion in
a way that takes us back to the initial state
It is possible to calculate the change of entropy for both reversible and irreversible processes
Letrsquos consider a reversible process first In a reversible process the entropy change is given
by the formula
dQS
T
where dQ is the amount of heat added to the system during a small step of the process
The total energy change during the process may then simply be calculated by integration
f f
f i i i
dQS S S dS
T
13
(Just as in the case of energy we are mainly interested in differences rather than in
absolute magnitudes)
This integration method does not work for irreversible processes For instance in the
rapid free expansion discussed above no heat is added to the system but its disorder
obviously goes up We need to find some other way to calculate the entropy The key
is the fact that the entropy of a system depends only on its current state and not at all
on how it arrived there So to calculate the entropy change in an irreversible process
first find out what the initial and final conditions of the process are Then invent a
reversible process that takes the system from the initial state to the final state Since
the entropy depends only on the state of system and not on the process the entropy
change for the reversible process is the same as that for the irreversible process
Entropy is thus like pressure or temperature if the pressure changes from Pl to to P2
then the pressure difference is P2 - Pl regardless of how the pressure change was
made This seems obvious for pressure but not for entropy only because you arent
comfortable with entropy yet
Entropy of an Ideal Gas
For n moles of an ideal gas the difference in entropy between a state with temperature
T and volume V and some standard state with temperature To and volume Vo is given
by
ln lno o Vo o
T VS T V S T V nC nR
T V
Entropy in a Phase Change
Calculating entropy change in a phase change like melting or boiling is easy It is
QT Where Q is in the latent heat for example the flows in to cause the melting and
T is the temperature of the phase change There is NO integral to do
14
Serway Chapter 35
Angle of Reflection
If plane waves are incident on a reflecting surface with the propagation direction of
the waves making angle θ1 with the normal direction to the surface then the reflected
angle θ1prime relative to the surface normal is simply
1 1
ie the incident angles and reflected angles are the same
Refraction
If plane waves traveling through medium 1 are incident on a plane interface between
medium 1 and medium 2 then the angle of incidence of the incoming wave θ1 and
the angle of refraction of the transmitted wave θ2 are related by Snellrsquos law
2 2
1 1
sin
sin
v
v
where v1 and v2 are the wave speeds in medium 1 and medium 2 The angle of
incidence and the angle of refraction are both measured between the wave propagation
direction and the normal to the interface In terms of indices of refraction in the case
of light waves Snellrsquos law takes the more familiar form
1 1 2 2sin sinn n
where ni = cvi
Total Internal Reflection
If a wave is incident from a medium of low wave speed into a medium of high wave
speed the law of refraction requires that the angle of refraction be greater than the
angle of incidence If the angle of refraction is required to be greater than 90deg then no
refracted wave can exist and total internal reflection occurs The critical incident
angle θc beyond which total internal reflection occurs is given by
2
1
sin c
n
n
15
Dispersion of Light
In addition to the speed of light varying from material to material it also varies with
wavelength within each material This means that the index of refraction is generally a
function of wavelength
c
nv
Since the wave speed is not constant such a medium is dispersive meaning in this
context that refraction actually disperses white light into its various colors because
Snellrsquos law gives a different angle for each wavelength In most materials the
variation with wavelength is quite small but this small effect is responsible for some
of the most spectacular color effects we ever see including rainbows a flashing
crystal chandelier and the colored fire of a diamond solitaire by candlelight
16
Serway Chapter 36
Real and Virtual Images
When light rays are focused at a certain plane producing an image if a sheet of white
paper is placed there we call the image a real image The images produced by film
projectors and overhead projectors are examples of real images
When light rays appear to come from a certain location but no image is produced
when a screen is placed there we say that there is a virtual image at that location For
instance when you look in a mirror it appears that someone is behind the mirror but
a screen placed back there in the dark would show nothing Your image in the mirror
is a virtual image
Ray Tracing
There are lots of rules about how to find the images in optical systems but the best
way to keep things straight is to learn how to draw the principal rays for curved
mirrors and lenses The rays for convex and concave mirrors are shown in Fig 3615
The rays for converging and diverging thin lenses are shown in Fig 3627 You should
memorize the principal rays and know how to use them to locate images
Curved Mirrors
The focal length of a curved mirror with radius of curvature R is given by
2
Rf
The relation between object distance p image distance q and focal length f is
1 1 1
p q f
If the mirror is a diverging mirror f should be negative and if q should turn out to be
negative the image is virtual
Lateral Magnification
The lateral magnification in an optical system is defined by the ratio of the image size
to the object size
17
Image height
Object height
qM
p
for curved mirrors
As usual there are sign conventions here too but this formula just gives the
magnitude It is better to keep keep track of upright images versus inverted images by
means of ray diagrams rather than by memorizing sign conventions
Thin Lenses
The focal length of a thin lens is related to the radii of curvature of the two faces R1
and R2 of the lens by
1 2
1 1 11n
f R R
Note that this formula differs from Eq (3611) in the text by not having a minus sign
between the two R-terms We like this form better because for a simple converging
lens like a magnifying glass we just use positive values of R for both surfaces If one
of the faces is concave producing divergence use a negative value for R And if a
surface is flat use R = infin If the face is flat the radius is infinite
The relation between the image object and focal distances for a thin lens is the same
as that for a curved mirror
1 1 1
q p f
Use a negative focal length if the lens is diverging
The lateral magnification for a thin lens is the same as for a curved mirror
Image height
Object height
qM
p
Camera
The lens system in a camera projects a real image of an object onto the film (or CCD
array in a digital camera) The position of the image is adjusted to be on the film by
moving the lens into or out of the camera
18
Eye
The eye is like a camera in that a real image is formed on the retina Unlike a camera
the image position is adjusted by changing the focal length of the lens This is done by
the ciliary muscle which squeezes the lens changing its shape
Near Point The near point is the closest distance from the eye for which the lens can
focus an image on the retina It is usually 18-25 cm for young persons
Far Point The far point is the greatest distance from the eye for which the lens can
focus an image on the retina For a person with normal vision the far point is at
infinity
Nearsightedness A person is nearsighted if their far point is at some finite distance
less than infinity This condition can be corrected with a lens that takes an object at
infinity and produces a virtual image at the personrsquos far point
Farsightedness and Presbyopia A person is farsighted if their near point is too far
away for comfortable near work like reading or knitting This can be corrected by a
lens which takes an object at a normal near point distance of 18-25 cm and produces a
virtual image at the personrsquos natural near point Presbyopia involves a similar
problem which nearly all people experience as they age The ciliary muscle becomes
too weak and the lens becomes too stiff to allow the eye to provide for both near and
far vision The solution for this problem is either reading glasses or bifocal lenses
Reading classes are just weak magnifying glasses mounted on eyeglass frames
Bifocal lenses are split into upper and lower halves The lower half is a lens which
gives the proper correction for near work and the upper half is a different lens for
proper focusing at infinity
Angular Size
When an object is brought closer to the eye it appears to be larger because the image
on the retina is larger The size of this image is directly proportional to the objectrsquos
angular size which is the angle subtended by the object measured from the center of
the lens of the eye In optical instruments which are to be used with the eye the
angular size of the final image is whatrsquos important because it determines how large
the image will appear to the viewer
19
Simple Magnifier
A simple magnifier is a single converging lens or magnifying glass It takes an object
closer to the eye than a normal near point and produces a virtual image at or beyond
this near point The angular magnification is defined to be the ratio of the angular size
when viewed through the lens to the angular size of the object when viewed at the
normal near point (without aid of the lens)
Microscope
This instrument has two lenses (1) The objective is near the object being viewed and
produces a greatly magnified real image (2) The eyepiece is a simple magnifier which
the viewer uses to closely examine the image from (1)
Telescope
This instrument also has two lenses (1) The objective at the front of the telescope
takes light from a distant object and produces a real inverted image (which is rather
small) near its focal point (2) This small real image is then examined by the eyepiece
functioning as a simple magnifier to produce a virtual image with a larger angular
size
20
Serway Chapter 37
Two-Slit Interference
If light is incident on two closely spaced narrow slits a pattern of light and dark
stripes is produced beyond the slits The bright stripes or fringes are caused by
constructive interference of the two waves coming from the slits Constructive
interference occurs whenever two waves arrive at a location in phase with each other
This occurs when the distance x1 from slit 1 to a point P on the screen and the
distance x2 from slit 2 to point P differ by in integral number of wavelengths
1 2 where 0 1 2x x m m
where λ is the wavelength of the light When x1 and x2 are much larger than the slit
spacing d this condition reduces to
sind m
where θ is the angle between the direction of the incident light and the direction of the
light arriving at the screen
Thin Films
When light is partially reflected and partially transmitted by a thin film of transparent
material it is possible to have interference between the wave reflected from the front
of the film and light reflected from the back of the film (The colored reflections from
the thin film of oil on the water in a rain-soaked parking lot are an example of this
effect) It is difficult to write down formulas that will work in all cases so we will
just review the important principles here
1 If the two reflected waves are in phase with each other the film has enhanced
reflection (constructive interference) but if the two reflected waves are out of phase
with each other reflection is diminished (destructive interference) Phase shifts occur
due to reflection and due to the extra path length through the film of the wave
reflected from the back of the film
2 The phase change due to reflection is determined by the difference in index of
refraction between the two media involved in the reflection If the wave is incident
21
from a medium with a low index of refraction into a medium with a high index of
refraction a phase change of 180deg occurs and the reflected wave is inverted If
incident from high to low no phase shift occurs and the reflected wave is non-
inverted
3 The extra path length through the film of the wave reflected from the back of the
film is equal to 2t where t is the thickness of the film (The incident light is assumed
to be normal to the surface of the film) The number of wavelengths contained in the
extra path length is equal to 2tλn where λn = λn is the wavelength of the light in the
film
4 Rules for reflection from thin films
If one ray is inverted and the other is not then we have
12 constructive
2 nt m
2 destructivent m
If either both rays are inverted or both are non-inverted we have
2 constructivent m
12 destructive
2 nt m
22
Serway Chapter 38
Diffraction Grating
A diffraction grating is simply a fancy version of two-slit interference with the two
slits replaced by thousands of slits Just as in the two-slit case the bright fringes
occur at angles given by
sin md
but in this case d the distance between neighboring slits is made to be very small
The effect of having many slits instead of two is to make each bright fringe highly
localized with wide dark regions between neighboring maxima
Single Slit Diffraction
When light passes through an opening in an opaque screen an interference pattern is
produced beyond the opening To understand why we may replace the single opening
by many small coherent sources of light These many sources interfere with each
other producing a pattern known as a diffraction pattern (Note that many authors do
not distinguish between interference and diffraction treating them as interchangeable
terms) If the opening is a slit of width a then the diffraction pattern far from the slit
will have a bright central maximum with a succession of minima and weaker maxima
on either side The angle between the incident direction and the minima is given by
sin ma
where m = plusmn1 plusmn2
If the opening is circular with diameter D the angle between the incident direction
and the first minimum is given by
sin 122D
Optical Resolution and Rayleighrsquos Criterion
Two point sources can just be resolved (distinguished from each other) if the peak of
the diffraction image of the first source overlies the first minimum of the diffraction
image of the second source For circular holes of the kind usually encountered in
23
optical devices this condition is approximately satisfied when the angular separation
between the two sources as viewed from the optical instrument is greater than or equal
to the critical angle
min 122D
where λ is the wavelength of the light and where D is the diameter of the aperture in
the instrument
Polarization
We say that an electromagnetic wave is polarized if its electric field vector doesnrsquot
change direction in a random fashion The simplest kind of polarization is linear
polarization in which the electric field vector oscillates back and forth along the
same axis in space Polarized light can be produced from normal unpolarized light by
selective absorption (as in Polaroid sunglasses) by passing light through crystals that
have different indices of refraction for different polarizations (double refraction) by
scattering (the blue sky is polarized) and by reflection (glare) Polarization by
reflection occurs when light reflects from a shiny insulating (non-metallic) surface
The amount of polarization is greatest for reflection at Brewsterrsquos angle
2
1
tan P
n
n
where θP is the incidence angle of light from medium 1 onto medium 2 and where n1
and n2 are the indices of refraction for the media (In our everyday experience n1 = 1
since the light comes in through the air and n2 is the index of refraction of the shiny
insulating material producing the glare eg water glass plastic paint etc)
Malusrsquos Law
The intensity of transmitted polarized light through a perfect polarizer is related to the
incident intensity of polarized light by Malusrsquos law
2cosoI I
where Io is the intensity of the incident light and where θ is the angle between the
electric field vector in the incident wave and the transmission axis the polarizer
24
Serway Chapter 39
Principles of Relativity
All of the weirdness of relativity flows from two simple principles
(1) The laws of physics must be the same in all inertial (non-accelerating) reference
frames
(2) The speed of light in vacuum has the same value in all inertial reference frames
Note that (1) seems reasonable but (2) is very odd It says that if two space ships are
approaching each other at nearly the speed of light and a laser pulse is shot from ship
1 toward ship 2 then when the people on ship 2 measure the speed of the pulse as it
goes by it is moving at 3 times 108 ms the same as if the ships were stationary
Simultaneity
If observer 1 sees two events in her own frame as simultaneous at two different
locations a moving observer 2 will see these two events happening at different times
Relativistic gamma
The factor γ (gamma) appears regularly in the formulas of relativity
2
2
1
1 vc
where v is the relative speed between two inertial frames
Time Dilation
If observer 1 sees two events at the same location in space separated by time Δtp in his
own frame then observer 2 moving at speed v relative to observer 1 will see these
two events separated by a longer time Δt
pt t
Moving clocks run slow
Length Contraction
If observer 1 measures the length of an object along the x-direction at some instant of
time in her frame to be Lp then observer 2 moving in the x-direction at speed v will
measure the length L of the object to be shorter
pLL
25
Moving meter sticks are short
Velocity Addition
If a particle is moving at velocity vac relative to frame c and if frame c is moving at
speed vcb with respect to frame b (with both velocities directed along the same line)
what would be the velocity of the particle in frame b In classical physics the answer is
ab ac cbv v v
but if any of these velocities are an appreciable fraction of the speed of light the
answer changes
21ac cb
abac cb
v Vv
v v c
In using this equation make sure that vac has the same sign in the numerator and in the
denominator and that vbc also has the same sign in both places Note that this equation
is different (and in our opinion easier to understand) than the equations in the text
Momentum and Energy
The momentum and total energy of a particle moving at velocity v (its speed is v v )
are given by
22
2 2
22
1 1 vvc c
m mcm E mc
vp v
Rest Energy
When a particle is at rest (v = 0 and γ = 1) then E = mc2 is called the rest energy It
includes the potential energy of the particle so that if we change the potential energy
of a particle we change its rest mass
Kinetic Energy The kinetic energy is the difference between the total energy and the
rest energy
2 2K mc mc
For v ltlt c this reduces to K = mv22
General Relativity
Einsteinrsquos principle of equivalence in the theory of general relativity is In the vicinity
of a point a gravitational field is equivalent to an accelerated frame of reference in
26
the absence of gravitational effects For example there is no difference whatever in
the physics experienced by an astronaut on earth and one who is in interstellar space
far away from any stars or planets but who has just turned on her spaceshiprsquos engines
and is accelerating at 98 ms2
27
Serway Chapter 40
Photons
We now know that even though light behaves like a wave it is better described as
consisting of small packets of energy called photons The energy of a photon is related
to its frequency by
E hf
where h is Planckrsquos constant
346626 10 J sh
Photons also carry momentum given by
hf hp
c
Since the amount of energy in each photon is fixed the difference between dim light
and intense light (of the same wavelength) is that dim light consists of fewer photons
than intense light
Photoelectric Effect
In the photoelectric effect light shining onto a metal surface gives the electrons in the
metal enough energy to escape and be detected It requires a certain amount of energy
(called the work function typically = 3 ndash 5 eV) just to kick the electron out of the
metal so the light must deliver at least this much energy to an electron to produce the
effect It is observed that red light no matter how intense never produces electrons
But ultraviolet light even if quite dim will eject electrons from the metal Since red
light consists of 2 eV photons and ultraviolet light has photons with energies around
4-6 eV the photon idea explains the behavior of the photoelectric effect The
maximum energy that an ejected electron can have is
maxK hf
where f is the photon frequency The minimum frequency that light can have and cause
electrons to be ejected is called the cutoff energy and is found by setting Kmax = 0
cf h
28
Compton Effect
When high-frequency light interacts with free electrons the scattered light does not
have the same wavelength as the incident light contrary to what classical
electromagnetism would predict But the photon picture in which photons have
momentum and energy allows us to treat scattering as a collision between the photon
and the electron In this collision the electron and the scattered photon both have a
different momentum and energy than they did before And since p = hλ for a photon
if the momentum is different the wavelength will also be different
after before 1 cose
h
m c
where the angle θ is the angle between the incoming photon direction and the
direction of the scattered photon
Particles are Waves
Since photons behave like particles sometimes it is not surprising that elementary
particles can behave like waves sometimes The wavelength of a particle with
momentum p is given by
h
p
Wave-particle Duality
Both photons and elementary particles have a dual nature sometimes they behave like
particles and sometimes they behave like waves It is hard for us to comprehend the
nature of such an object by making mental pictures but experiments definitively show
that this is the case Since we have no direct experience with photons and elementary
particles (because their energies are so much smaller than the energies of the everyday
objects we encounter) it is perhaps not surprising that we have a hard time forming a
mental picture of how they behave
Electron Interference
Since an electron is both a particle and a wave just like a photon it should be able to
produce an interference pattern This is observed If an electron beam is shot at two
closely-spaced slits and if the electrons that pass through the slits are detected
downstream it is found that there are some locations where electrons are never detected
29
and others where lots of electrons are detected The pattern is exactly the same as the
one observed for light waves provided that we use the electron wavelength λ = hp in
place of the wavelength of light This pattern is observed even though each electron is
detected as a single dot on the screen Only after many such dots are collected does the
pattern emerge And if we try to understand how this effect could possibly work by
looking closely at each slit to see which one the electron came through the pattern
disappears the act of measurement destroys the interference This means that each
single electron somehow comes through both slits (as a wave would)
Uncertainty Principle
In classical physics we always imagine that the positions and momenta of moving
particles have definite values It might be hard to measure them but surely at each
instant of time a particle should be precisely located at some point in space and have a
similarly precise momentum This turns out not to be true Instead both position and
momentum are required to be uncertain with their uncertainties Δx and Δp satisfying
the Heisenberg uncertainty relation
2x p
So if the particle were known to be precisely at some particular location (so that Δx =
0) we couldnrsquot know anything about its momentum (Δp = infin) and if its momentum
were exactly known we couldnrsquot know its position
There is a similar relation involving the particlersquos energy E and the time interval Δt
over which this energy is measured
2E t
This means that energy is not actually conserved if we are considering very small time
intervals and this brief non-conservation of energy has been observed
30
Serway Chapter 41
Wave Function ψ and Probability
Quantum mechanics does not predict exactly what an electron or a photon will do Instead
it specifies the wave function or probability amplitude ψ of an electron or a photon
This wave function is a complex-valued function of space and time whose squared
magnitude is the probability density P for finding a particle at a particular place in
space at a certain time
2P
where is the complex conjugate of ψ
For example the wave function of an electron with perfectly specified momentum p
would have a wavelength given by
h
p
and its probability amplitude would be proportional to
2i x ipxe e
The corresponding probability density would then be
21ipx ip ipxP e e e
which means that the electron is equally probable to be anywhere along the x axis
This is in accord with the uncertainty principle since we specified the momentum
precisely we canrsquot have any idea about the position of the electron
As another example you have probably seen ldquofuzzy ballrdquo drawings of electron
orbitals in a chemistry book These fuzzy balls are meant to indicate the distribution
of the probability density 2
P in the orbital
Particle in a Box
A simple example in which we can calculate the wave function is the case of a particle of
mass m constrained to be inside a 1-dimensional box between x = 0 and x = L In this
case the general wave function is a linear superposition of wave functions ψn of the form
sin niE tn
n xx t A e
L
31
where A is a positive constant where n = 1 2 3 and where the energy associated
with each of the quantum states ψn is given by
22
28n
hE n
mL
This wave function is zero at x = 0 and x = L which means that the particle will never
be found at the walls of the box The wave function has maximum values in the
interior and at these places the particle is most likely to be found and it also has
places where it is zero and at these places the particle will also never be found as
expressed by the formula
2sinn x
PL
The particle in the box is interfering with itself producing a probability interference
pattern across the box just like the interference patterns we studied with light and
sound
32
Serway Chapter 42
Atoms
Many people picture an atom as a miniature solar system where electrons orbit around
a massive nucleus at the center This picture is misleading because in atoms the wave
nature of electrons dominates The electrons form 3-dimensional standing waves
(called orbitals) centered at the nucleus And if you ask what is it thatrsquos waving the
answer is ψ the probability amplitude (see Chapter 41)
Standing waves on a string can occur only for certain frequencies (the fundamental
and higher harmonics) Similarly atomic orbitals occur only for certain energies For
the hydrogen atom the energies of the orbitals have the a particularly simple form
2
1136 eVnE
n
where n = 1 2 3 is called the principal quantum number For other atoms the
determination of the orbital energies requires numerical calculation by computers
Atomic Spectra
If an electron is somehow given extra extra energy (we say that it is excited) so that it
occupies a higher orbital it will eventually ldquofallrdquo back down to a lower orbital Each
time an electron falls to a lower orbital it loses the difference in energy between the
two orbitals in the form of a photon Since the orbital energies are discrete so are the
energy differences and so are the wavelengths of the emitted light The entire set of
these discrete wavelengths is called the atomic spectrum and it is unique to each type
of atom For hydrogen the spectrum can be simply written as
H 2 2
1 1 1
f i
Rn n
where RH = 10973732 times 107 m-1 For other atoms the spectrum cannot be expressed as
a formula
Orbital Angular Momentum Electrons in atoms also have quantized values of
angular momentum The orbital quantum number ℓ specifies the value of this
quantized angular momentum through the formula
1L
33
If we want to know the value of the angular momentum along some direction in space
say the z direction the answer is not L but rather
zL m
where mℓ is another quantum number which runs from
1 1m
This quantum number is important when an atom sits in a magnetic field
Spin Angular Momentum It has been found experimentally that electrons and other
charged particles also carry internal angular momentum which we call spin
Electrons have an intrinsic spin angular momentum s along a specified axis that is
extremely quantized it can only take on 2 possible values
1
2zs s
We thus say that electrons have spin s = 12 or that electrons are rdquospin one-half parti-
cles Since s is an angular momentum it obeys the same rule as that for orbital angular
momentum namely that if its value along some axis is s then its total magnitude is
given by
31
2S s s
I know this seems weird but quantum mechanics is weird The only excuse for this
bizarre way of looking at the world is that it predicts what happens in experiments
Exclusion Principle The answer to the question of how many electrons (or any other
spin one-half particle) can be in one particular quantum state was discovered by
Wolfgang Pauli and is called the exclusion principle
ldquoNo two electrons can ever be in the same quantum state therefore no two electrons
in the same atom can have the same set of quantum numbersrdquo
This is the reason that we have atoms with different properties instead of every atom
simple having all of its electrons in the ground state All of the variety we see around
us in the world is the result of chemical differences and these differences would not
exist unless electrons obeyed this important principle The entire structure of the
periodic table (see pages 1377-1379 in Serway) is an expression of this principle
34
Serway Chapter 44
Nuclear Properties
The nucleus is composed of protons with charge +e and mass mp = 1007226 u and of
neutrons with zero charge and mass mn = 1008665 u where u is one atomic mass unit
-271 u=1660540 10 kg
The atomic number Z counts the number of protons in a nucleus while the neutron
number N counts the number of neutrons The mass number A is the sum of the two
A N Z
Protons and nuetrons have about the same mass and some times called baryons (heavy
ones) or nucleons (partcles in the nucleus) A is sometimes called the baryon
number The chemical elements are distinguished by Z ie hydrogen has Z = 1 iron
has Z = 26 uranium has Z = 92 etc But for a given Z there might be several nuclei
with different numbers of neutrons These nuclear siblings are called isotopes
These numbers are used to label nuclei according to the pattern
5626 ie FeA
Z X
denotes the isotope of iron with N = 56 ndash 26 = 30 neutrons
The nucleus is roughly spherical with a radius given approximately by
1 3 150 0where 12 10 mr r A r
Nuclear Stability
Because the positively charged protons electrically repel each other with an enormous
force at distances as small as 10-15 m there must be some really strong force that that
overcomes electrical repulsion to hold protons and neutrons together This force is
called with some lack of imagination the strong force It is a very short-range force
(it only acts over a distance of about 2 x 10-15 m) and attracts protons to protons
neutrons to neutrons and neutrons to protons But in spite of this strong nuclear force
the coulomb repulsion of the protons is still present so anything that might keep the
protons from being right next to each other would help keep the nucleus from
35
exploding This role is played by the neutrons and for nuclei with Z le 20 the stable
nuclei roughly have N = Z
For Z gt 20 the coulomb repulsion force is more powerful and more neutrons are
needed to dilute it up to about N = 15Z around Z = 80 For Z gt 83 no amount of
neutrons can help and these nuclei are unstable (radioactive) There is an ldquoisland of
(relative) stabilityrdquo around Z=90-92 (thorium and uranium respectively) These
elements have one or two isotopes with half-lifes of billions of years so there are
substantial amounts of such elements on Earth This fact makes it possible to have
practical fission devices
Radioactive Decay
There are three types of radioactive decay
Alpha decay the nucleus kicks out a helium nucleus (N = 2 Z = 2 A = 4) So if the
original nucleus is called X and the new nucleus is called Y then the decay would look
like this
4 42 2X Y+ HeA A
Z Z
Beta decay the nucleus either kicks out an electron (endash) or its positively-charged
antimatter twin the positron (e+) plus either an electron neutrino v or an electron
anti-neutrino v
1X Y+e electron decayA AZ Z v
1X Y+e positron decayA AZ Z v
A neutrino is a particle with no charge hardly any mass (much less than the electron
mass) and interacts so weakly with matter that most neutrinos upon encountering the
planet earth just pass right through it as if it werenrsquot there
Gamma decay the nucleons in the nucleus X are in an excited energy state X
(perhaps as a result of having undergone alpha or beta decay) and they drop down to a
lower energy state shedding the energy as a high frequency photon
X XA AZ Z
36
This process is exactly analogous to the way that the electrons in atoms emit photons
Decay Rate and Half Life
There is no way to predict exactly when an unstable or excited nucleus will decay but
there is an average rate at which this decay occurs called the decay constant λ The
meaning of this constant is that if there are a large number N of nuclei in a sample
then the number of decays per second that will be observed (called the decay rate R)
is R = λN In mathematical language
dMR N
dt
This simple differential equation has for its solution
0tN t N e
where N0 is the number of nuclei in the sample at time t = 0
The half-life is the time it takes for half of the nuclei in the sample to decay and is
related to the decay constant by
1 2
ln 2 0693T
Disintegration Energy
When a nucleus decays it is making a transition to an overall state of lower energy
which means according to Einsteinrsquos famous formula E = mc2 that the sum of the
masses after the decay must be less than the mass before with the lost mass appearing
as kinetic energy among the decay products For example in alpha decay this kinetic
energy called the disintegration energy Q is given by
2X YQ M M M c
37
Serway Chapter 45
Nuclear energy
There are two ways to extract energy from the nucleus fission and fusion For nuclei
with Z greater than 26 breaking the nucleus apart into pieces leads to a lower
2mc energy than the original nucleus so energy can be extracted by fission For
nuclei with Z less than 26 a lower energy is achieved by combining nuclei so energy
can be extracted by fusion
Fission
Since neutrons have no charge they are not repelled from nuclei as protons are for
this reason their behavior is the key to understanding how fission works
Fast neutrons mostly bounce off other nuclei with each collision slowing the neutron
down This elastic energy loss is most effective if the other nuclei have low mass (like
hydrogen) and these materials are called moderators because of their ability to slow
down fast neutrons
The reason that slowing neutrons is important is that slow neutrons are much more
likely to be absorbed by a nucleus which then leads to nuclear reactions of various
kinds For a few very large nuclei like uranium-235 and some plutonium isotopes
absorption of a slow neutron causes the nucleus to split into two large fragments plus
2 or 3 fast neutrons (energy is released in the process too) Thus one slow neutron
can produce energy plus 2 or 3 more neutrons which if moderated from fast to slow
can split 2 or 3 more nuclei leading to a runaway chain reaction If the reaction is
allowed to proceed unchecked it produces a large explosion If a neutron absorbing
material is added to the mix (like the cadmium in reactor control rods) it is possible to
keep the reaction under control and to extract the released energy as heat to drive
steam turbines and produce electricity
Fusion
Fusion involves mashing two nuclei together and since they are both charged and repel
each other this reaction is much harder to make go The nuclei must have enough energy to
overcome the coulomb repulsion which is why this reaction requires a high temperature
(hundreds of millions of degrees K) like that in the sun (which is burning hydrogen to
helium via fusion) or like that in the center of the fission explosion that is used to detonate
38
a hydrogen bomb
This reaction is of interest for power production in spite of this difficult temperature
requirement because of the abundance of fusion fuel on the planet There are about
012 g of deuterium ( 21H ) in every gallon of water on earth and it only costs about 4
cents to extract it The fusion energy available from this minuscule amount of
deuterium would run a 1000-Megawatt power plant for 10 seconds Water is so
abundant on earth that if fusion were to work we would have an essentially
inexhaustible source of energy
So why donrsquot we have fusion power plants Well the fuel is cheap but the match is
incredibly expensive The only way we know to control this difficult high-temperature
reaction is with large and expensive pieces of equipment involving either large
magnetic fields and complex high-power electromagnetic antennas or with gigantic
(football-fieldsized) laser facilities involving more than a hundred of the highest-
energy lasers ever built Power plants based on these current methods for controlling
fusion are unattractive to the fiscally-minded people who run the electric power
industry Hopefully better designs will be discovered as experiments continue
39
c actinium 68 Er erbium 101 Md mendelevium 104 Rf rutherfordium l aluminum 63 Eu europium 80 Hg mercury 62 Sm samarium m americium 100 Fm fermium 42 Mo molybdenum 21 Sc scandium b antimony 9 F fluorine 60 Nd neodymium 106 Sg seaborgium r argon 87 Fr francium 10 Ne neon 34 Se selenium s arsenic 64 Gd gadolinium 93 Np neptunium 14 Si silicon t astatine 31 Ga gallium 28 Ni nickel 47 Ag silver a barium 32 Ge germanium 41 Nb niobium 11 Na sodium k berkelium 79 Au gold 7 N nitrogen 38 Sr strontium e beryllium 72 Hf hafnium 102 No nobelium 16 S sulfur i bismuth 108 Hs hassium 76 Os osmium 73 Ta tantalum h bohrium 2 He helium 8 O oxygen 43 Tc technetium
boron 67 Ho holmium 46 Pd palladium 52 Te tellurium r bromine 1 H hydrogen 15 P phosphorus 65 Tb terbium d cadmium 49 In indium 78 Pt platinum 81 Tl thallium a calcium 53 I iodine 94 Pu plutonium 90 Th thorium f californium 77 Ir iridium 84 Po polonium 69 Tm thulium
carbon 26 Fe iron 19 K potassium 50 Sn tin e cerium 36 Kr krypton 59 Pr praseodymium 22 Ti titanium s cesium 57 La lanthanum 61 Pm promethium 74 W tungsten l chlorine 103 Lr lawrencium 91 Pa protactinium 92 U uranium r chromium 82 Pb lead 88 Ra radium 23 V vanadium o cobalt 3 Li lithium 86 Rn radon 54 Xe xenon u copper 71 Lu lutetium 75 Re rhenium 70 Yb ytterbium m curium 12 Mg magnesium 45 Rh rhodium 39 Y yttrium b dubnium 25 Mn manganese 37 Rb rubidium 30 Zn zinc y dysprosium 109 Mt meitnerium 44 Ru ruthenium 40 Zr zirconium s einsteinium
40
UNITS The SI units are given in parentheses Other commonly used units are given in terms of the SI units Symbols conform with the recommendations of the American National Standards Institute (ANSI) and the American Institute of Physics (AIP) Commonly used multiples of SI units
prefix symbol factor tera T 1012 giga G 109
mega M 106 kilo k 103
centi c 10-2 milli m 10-3
micro μ 10-6 nano n 10-9 pico p 10-12
femto f 10-15
length meter (m) angstrom 1 Ǻ = 10-10 m inch 1 in = 254 x 10-2 m foot 1 ft = 03048 m mile 1 mi = 1609 m
mass kilogram (kg) atomic mass unit 1 u = 1661 x 10-27 kgslug 1 slug = 1459 kg
force newton (N = kg bull ms2) dyne 1 dyn = 10-5 N pound 1 lb = 4448 N
pressure Pascal (Pa = kgm bull s2) atmosphere 1 atm = 1013 x 105 Pa poundssquare inch 1 psi = 6895 Pa cm of mercury 1 cm Hg = 1333 Pa bar 1 bar = 1000 x 105 Pa torr 1 torr = 1333 Pa
time second (s) minute 1 min = 60 s hour 1 h =3600 s
frequency hertz (Hz = s-1) radianssecond 1 rads = 12π Hz
energy joule (J = kg bull m2s2) erg 1 erg = 10-7 J electron volt 1 eV = 1602 x 10-19 J calorie 1 cal = 4187 J kilowatt-hour 1 kWh = 36 x 106 J British thermal unit 1 Btu = 1055 J
power watt (W = kg bull m2s3) horsepower 1 hp = 7457 W
charge coulomb (C = A bull s)
electric potential volt (V = kg bull m2 s3 bull A)
current ampere (A)
resistance ohm (Ω = kg bull m2s3 bull A2)
capacitance farad (F = s4 bull A2kg bull m2)
magnetic field tesla (T = kgs2 bull A) gauss 1 G = 10-4 T
magnetic flux weber (Wb = kg m2s2 bull A) maxwell 1 Mx = 10-8 Wb
magnetic inductance henry (H = kg bull m2s2 bullA2)
temperature kelvin (K) degrees Celsius 0degC = 27315 K
angle radian (rad) degree 1 deg = π180 rad revolution 1 rev = 2π rad
41
Some Physical Constantsa Quantity Symbol Valueb Atomic mass unit μ 1660 538 73 (13) x 10-27 kg
931494 013 (37) MeVc2 Avogadros number NA 6022 141 99 (47) x 1023 particlesmol Avogadrorsquos number 6022 x 1023 mol Bohr magneton
2B e
em 9274 008 99 (37) x 10-24 JT
Bohr magneton μB 927 x 10-24 JT Bohr radius
0
2
2e em e k
a 5291 772 083 (19) x 10-11 m
Boltzmanns constant B A
RNk 1380 650 3 (24) x 10-23 JK
Boltzmannrsquos constant kB 1380 x 10-23 JK Compton wavelength
C eh
m c 2426 310 215 (18) x 10-12 m
Deuteron mass md 3343 583 09 (26) x 10-27 kg 2013 553 212 71 (35) u
electron charge e 1602 x 10-19 C Electron mass me 9109 381 88 (72) x 10-31 kg
5485 799 110 (12) x 10-4 u 0510 998 902 (21) MeVc2
electron mass 911 x 10-31 kg Electron volt eV 1602 176 462 (63) x 10-19 J Elementary charge e 1602 176 462 (63) x 10-19 C Gas constant R 8314 472 (15) JKmol Gravitational constant G 6673 (10) x 10-11 Nm2kg2 Neutron mass mn 1674 927 16 (13) x 10-27 kg
1008 664 915 78 (55) u 939565 330 (38) MeVc2
neutron mass 1675 x 10-27 kg Nuclear magneton
2n p
em 5050 783 17 (20) x 10-27 JT
permeability constant μ0 1257 x 10-6 Hm permittivity constant ε0 8854 x 10-12 Fm Plancks constant h
2h
6626 068 76 (52) x 10-34 Js 1054 571 596 (82) x 10-34 Js
Planckrsquos constant h ħ
6626 x 10-34 Js 1055 x 10-34 Js
Proton mass mp 1672621 58 (13) x 10-27 kg 1007 276 466 88 (13) u 938271 998 (38) MeVc2
proton mass 1673 x 10-27 kg Rydberg constant RH 1097 373 156 854 9 (83) x 107 m-1 Speed of light in vacuum c 2997 92458 x 108 ms (exact) speed of light c 300 x 108 ms a
These constants are the values recommended in 1998 by CODATA based on a least-squares adjustment of data from different measurements For a more complete list see P J Mohr and B N Taylor Rev Mod Phys 72351 2000 b The numbers in parentheses for the values above represent the uncertainties of the last two digits
NOTE The ones in red are the ones from Appendix 13 The ones in yellow are the ones that are from the Serway book table but that were already on the Appendix 13 table
42
INDEX
Absolute zero 9
Adiabatic 17
Adiabatic exponent 18
Adiabatic process 17
Alpha decay 46
Amplitude 4
Angle of reflection 25
Angular frequency 3
Angular magnification 30
Angular momentum atomic 43
Angular size 29
Archimedes Principle 1
Atomic spectra 43
Atoms 43
Avogadrorsquos number 10
Beats 8
Bernoullirsquos Equation 2
Beta decay 46
Bifocals 29
Brewsterrsquos angle 34
British Thermal Unit Btu 11
Buoyancy 2
calorie 11
Calorie 11
Camera 28
Carnot cycle 21
Carnot efficiency 22
Celsius scale 9
Ciliary muscle 29
Coefficient of performance 21
Compton effect 39
Constructive interference 7
Continuity equation of 2
Contraction length 35
Convection 14
Curved mirrors 27
Decay constant 46
Decay rate radioactive 46
Decibel scale 5
Degrees of freedom 16
Density 1
Destructive interference 7
Diffraction single slit 33
Diffraction grating 33
Dilation time 35
Disintegration energy 47
Dispersion of light 26
Doppler effect 6
Double slit interference 31
Efficiency 20
Electron interference 39
Emissivity 15
Energy nuclear decay 47
Energy relativistic 36
43
Engineering work 20
Entropy 22
Entropy ideal gas 24
Equipartition of energy 18
Equivalence principle 36
Exclusion principle 44
Expansion thermal 9
Expansion coefficient linear 9
Eye 29
Fahrenheit scale 9
Far point 29
Farsightedness 29
First Law of Thermodynamics 12
Fission 48
Flux volume 2
Free expansion 23
Fringes 31
Fusion 48
Fusion heat of 11
Gamma relativistic 35
Gamma decay 46
General relativity 36
Half-life 46
Heat 11
Heat capacity 11
Heat conduction 13
Heat engine 20
Heat of fusion 11
Heat of vaporization 11
Heat pump 20
Heats of transformation 11
Hydrostatics 1
Ideal Gas Law 10
Images real and virtual 27
Intensity sound 5
Interference 7
Interference two-slit 31
Internal energy 12
Internal energy degrees of freedom 16
Irreversible process 19 23
Isotopes 45
Joule 11
Kelvin scale 9
Kinetic energy relativity 36
Kinetic theory 16
Latent heat 11
Length contraction 35
Linear expansion coefficient 9
Linear polarization 34
Linear superposition 7
Longitudinal wave 3
Loudness 5
Magnification lateral 27
Magnifying glass 30
Malusrsquos law 34
Microscope 30
Momentum relativistic 36
Muscle ciliary 29
44
Musical instruments 8
Musical scale 8
Near point 29
Nearsightedness 29
Nonlinear 7
Nuclear energy 48
Nuclear properties 45
Nuclear stability 45
Octave 8
Optical resolution 33
Orbital quantum number ℓ 43
Orbitals 43
Particle in a Box 41
Particles are waves 39
Pascalrsquos Principle 1
Period 3
Photoelectric effect 38
Photons 38
Pitch 8
Polarization 34
Power sound 5
Presbyopia 29
Pressure 1
Principal quantum number n 43
Principle of equivalence 36
Principle of linear superposition 7
Probability amplitude ψ 41
Processes thermodynamic 13
R-value 14
Radiation thermal 14
Radioactive decay 46
Radioactive decay rate 46
Ray tracing 27
Rayleighrsquos criterion 33
Reading glasses 29
Real image 27
Refraction 25
Refrigerator 20
Relativistic gamma 35
Relativity principles 35
Resolved for light sources 33
Rest energy 36
Reversible process 19 22
Rope wave speed 4
Second Law of Thermodynamics 19
Shock waves 6
Simple magnifier 30
Simultaneity 35
Single slit diffraction 33
Snellrsquos law 25
Sound speed 5
Specific heat 11
Spin Angular momentum 44
Standing waves 7
Stefanrsquos law 14
Strong force 45
Telescope 30
Temperature 9
45
Temperature Scales 9
Thermal conductivity 14
Thermal energy 12
Thermal expansion 9
Thin film interference 31
Thin lenses 28
Time dilation 35
Tone musical 8
Total internal reflection 25
Transverse wave 3
Traveling Waves 3
Two-slit interference 31
Uncertainty principle 40
Vaporization heat of 11
Velocity addition relativity 36
Virtual image 27
Volume flux 2
Wave function 41
Wave Function ψ and Probability 41
Wave speed 3
Wave-particle duality 39
Wavelength 3
Wavenumber 3
Wien Displacement Law 16
Work 11
1
Path A-B B-C C-D D-A A to A net
Q
W
ΔU
ΔS
Internal Energy
The energy stored in a substance is called its internal energy Eint This energy may be
stored as random kinetic energy or as potential energy in each molecule (stretched
chemical bonds electrons in excited states etc) For ideal gases all states with the
same temperature will have the same Eint
First Law of Thermodynamics
The change ΔEint in the internal energy of a system is given by
intE Q W
where Q is the heat absorbed by the system and where W is the work done on the
system Hence if a system absorbs heat (and if Wge0) the internal energy increases
Likewise if the system does work (W on the system is negative) and if Qge0 the
internal energy decreases Potential Pitfall Many times people talk about work done
by the system It is the minus of W on the system Donrsquot get tripped up
Processes
Adiabatic process No heat is exchanged Q = 0 requiring that the system be insulated
from the environment A process may be approximately adiabatic if it happens so
rapidly that heat does not have time to enter or leave the system Work + or ndash is done
and ΔEint = W
Adiabatic free expansion A gas is allowed to expand into a vacuum without pushing
on anythingndashit just rushes into the vacuum In this process Q = W = 0 so the internal
energy and temperature does not change (Note the difference between an adiabatic
process and a free expansion is that NO work is done in the adiabatic free expansion)
Isobaric process The pressure is held fixed ΔP = 0 For example usually the
pressure increases when a gas is heated but if it were allowed to expand during the
2
heating process in just the right way its pressure could remain fixed In isobaric
processes both Q and W are non-zero (Work is eacy = ndashP(Vf ndash Vi)
Isovolumetric process The volume is held fixed ΔV = 0 The work done by the gas is
then zero and so we have ΔEint = Q
Isothermal process The temperature is held fixed ΔT = 0 There is no change in Eint
so Q = ndashW for isothermal processes For an ideal gas PV = nRT so PV = constant in
an isothermal process The work done on the gas is then given by
lnf
i
Vi
Vf
VW PdV nRT
V
Heat Conduction
The quantity P is defined to be the rate at which heat flows through an object and is a
power having units of watts It is analogous to electric current which is the rate at
which charge flows through an object If the flow of heat through a slab of length L
and cross-sectional area A is steady in time then P is given by the equation
h cT TdQkA
dt L
P =
where Th and Tc are the (hot and cold) temperatures of the two ends of the slab The
heat flows of course because of this temperature difference The quantity k is called
the thermal conductivity and is a constant that is characteristic of the material It is
analogous to the electrical conductivity h cT TL is sometimes called the temperature
gradient and is written dTd dTdx
R-Values
It is common to have the heat-conducting properties of materials described by their R-
values especially for insulating materials like fiberglass batting The connection
between k and R is R = Lk where L is the material thickness In this country R-values
always have units of 2ft F hourBtu
Polyurethane foam 6 (1 in thick) Air space 1 (35 in thick) Fiber glass batting 11
(35 in thick) Wood about 1 (1 in thick) Glass 09 (0125 in thick)
The heat flow rate through a slab of area A is given by
3
h cT TA
R
P
in units of Btuhour Note that A must be in square feet and the temperatures must be
in degrees Fahrenheit
Convection
Convection is the transfer of thermal energy by flow of material For instance a home
furnace doesnrsquot heat a house by waiting for the heat from the burner to slowly conduct
throughout the house instead it quickly pumps warm air to all of the rooms
Generally convection is a much faster way to transfer heat than conduction
Radiation
Electromagnetic radiation can also transfer heat When you warm yourself near a
campfire which has burned itself down into a bed of glowing embers you are
receiving radiant heat from the infrared portion of the electromagnetic spectrum The
rate at which an object emits radiant heat is given by Stefanrsquos law
4AeTP
where P is the radiated power in watts σ is a constant
8 2 45696 10 W m K
A is the surface area of the object in m2 and T is the temperature in kelvins The
constant e is called the emissivity and it varies from substance to substance A perfect
absorber (think black velvet) has e = 1 while a perfect reflector (think mirror) has e =
0 Hence black objects radiate very well while shiny ones do not Also an object that
is hotter than its surroundings radiates more energy than it absorbs whereas an object
that is cooler than its surroundings absorbs more energy than it radiates
Terminology
Transfer variables vs state variables
Energy transfer by heat as well as work done depends on the initial final and
intermediate states of the system They are transfer variables But their sum (Q + W =
4
Eint) is a state variable
Figure 205
5
Serway Chapter 21
Kinetic Theory
The ideal gas law works for all atoms and molecules at low pressure It is rather
amazing that it does Kinetic theory explains why The properties of an ideal gas can
be understood by thinking of it as N rapidly moving particles of mass m As these
particles collide with the container walls momentum is imparted to the walls which
we call the force of gas pressure In this picture the pressure is related to the average
of the square of the particle velocity 2v by
22 1( )
3 2
NP mv
V
Using the ideal gas law we obtain the average translational kinetic energy per
molecule
21 3
2 2 Bmv k T
The rms speed is then given by
2rms
3 3Bk T RTv v
m M
where M is the molecular mass in kgmol
Degrees of Freedom
Roughly speaking a degree of freedom is a way in which a molecule can store energy
For instance since there are three different directions in space along which a molecule
can move there are three degrees of freedom for the translational kinetic energy
There are also three different axes of rotation about which a polyatomic molecule can
spin so we say there are three degrees of freedom for the rotational kinetic energy
There are even degrees of freedom associated with the various ways in which a
molecule can vibrate and with the different energy levels in which the electrons of
the molecule can exist
Internal Energy and Degrees of Freedom The internal energy of an ideal gas made
up of molecules with J degrees of freedom is given by
int 2 2 B
J JE nRT Nk T
6
Heat Capacities of an Ideal Gas The heat capacity of a gas is described by means of
molar heat capacities CV and CP These are the heat capacities per mole and the
subscript V on CV means that the volume is being held constant while for CP the
pressure is held constant For example to raise the temperature of n moles of a gas
whose pressure is held constant by 10 K we would have to supply an amount of heat Q
= nCP (10) K
Molar Specific Heat of an Ideal Gas at Constant Volume
VQ nC T
3monatomic
2VC R
5diatomic
2VC R
5polyatomic
2VC
Real gases deviate from these formulas because in addition to the translational and ro-
tational degrees of freedom they also have vibrational and electronic degrees of
freedom These are unimportant at low temperatures due to quantum mechanical
effects but become increasingly important at higher temperatures The rough rule is
No of degrees of freedom
2VC R
Molar Specific Heat of an Ideal Gas at Constant Pressure
PQ nC T
P VC C R
The internal energy of an ideal gas depends only on the temperature
int VE nC T
Adiabatic Processes in an Ideal Gas
7
An adiabatic process is one in which no heat is exchanged between the system and the
environment When an ideal gas expands or contracts adiabatically not only does its
pressure change as expected from the ideal gas law but its temperature changes as
well Under these conditions the final pressure Pf can be computed from the initial
pressure Pi and from the final and initial volumes Vf and Vi by
or constantf f i iP V PV PV
where γ = CPCV The quantity γ is called the adiabatic exponent Note that this
doesnrsquot mean that the ideal gas law no longer holds it does and in fact it can be
combined with the adiabatic law for pressure given above to obtain the adiabatic law
for temperatures
1 constantTV
Compressions in sound waves are adiabatic because they happen too rapidly for any
appreciable amount of heat to flow This is why the adiabatic exponent appears in the
formula for the speed of sound in an ideal gas
RTv
M
Note that v depends only on T and not on P Because it depends only on the
temperature the speed of sound is the same in Provo as at sea level in spite of the
lower pressure here due to the difference in elevation
Equipartition of Energy
Every kind of molecule has a certain number of degrees of freedom which are
independent ways in which it can store energy Each such degree of freedom has
associated with it ndash on average ndash an energy of 12 Bk T per molecule (or 1
2 RT per mole)
(Note since a molecule has so many possible degrees of freedom it would seem that
there should be a lot of 12 sBk T to spread around But because energy is quantized
some of these degrees of freedom are not ldquoactiverdquo until the temperature becomes high
enough that 12 Bk T is as big as the lowest quantum of energy
8
Serway Chapter 22
Second Law of Thermodynamics
There are several equivalent forms of this important law
Kelvin It is not possible to change heat completely into work with no other change
taking place Or in other words there are no perfect heat engines
Clausius It is not possible for heat to flow from one body to another body at a higher
temperature with no other change taking place Or in other words there are no
perfect refrigerators
Entropy In any thermodynamic process that proceeds from one equilibrium state to
another the entropy of the system + environment either remains unchanged or
increases The total entropy never decreases This law is a bit of an oddity among the
laws of physics because it is not absolute Things are forbidden by the second law not
because it is impossible for them to happen but because it is extremely unlikely for
them to happen (See below for more information about entropy)
Reversible and Irreversible Processes
A reversible process is one which occurs so slowly that it is in thermal equilibrium (or
very nearly so) at all times A hallmark of such processes is that a motion picture of
them looks perfectly normal whether run forward or backward Imagine for instance
the slow expansion of a gas at constant temperature in a cylinder whose volume is being
increased by a slowly moving piston Run the movie backwards and what do you see
You see the slow compression of a gas at constant temperature which looks perfectly
normal
An irreversible process is one which occurs in such a way that thermal equilibrium is
not maintained throughout the process The mark of this kind of process is that a motion
picture of it looks very odd when run backward Imagine the sudden expansion of a gas
into a previously evacuated chamber because a hole was punched in the wall between a
pressurized chamber and the evacuated one Run the movie backward and what do you
see You see the gas in the soon-to-be-evacuated chamber gather itself together and
stream through a tiny hole into a chamber in which there is already plenty of gas If you
have ever seen this happen get in touch with the support group for those who have
witnessed the spontaneous combustion of large mammals by calling 1-800-PYROCOW
9
Heat Engines
Because of the vexing difference in sign between work done on and system and work done
by a system we will invent a new work variable Weng Heat engines do work and so the net
W for these engines is negative But in engineering applications hidden minus signs are
regarded as evil so for heat engines we donrsquot talk about W instead we talk about its
magnitude engW W So for heat engines the first law is
int engE Q W
But for heat pumps and refrigerators work is done on the system so we use the usual work
W when we talk about these systems
A heat engine is a machine that absorbs heat converts part of it to work and exhausts the
rest The heat must be absorbed at high temperature and exhausted at low temperature If the
absorbed heat is Qh the exhausted heat is Qc and the work done by the engine is Weng then
eng h cW Q Q
and the efficiency of the engine is defined to be
eng
h
We
Q
A perfect engine would convert the heat hQ completely into work Weng giving an effi-
ciency of e = 1 Energy conservation alone allows a perfect engine but the second law
requires e lt 1
Refrigerators and Heat Pumps
A refrigerator is a machine that absorbs heat at low temperature and exhausts it at high
temperature the ldquobackwardsrdquo heat transfer being driven by the work done on the machine
by some source of power A heat pump is a machine that either works like a refrigerator
keeping a place cold by transferring heat from this cold place to a higher temperature
environment (cooling mode like an air conditioner) or it functions as a heater
transferring heat into a warm place from a cooler one (heating mode like a window unit
that heats a house by extracting thermal energy from the cold outdoors) The coefficient
of performance of a refrigerator or of a heat pump in cooling mode is defined to be
COP cooling modecQ
W
10
For a heat pump in heating mode the coefficient of performance is
COP heating modehQ
W
Note that we donrsquot have to use the engineering work here because in these systems
work is done on the system and W is naturally positive
A perfect refrigerator would take in heat Qc and exhaust the same amount of heat without
doing any work giving an infinite coefficient of performance Energy conservation alone
allows a perfect refrigerator but the second law requires COP lt infin A perfect heat pump
in heating mode would transfer Qh into the house without doing any work and so would
also have an infinite coefficient of performance The second law forbids this too A good
coefficient of performance for a real device would be around 5 or 6
Carnot Cycle
The most efficient of all possible engines is one that uses the Carnot cycle This cycle
employs an ideal gas has no friction and operates very slowly so that the gas can be in
thermal equilibrium at all parts of the cycle This means of course that it canrsquot
possibly be built and even if it could be built it would not run fast enough to be useful
Nevertheless this cycle is very important because it gives an upper bound on the
efficiency of real engines There cannot possibly be an engine that is more efficient
than one based on the Carnot cycle This cycle consists of the following four steps
1 The ideal gas absorbs heat Qh at constant temperature Th while the gas increases its
volume The reason that heat is absorbed is that expansion tends to cool the gas but
thermal contact with the environment at Th keeps the temperature high by heat
conduction into the ideal gas
2 The ideal gas further increases its volume by an adiabatic expansion This expansion
causes the gas to cool so at the end of this part of the cycle the gas is at temperature Tc
3 The gas exhausts heat Qc at constant temperature Tc while the gas decreases in
volume The reason that heat is exhausted is that compression tends to heat the gas
but thermal contact with the environment at Tc keeps the temperature low by heat
conduction out of the ideal gas
4 The gas is adiabatically compressed back to its original volume (the volume it started
with in step 1) This compression heats the gas from Tc up to Th
11
The efficiency of a Carnot engine is given by the very simple formula
1 cC
h
Te
T
where the temperatures must in be Kelvin No real engine can be more efficient than
this
The coefficient of performance of a Carnot refrigerator or heat pump in cooling mode
is given by
COP cooling mode cC
h c
T
T T
and no real refrigerator can have a coefficient of performance greater than this
The coefficient of performance of a Carnot heat pump in heating mode is
COP heating mode hC
h c
T
T T
So why donrsquot we just use these wonderful Carnot engines and have perfect efficiency
To make the reversible steps in the cycle really reversible they would have to occur
infinitely slowly So the price you pay for making a perfect engine is that it takes
forever to get it to do any work
Entropy
The entropy of a system is defined in terms of its molecular makeup and measures
roughly the disorder of the system If the system is packed into a very small volume
then it is quite ordered and the entropy will be low If it occupies a large volume the
entropy is high (To see what this has to do with disorder note that socks in a drawer
occupy a small volume while socks on the bed in the corner by the door and
hanging from the chandelier occupy a large volume) If the system is very cold then
the molecules hardly move and may even reach out to each other and form a crystal
This is a highly ordered state and therefore has low entropy If the system is very hot
with rapidly speeding molecules crashing into the container walls and bouncing off
each other things are disordered and the entropy is high
It is possible to calculate the entropy of a system in terms of its macroscopic thermody-
namic properties ie pressure volume temperature number of moles etc The key to this
calculation is the concept of a reversible process A reversible process is one that is
12
carried out without friction and so slowly that the process can be reversed at any stage by
making an infinitesimal change in the environment of the system The slow expansion of the
gas at Th in the Carnot cycle with tiny amounts of heat being transferred to the environment
is reversible If we turned around and began slowly to compress the gas would just slowly
exhaust heat to the environment in the exact reverse way that it absorbed it during expansion
Most processes however are irreversible For example if a gas-filled box were suddenly
increased in size so that the particles were free to wander into the void created by the sudden
expansion then the gas would eventually fill the new volume uniformly at the same
temperature as before the expansion (The temperature is unchanged in this imaginary
process because the kinetic energy of the molecules would be unaffected by such an
instantaneous expansion of the container walls) This imaginary but highly thought-
stimulating process is called a free expansion and it is impossible to reverse it During the
expansion we didnrsquot push on any of the molecules so reversing this process would mean
making them go back into their original volume without pushing on them they simply will
not cooperate to this extent Another way to see that just pushing them back where they came
from does not reverse the free expansion is to think about what would happen if we just
compressed either adiabatically or isothermally An adiabatic compression back to the
original volume would heat the gas above its original temperature and an isothermal
compression would require that heat be exhausted to the environment But the free expansion
involved neither temperature changes nor heat exchanges so neither of these two processes
is the reverse of the free expansion It is simply impossible to reverse this rapid expansion in
a way that takes us back to the initial state
It is possible to calculate the change of entropy for both reversible and irreversible processes
Letrsquos consider a reversible process first In a reversible process the entropy change is given
by the formula
dQS
T
where dQ is the amount of heat added to the system during a small step of the process
The total energy change during the process may then simply be calculated by integration
f f
f i i i
dQS S S dS
T
13
(Just as in the case of energy we are mainly interested in differences rather than in
absolute magnitudes)
This integration method does not work for irreversible processes For instance in the
rapid free expansion discussed above no heat is added to the system but its disorder
obviously goes up We need to find some other way to calculate the entropy The key
is the fact that the entropy of a system depends only on its current state and not at all
on how it arrived there So to calculate the entropy change in an irreversible process
first find out what the initial and final conditions of the process are Then invent a
reversible process that takes the system from the initial state to the final state Since
the entropy depends only on the state of system and not on the process the entropy
change for the reversible process is the same as that for the irreversible process
Entropy is thus like pressure or temperature if the pressure changes from Pl to to P2
then the pressure difference is P2 - Pl regardless of how the pressure change was
made This seems obvious for pressure but not for entropy only because you arent
comfortable with entropy yet
Entropy of an Ideal Gas
For n moles of an ideal gas the difference in entropy between a state with temperature
T and volume V and some standard state with temperature To and volume Vo is given
by
ln lno o Vo o
T VS T V S T V nC nR
T V
Entropy in a Phase Change
Calculating entropy change in a phase change like melting or boiling is easy It is
QT Where Q is in the latent heat for example the flows in to cause the melting and
T is the temperature of the phase change There is NO integral to do
14
Serway Chapter 35
Angle of Reflection
If plane waves are incident on a reflecting surface with the propagation direction of
the waves making angle θ1 with the normal direction to the surface then the reflected
angle θ1prime relative to the surface normal is simply
1 1
ie the incident angles and reflected angles are the same
Refraction
If plane waves traveling through medium 1 are incident on a plane interface between
medium 1 and medium 2 then the angle of incidence of the incoming wave θ1 and
the angle of refraction of the transmitted wave θ2 are related by Snellrsquos law
2 2
1 1
sin
sin
v
v
where v1 and v2 are the wave speeds in medium 1 and medium 2 The angle of
incidence and the angle of refraction are both measured between the wave propagation
direction and the normal to the interface In terms of indices of refraction in the case
of light waves Snellrsquos law takes the more familiar form
1 1 2 2sin sinn n
where ni = cvi
Total Internal Reflection
If a wave is incident from a medium of low wave speed into a medium of high wave
speed the law of refraction requires that the angle of refraction be greater than the
angle of incidence If the angle of refraction is required to be greater than 90deg then no
refracted wave can exist and total internal reflection occurs The critical incident
angle θc beyond which total internal reflection occurs is given by
2
1
sin c
n
n
15
Dispersion of Light
In addition to the speed of light varying from material to material it also varies with
wavelength within each material This means that the index of refraction is generally a
function of wavelength
c
nv
Since the wave speed is not constant such a medium is dispersive meaning in this
context that refraction actually disperses white light into its various colors because
Snellrsquos law gives a different angle for each wavelength In most materials the
variation with wavelength is quite small but this small effect is responsible for some
of the most spectacular color effects we ever see including rainbows a flashing
crystal chandelier and the colored fire of a diamond solitaire by candlelight
16
Serway Chapter 36
Real and Virtual Images
When light rays are focused at a certain plane producing an image if a sheet of white
paper is placed there we call the image a real image The images produced by film
projectors and overhead projectors are examples of real images
When light rays appear to come from a certain location but no image is produced
when a screen is placed there we say that there is a virtual image at that location For
instance when you look in a mirror it appears that someone is behind the mirror but
a screen placed back there in the dark would show nothing Your image in the mirror
is a virtual image
Ray Tracing
There are lots of rules about how to find the images in optical systems but the best
way to keep things straight is to learn how to draw the principal rays for curved
mirrors and lenses The rays for convex and concave mirrors are shown in Fig 3615
The rays for converging and diverging thin lenses are shown in Fig 3627 You should
memorize the principal rays and know how to use them to locate images
Curved Mirrors
The focal length of a curved mirror with radius of curvature R is given by
2
Rf
The relation between object distance p image distance q and focal length f is
1 1 1
p q f
If the mirror is a diverging mirror f should be negative and if q should turn out to be
negative the image is virtual
Lateral Magnification
The lateral magnification in an optical system is defined by the ratio of the image size
to the object size
17
Image height
Object height
qM
p
for curved mirrors
As usual there are sign conventions here too but this formula just gives the
magnitude It is better to keep keep track of upright images versus inverted images by
means of ray diagrams rather than by memorizing sign conventions
Thin Lenses
The focal length of a thin lens is related to the radii of curvature of the two faces R1
and R2 of the lens by
1 2
1 1 11n
f R R
Note that this formula differs from Eq (3611) in the text by not having a minus sign
between the two R-terms We like this form better because for a simple converging
lens like a magnifying glass we just use positive values of R for both surfaces If one
of the faces is concave producing divergence use a negative value for R And if a
surface is flat use R = infin If the face is flat the radius is infinite
The relation between the image object and focal distances for a thin lens is the same
as that for a curved mirror
1 1 1
q p f
Use a negative focal length if the lens is diverging
The lateral magnification for a thin lens is the same as for a curved mirror
Image height
Object height
qM
p
Camera
The lens system in a camera projects a real image of an object onto the film (or CCD
array in a digital camera) The position of the image is adjusted to be on the film by
moving the lens into or out of the camera
18
Eye
The eye is like a camera in that a real image is formed on the retina Unlike a camera
the image position is adjusted by changing the focal length of the lens This is done by
the ciliary muscle which squeezes the lens changing its shape
Near Point The near point is the closest distance from the eye for which the lens can
focus an image on the retina It is usually 18-25 cm for young persons
Far Point The far point is the greatest distance from the eye for which the lens can
focus an image on the retina For a person with normal vision the far point is at
infinity
Nearsightedness A person is nearsighted if their far point is at some finite distance
less than infinity This condition can be corrected with a lens that takes an object at
infinity and produces a virtual image at the personrsquos far point
Farsightedness and Presbyopia A person is farsighted if their near point is too far
away for comfortable near work like reading or knitting This can be corrected by a
lens which takes an object at a normal near point distance of 18-25 cm and produces a
virtual image at the personrsquos natural near point Presbyopia involves a similar
problem which nearly all people experience as they age The ciliary muscle becomes
too weak and the lens becomes too stiff to allow the eye to provide for both near and
far vision The solution for this problem is either reading glasses or bifocal lenses
Reading classes are just weak magnifying glasses mounted on eyeglass frames
Bifocal lenses are split into upper and lower halves The lower half is a lens which
gives the proper correction for near work and the upper half is a different lens for
proper focusing at infinity
Angular Size
When an object is brought closer to the eye it appears to be larger because the image
on the retina is larger The size of this image is directly proportional to the objectrsquos
angular size which is the angle subtended by the object measured from the center of
the lens of the eye In optical instruments which are to be used with the eye the
angular size of the final image is whatrsquos important because it determines how large
the image will appear to the viewer
19
Simple Magnifier
A simple magnifier is a single converging lens or magnifying glass It takes an object
closer to the eye than a normal near point and produces a virtual image at or beyond
this near point The angular magnification is defined to be the ratio of the angular size
when viewed through the lens to the angular size of the object when viewed at the
normal near point (without aid of the lens)
Microscope
This instrument has two lenses (1) The objective is near the object being viewed and
produces a greatly magnified real image (2) The eyepiece is a simple magnifier which
the viewer uses to closely examine the image from (1)
Telescope
This instrument also has two lenses (1) The objective at the front of the telescope
takes light from a distant object and produces a real inverted image (which is rather
small) near its focal point (2) This small real image is then examined by the eyepiece
functioning as a simple magnifier to produce a virtual image with a larger angular
size
20
Serway Chapter 37
Two-Slit Interference
If light is incident on two closely spaced narrow slits a pattern of light and dark
stripes is produced beyond the slits The bright stripes or fringes are caused by
constructive interference of the two waves coming from the slits Constructive
interference occurs whenever two waves arrive at a location in phase with each other
This occurs when the distance x1 from slit 1 to a point P on the screen and the
distance x2 from slit 2 to point P differ by in integral number of wavelengths
1 2 where 0 1 2x x m m
where λ is the wavelength of the light When x1 and x2 are much larger than the slit
spacing d this condition reduces to
sind m
where θ is the angle between the direction of the incident light and the direction of the
light arriving at the screen
Thin Films
When light is partially reflected and partially transmitted by a thin film of transparent
material it is possible to have interference between the wave reflected from the front
of the film and light reflected from the back of the film (The colored reflections from
the thin film of oil on the water in a rain-soaked parking lot are an example of this
effect) It is difficult to write down formulas that will work in all cases so we will
just review the important principles here
1 If the two reflected waves are in phase with each other the film has enhanced
reflection (constructive interference) but if the two reflected waves are out of phase
with each other reflection is diminished (destructive interference) Phase shifts occur
due to reflection and due to the extra path length through the film of the wave
reflected from the back of the film
2 The phase change due to reflection is determined by the difference in index of
refraction between the two media involved in the reflection If the wave is incident
21
from a medium with a low index of refraction into a medium with a high index of
refraction a phase change of 180deg occurs and the reflected wave is inverted If
incident from high to low no phase shift occurs and the reflected wave is non-
inverted
3 The extra path length through the film of the wave reflected from the back of the
film is equal to 2t where t is the thickness of the film (The incident light is assumed
to be normal to the surface of the film) The number of wavelengths contained in the
extra path length is equal to 2tλn where λn = λn is the wavelength of the light in the
film
4 Rules for reflection from thin films
If one ray is inverted and the other is not then we have
12 constructive
2 nt m
2 destructivent m
If either both rays are inverted or both are non-inverted we have
2 constructivent m
12 destructive
2 nt m
22
Serway Chapter 38
Diffraction Grating
A diffraction grating is simply a fancy version of two-slit interference with the two
slits replaced by thousands of slits Just as in the two-slit case the bright fringes
occur at angles given by
sin md
but in this case d the distance between neighboring slits is made to be very small
The effect of having many slits instead of two is to make each bright fringe highly
localized with wide dark regions between neighboring maxima
Single Slit Diffraction
When light passes through an opening in an opaque screen an interference pattern is
produced beyond the opening To understand why we may replace the single opening
by many small coherent sources of light These many sources interfere with each
other producing a pattern known as a diffraction pattern (Note that many authors do
not distinguish between interference and diffraction treating them as interchangeable
terms) If the opening is a slit of width a then the diffraction pattern far from the slit
will have a bright central maximum with a succession of minima and weaker maxima
on either side The angle between the incident direction and the minima is given by
sin ma
where m = plusmn1 plusmn2
If the opening is circular with diameter D the angle between the incident direction
and the first minimum is given by
sin 122D
Optical Resolution and Rayleighrsquos Criterion
Two point sources can just be resolved (distinguished from each other) if the peak of
the diffraction image of the first source overlies the first minimum of the diffraction
image of the second source For circular holes of the kind usually encountered in
23
optical devices this condition is approximately satisfied when the angular separation
between the two sources as viewed from the optical instrument is greater than or equal
to the critical angle
min 122D
where λ is the wavelength of the light and where D is the diameter of the aperture in
the instrument
Polarization
We say that an electromagnetic wave is polarized if its electric field vector doesnrsquot
change direction in a random fashion The simplest kind of polarization is linear
polarization in which the electric field vector oscillates back and forth along the
same axis in space Polarized light can be produced from normal unpolarized light by
selective absorption (as in Polaroid sunglasses) by passing light through crystals that
have different indices of refraction for different polarizations (double refraction) by
scattering (the blue sky is polarized) and by reflection (glare) Polarization by
reflection occurs when light reflects from a shiny insulating (non-metallic) surface
The amount of polarization is greatest for reflection at Brewsterrsquos angle
2
1
tan P
n
n
where θP is the incidence angle of light from medium 1 onto medium 2 and where n1
and n2 are the indices of refraction for the media (In our everyday experience n1 = 1
since the light comes in through the air and n2 is the index of refraction of the shiny
insulating material producing the glare eg water glass plastic paint etc)
Malusrsquos Law
The intensity of transmitted polarized light through a perfect polarizer is related to the
incident intensity of polarized light by Malusrsquos law
2cosoI I
where Io is the intensity of the incident light and where θ is the angle between the
electric field vector in the incident wave and the transmission axis the polarizer
24
Serway Chapter 39
Principles of Relativity
All of the weirdness of relativity flows from two simple principles
(1) The laws of physics must be the same in all inertial (non-accelerating) reference
frames
(2) The speed of light in vacuum has the same value in all inertial reference frames
Note that (1) seems reasonable but (2) is very odd It says that if two space ships are
approaching each other at nearly the speed of light and a laser pulse is shot from ship
1 toward ship 2 then when the people on ship 2 measure the speed of the pulse as it
goes by it is moving at 3 times 108 ms the same as if the ships were stationary
Simultaneity
If observer 1 sees two events in her own frame as simultaneous at two different
locations a moving observer 2 will see these two events happening at different times
Relativistic gamma
The factor γ (gamma) appears regularly in the formulas of relativity
2
2
1
1 vc
where v is the relative speed between two inertial frames
Time Dilation
If observer 1 sees two events at the same location in space separated by time Δtp in his
own frame then observer 2 moving at speed v relative to observer 1 will see these
two events separated by a longer time Δt
pt t
Moving clocks run slow
Length Contraction
If observer 1 measures the length of an object along the x-direction at some instant of
time in her frame to be Lp then observer 2 moving in the x-direction at speed v will
measure the length L of the object to be shorter
pLL
25
Moving meter sticks are short
Velocity Addition
If a particle is moving at velocity vac relative to frame c and if frame c is moving at
speed vcb with respect to frame b (with both velocities directed along the same line)
what would be the velocity of the particle in frame b In classical physics the answer is
ab ac cbv v v
but if any of these velocities are an appreciable fraction of the speed of light the
answer changes
21ac cb
abac cb
v Vv
v v c
In using this equation make sure that vac has the same sign in the numerator and in the
denominator and that vbc also has the same sign in both places Note that this equation
is different (and in our opinion easier to understand) than the equations in the text
Momentum and Energy
The momentum and total energy of a particle moving at velocity v (its speed is v v )
are given by
22
2 2
22
1 1 vvc c
m mcm E mc
vp v
Rest Energy
When a particle is at rest (v = 0 and γ = 1) then E = mc2 is called the rest energy It
includes the potential energy of the particle so that if we change the potential energy
of a particle we change its rest mass
Kinetic Energy The kinetic energy is the difference between the total energy and the
rest energy
2 2K mc mc
For v ltlt c this reduces to K = mv22
General Relativity
Einsteinrsquos principle of equivalence in the theory of general relativity is In the vicinity
of a point a gravitational field is equivalent to an accelerated frame of reference in
26
the absence of gravitational effects For example there is no difference whatever in
the physics experienced by an astronaut on earth and one who is in interstellar space
far away from any stars or planets but who has just turned on her spaceshiprsquos engines
and is accelerating at 98 ms2
27
Serway Chapter 40
Photons
We now know that even though light behaves like a wave it is better described as
consisting of small packets of energy called photons The energy of a photon is related
to its frequency by
E hf
where h is Planckrsquos constant
346626 10 J sh
Photons also carry momentum given by
hf hp
c
Since the amount of energy in each photon is fixed the difference between dim light
and intense light (of the same wavelength) is that dim light consists of fewer photons
than intense light
Photoelectric Effect
In the photoelectric effect light shining onto a metal surface gives the electrons in the
metal enough energy to escape and be detected It requires a certain amount of energy
(called the work function typically = 3 ndash 5 eV) just to kick the electron out of the
metal so the light must deliver at least this much energy to an electron to produce the
effect It is observed that red light no matter how intense never produces electrons
But ultraviolet light even if quite dim will eject electrons from the metal Since red
light consists of 2 eV photons and ultraviolet light has photons with energies around
4-6 eV the photon idea explains the behavior of the photoelectric effect The
maximum energy that an ejected electron can have is
maxK hf
where f is the photon frequency The minimum frequency that light can have and cause
electrons to be ejected is called the cutoff energy and is found by setting Kmax = 0
cf h
28
Compton Effect
When high-frequency light interacts with free electrons the scattered light does not
have the same wavelength as the incident light contrary to what classical
electromagnetism would predict But the photon picture in which photons have
momentum and energy allows us to treat scattering as a collision between the photon
and the electron In this collision the electron and the scattered photon both have a
different momentum and energy than they did before And since p = hλ for a photon
if the momentum is different the wavelength will also be different
after before 1 cose
h
m c
where the angle θ is the angle between the incoming photon direction and the
direction of the scattered photon
Particles are Waves
Since photons behave like particles sometimes it is not surprising that elementary
particles can behave like waves sometimes The wavelength of a particle with
momentum p is given by
h
p
Wave-particle Duality
Both photons and elementary particles have a dual nature sometimes they behave like
particles and sometimes they behave like waves It is hard for us to comprehend the
nature of such an object by making mental pictures but experiments definitively show
that this is the case Since we have no direct experience with photons and elementary
particles (because their energies are so much smaller than the energies of the everyday
objects we encounter) it is perhaps not surprising that we have a hard time forming a
mental picture of how they behave
Electron Interference
Since an electron is both a particle and a wave just like a photon it should be able to
produce an interference pattern This is observed If an electron beam is shot at two
closely-spaced slits and if the electrons that pass through the slits are detected
downstream it is found that there are some locations where electrons are never detected
29
and others where lots of electrons are detected The pattern is exactly the same as the
one observed for light waves provided that we use the electron wavelength λ = hp in
place of the wavelength of light This pattern is observed even though each electron is
detected as a single dot on the screen Only after many such dots are collected does the
pattern emerge And if we try to understand how this effect could possibly work by
looking closely at each slit to see which one the electron came through the pattern
disappears the act of measurement destroys the interference This means that each
single electron somehow comes through both slits (as a wave would)
Uncertainty Principle
In classical physics we always imagine that the positions and momenta of moving
particles have definite values It might be hard to measure them but surely at each
instant of time a particle should be precisely located at some point in space and have a
similarly precise momentum This turns out not to be true Instead both position and
momentum are required to be uncertain with their uncertainties Δx and Δp satisfying
the Heisenberg uncertainty relation
2x p
So if the particle were known to be precisely at some particular location (so that Δx =
0) we couldnrsquot know anything about its momentum (Δp = infin) and if its momentum
were exactly known we couldnrsquot know its position
There is a similar relation involving the particlersquos energy E and the time interval Δt
over which this energy is measured
2E t
This means that energy is not actually conserved if we are considering very small time
intervals and this brief non-conservation of energy has been observed
30
Serway Chapter 41
Wave Function ψ and Probability
Quantum mechanics does not predict exactly what an electron or a photon will do Instead
it specifies the wave function or probability amplitude ψ of an electron or a photon
This wave function is a complex-valued function of space and time whose squared
magnitude is the probability density P for finding a particle at a particular place in
space at a certain time
2P
where is the complex conjugate of ψ
For example the wave function of an electron with perfectly specified momentum p
would have a wavelength given by
h
p
and its probability amplitude would be proportional to
2i x ipxe e
The corresponding probability density would then be
21ipx ip ipxP e e e
which means that the electron is equally probable to be anywhere along the x axis
This is in accord with the uncertainty principle since we specified the momentum
precisely we canrsquot have any idea about the position of the electron
As another example you have probably seen ldquofuzzy ballrdquo drawings of electron
orbitals in a chemistry book These fuzzy balls are meant to indicate the distribution
of the probability density 2
P in the orbital
Particle in a Box
A simple example in which we can calculate the wave function is the case of a particle of
mass m constrained to be inside a 1-dimensional box between x = 0 and x = L In this
case the general wave function is a linear superposition of wave functions ψn of the form
sin niE tn
n xx t A e
L
31
where A is a positive constant where n = 1 2 3 and where the energy associated
with each of the quantum states ψn is given by
22
28n
hE n
mL
This wave function is zero at x = 0 and x = L which means that the particle will never
be found at the walls of the box The wave function has maximum values in the
interior and at these places the particle is most likely to be found and it also has
places where it is zero and at these places the particle will also never be found as
expressed by the formula
2sinn x
PL
The particle in the box is interfering with itself producing a probability interference
pattern across the box just like the interference patterns we studied with light and
sound
32
Serway Chapter 42
Atoms
Many people picture an atom as a miniature solar system where electrons orbit around
a massive nucleus at the center This picture is misleading because in atoms the wave
nature of electrons dominates The electrons form 3-dimensional standing waves
(called orbitals) centered at the nucleus And if you ask what is it thatrsquos waving the
answer is ψ the probability amplitude (see Chapter 41)
Standing waves on a string can occur only for certain frequencies (the fundamental
and higher harmonics) Similarly atomic orbitals occur only for certain energies For
the hydrogen atom the energies of the orbitals have the a particularly simple form
2
1136 eVnE
n
where n = 1 2 3 is called the principal quantum number For other atoms the
determination of the orbital energies requires numerical calculation by computers
Atomic Spectra
If an electron is somehow given extra extra energy (we say that it is excited) so that it
occupies a higher orbital it will eventually ldquofallrdquo back down to a lower orbital Each
time an electron falls to a lower orbital it loses the difference in energy between the
two orbitals in the form of a photon Since the orbital energies are discrete so are the
energy differences and so are the wavelengths of the emitted light The entire set of
these discrete wavelengths is called the atomic spectrum and it is unique to each type
of atom For hydrogen the spectrum can be simply written as
H 2 2
1 1 1
f i
Rn n
where RH = 10973732 times 107 m-1 For other atoms the spectrum cannot be expressed as
a formula
Orbital Angular Momentum Electrons in atoms also have quantized values of
angular momentum The orbital quantum number ℓ specifies the value of this
quantized angular momentum through the formula
1L
33
If we want to know the value of the angular momentum along some direction in space
say the z direction the answer is not L but rather
zL m
where mℓ is another quantum number which runs from
1 1m
This quantum number is important when an atom sits in a magnetic field
Spin Angular Momentum It has been found experimentally that electrons and other
charged particles also carry internal angular momentum which we call spin
Electrons have an intrinsic spin angular momentum s along a specified axis that is
extremely quantized it can only take on 2 possible values
1
2zs s
We thus say that electrons have spin s = 12 or that electrons are rdquospin one-half parti-
cles Since s is an angular momentum it obeys the same rule as that for orbital angular
momentum namely that if its value along some axis is s then its total magnitude is
given by
31
2S s s
I know this seems weird but quantum mechanics is weird The only excuse for this
bizarre way of looking at the world is that it predicts what happens in experiments
Exclusion Principle The answer to the question of how many electrons (or any other
spin one-half particle) can be in one particular quantum state was discovered by
Wolfgang Pauli and is called the exclusion principle
ldquoNo two electrons can ever be in the same quantum state therefore no two electrons
in the same atom can have the same set of quantum numbersrdquo
This is the reason that we have atoms with different properties instead of every atom
simple having all of its electrons in the ground state All of the variety we see around
us in the world is the result of chemical differences and these differences would not
exist unless electrons obeyed this important principle The entire structure of the
periodic table (see pages 1377-1379 in Serway) is an expression of this principle
34
Serway Chapter 44
Nuclear Properties
The nucleus is composed of protons with charge +e and mass mp = 1007226 u and of
neutrons with zero charge and mass mn = 1008665 u where u is one atomic mass unit
-271 u=1660540 10 kg
The atomic number Z counts the number of protons in a nucleus while the neutron
number N counts the number of neutrons The mass number A is the sum of the two
A N Z
Protons and nuetrons have about the same mass and some times called baryons (heavy
ones) or nucleons (partcles in the nucleus) A is sometimes called the baryon
number The chemical elements are distinguished by Z ie hydrogen has Z = 1 iron
has Z = 26 uranium has Z = 92 etc But for a given Z there might be several nuclei
with different numbers of neutrons These nuclear siblings are called isotopes
These numbers are used to label nuclei according to the pattern
5626 ie FeA
Z X
denotes the isotope of iron with N = 56 ndash 26 = 30 neutrons
The nucleus is roughly spherical with a radius given approximately by
1 3 150 0where 12 10 mr r A r
Nuclear Stability
Because the positively charged protons electrically repel each other with an enormous
force at distances as small as 10-15 m there must be some really strong force that that
overcomes electrical repulsion to hold protons and neutrons together This force is
called with some lack of imagination the strong force It is a very short-range force
(it only acts over a distance of about 2 x 10-15 m) and attracts protons to protons
neutrons to neutrons and neutrons to protons But in spite of this strong nuclear force
the coulomb repulsion of the protons is still present so anything that might keep the
protons from being right next to each other would help keep the nucleus from
35
exploding This role is played by the neutrons and for nuclei with Z le 20 the stable
nuclei roughly have N = Z
For Z gt 20 the coulomb repulsion force is more powerful and more neutrons are
needed to dilute it up to about N = 15Z around Z = 80 For Z gt 83 no amount of
neutrons can help and these nuclei are unstable (radioactive) There is an ldquoisland of
(relative) stabilityrdquo around Z=90-92 (thorium and uranium respectively) These
elements have one or two isotopes with half-lifes of billions of years so there are
substantial amounts of such elements on Earth This fact makes it possible to have
practical fission devices
Radioactive Decay
There are three types of radioactive decay
Alpha decay the nucleus kicks out a helium nucleus (N = 2 Z = 2 A = 4) So if the
original nucleus is called X and the new nucleus is called Y then the decay would look
like this
4 42 2X Y+ HeA A
Z Z
Beta decay the nucleus either kicks out an electron (endash) or its positively-charged
antimatter twin the positron (e+) plus either an electron neutrino v or an electron
anti-neutrino v
1X Y+e electron decayA AZ Z v
1X Y+e positron decayA AZ Z v
A neutrino is a particle with no charge hardly any mass (much less than the electron
mass) and interacts so weakly with matter that most neutrinos upon encountering the
planet earth just pass right through it as if it werenrsquot there
Gamma decay the nucleons in the nucleus X are in an excited energy state X
(perhaps as a result of having undergone alpha or beta decay) and they drop down to a
lower energy state shedding the energy as a high frequency photon
X XA AZ Z
36
This process is exactly analogous to the way that the electrons in atoms emit photons
Decay Rate and Half Life
There is no way to predict exactly when an unstable or excited nucleus will decay but
there is an average rate at which this decay occurs called the decay constant λ The
meaning of this constant is that if there are a large number N of nuclei in a sample
then the number of decays per second that will be observed (called the decay rate R)
is R = λN In mathematical language
dMR N
dt
This simple differential equation has for its solution
0tN t N e
where N0 is the number of nuclei in the sample at time t = 0
The half-life is the time it takes for half of the nuclei in the sample to decay and is
related to the decay constant by
1 2
ln 2 0693T
Disintegration Energy
When a nucleus decays it is making a transition to an overall state of lower energy
which means according to Einsteinrsquos famous formula E = mc2 that the sum of the
masses after the decay must be less than the mass before with the lost mass appearing
as kinetic energy among the decay products For example in alpha decay this kinetic
energy called the disintegration energy Q is given by
2X YQ M M M c
37
Serway Chapter 45
Nuclear energy
There are two ways to extract energy from the nucleus fission and fusion For nuclei
with Z greater than 26 breaking the nucleus apart into pieces leads to a lower
2mc energy than the original nucleus so energy can be extracted by fission For
nuclei with Z less than 26 a lower energy is achieved by combining nuclei so energy
can be extracted by fusion
Fission
Since neutrons have no charge they are not repelled from nuclei as protons are for
this reason their behavior is the key to understanding how fission works
Fast neutrons mostly bounce off other nuclei with each collision slowing the neutron
down This elastic energy loss is most effective if the other nuclei have low mass (like
hydrogen) and these materials are called moderators because of their ability to slow
down fast neutrons
The reason that slowing neutrons is important is that slow neutrons are much more
likely to be absorbed by a nucleus which then leads to nuclear reactions of various
kinds For a few very large nuclei like uranium-235 and some plutonium isotopes
absorption of a slow neutron causes the nucleus to split into two large fragments plus
2 or 3 fast neutrons (energy is released in the process too) Thus one slow neutron
can produce energy plus 2 or 3 more neutrons which if moderated from fast to slow
can split 2 or 3 more nuclei leading to a runaway chain reaction If the reaction is
allowed to proceed unchecked it produces a large explosion If a neutron absorbing
material is added to the mix (like the cadmium in reactor control rods) it is possible to
keep the reaction under control and to extract the released energy as heat to drive
steam turbines and produce electricity
Fusion
Fusion involves mashing two nuclei together and since they are both charged and repel
each other this reaction is much harder to make go The nuclei must have enough energy to
overcome the coulomb repulsion which is why this reaction requires a high temperature
(hundreds of millions of degrees K) like that in the sun (which is burning hydrogen to
helium via fusion) or like that in the center of the fission explosion that is used to detonate
38
a hydrogen bomb
This reaction is of interest for power production in spite of this difficult temperature
requirement because of the abundance of fusion fuel on the planet There are about
012 g of deuterium ( 21H ) in every gallon of water on earth and it only costs about 4
cents to extract it The fusion energy available from this minuscule amount of
deuterium would run a 1000-Megawatt power plant for 10 seconds Water is so
abundant on earth that if fusion were to work we would have an essentially
inexhaustible source of energy
So why donrsquot we have fusion power plants Well the fuel is cheap but the match is
incredibly expensive The only way we know to control this difficult high-temperature
reaction is with large and expensive pieces of equipment involving either large
magnetic fields and complex high-power electromagnetic antennas or with gigantic
(football-fieldsized) laser facilities involving more than a hundred of the highest-
energy lasers ever built Power plants based on these current methods for controlling
fusion are unattractive to the fiscally-minded people who run the electric power
industry Hopefully better designs will be discovered as experiments continue
39
c actinium 68 Er erbium 101 Md mendelevium 104 Rf rutherfordium l aluminum 63 Eu europium 80 Hg mercury 62 Sm samarium m americium 100 Fm fermium 42 Mo molybdenum 21 Sc scandium b antimony 9 F fluorine 60 Nd neodymium 106 Sg seaborgium r argon 87 Fr francium 10 Ne neon 34 Se selenium s arsenic 64 Gd gadolinium 93 Np neptunium 14 Si silicon t astatine 31 Ga gallium 28 Ni nickel 47 Ag silver a barium 32 Ge germanium 41 Nb niobium 11 Na sodium k berkelium 79 Au gold 7 N nitrogen 38 Sr strontium e beryllium 72 Hf hafnium 102 No nobelium 16 S sulfur i bismuth 108 Hs hassium 76 Os osmium 73 Ta tantalum h bohrium 2 He helium 8 O oxygen 43 Tc technetium
boron 67 Ho holmium 46 Pd palladium 52 Te tellurium r bromine 1 H hydrogen 15 P phosphorus 65 Tb terbium d cadmium 49 In indium 78 Pt platinum 81 Tl thallium a calcium 53 I iodine 94 Pu plutonium 90 Th thorium f californium 77 Ir iridium 84 Po polonium 69 Tm thulium
carbon 26 Fe iron 19 K potassium 50 Sn tin e cerium 36 Kr krypton 59 Pr praseodymium 22 Ti titanium s cesium 57 La lanthanum 61 Pm promethium 74 W tungsten l chlorine 103 Lr lawrencium 91 Pa protactinium 92 U uranium r chromium 82 Pb lead 88 Ra radium 23 V vanadium o cobalt 3 Li lithium 86 Rn radon 54 Xe xenon u copper 71 Lu lutetium 75 Re rhenium 70 Yb ytterbium m curium 12 Mg magnesium 45 Rh rhodium 39 Y yttrium b dubnium 25 Mn manganese 37 Rb rubidium 30 Zn zinc y dysprosium 109 Mt meitnerium 44 Ru ruthenium 40 Zr zirconium s einsteinium
40
UNITS The SI units are given in parentheses Other commonly used units are given in terms of the SI units Symbols conform with the recommendations of the American National Standards Institute (ANSI) and the American Institute of Physics (AIP) Commonly used multiples of SI units
prefix symbol factor tera T 1012 giga G 109
mega M 106 kilo k 103
centi c 10-2 milli m 10-3
micro μ 10-6 nano n 10-9 pico p 10-12
femto f 10-15
length meter (m) angstrom 1 Ǻ = 10-10 m inch 1 in = 254 x 10-2 m foot 1 ft = 03048 m mile 1 mi = 1609 m
mass kilogram (kg) atomic mass unit 1 u = 1661 x 10-27 kgslug 1 slug = 1459 kg
force newton (N = kg bull ms2) dyne 1 dyn = 10-5 N pound 1 lb = 4448 N
pressure Pascal (Pa = kgm bull s2) atmosphere 1 atm = 1013 x 105 Pa poundssquare inch 1 psi = 6895 Pa cm of mercury 1 cm Hg = 1333 Pa bar 1 bar = 1000 x 105 Pa torr 1 torr = 1333 Pa
time second (s) minute 1 min = 60 s hour 1 h =3600 s
frequency hertz (Hz = s-1) radianssecond 1 rads = 12π Hz
energy joule (J = kg bull m2s2) erg 1 erg = 10-7 J electron volt 1 eV = 1602 x 10-19 J calorie 1 cal = 4187 J kilowatt-hour 1 kWh = 36 x 106 J British thermal unit 1 Btu = 1055 J
power watt (W = kg bull m2s3) horsepower 1 hp = 7457 W
charge coulomb (C = A bull s)
electric potential volt (V = kg bull m2 s3 bull A)
current ampere (A)
resistance ohm (Ω = kg bull m2s3 bull A2)
capacitance farad (F = s4 bull A2kg bull m2)
magnetic field tesla (T = kgs2 bull A) gauss 1 G = 10-4 T
magnetic flux weber (Wb = kg m2s2 bull A) maxwell 1 Mx = 10-8 Wb
magnetic inductance henry (H = kg bull m2s2 bullA2)
temperature kelvin (K) degrees Celsius 0degC = 27315 K
angle radian (rad) degree 1 deg = π180 rad revolution 1 rev = 2π rad
41
Some Physical Constantsa Quantity Symbol Valueb Atomic mass unit μ 1660 538 73 (13) x 10-27 kg
931494 013 (37) MeVc2 Avogadros number NA 6022 141 99 (47) x 1023 particlesmol Avogadrorsquos number 6022 x 1023 mol Bohr magneton
2B e
em 9274 008 99 (37) x 10-24 JT
Bohr magneton μB 927 x 10-24 JT Bohr radius
0
2
2e em e k
a 5291 772 083 (19) x 10-11 m
Boltzmanns constant B A
RNk 1380 650 3 (24) x 10-23 JK
Boltzmannrsquos constant kB 1380 x 10-23 JK Compton wavelength
C eh
m c 2426 310 215 (18) x 10-12 m
Deuteron mass md 3343 583 09 (26) x 10-27 kg 2013 553 212 71 (35) u
electron charge e 1602 x 10-19 C Electron mass me 9109 381 88 (72) x 10-31 kg
5485 799 110 (12) x 10-4 u 0510 998 902 (21) MeVc2
electron mass 911 x 10-31 kg Electron volt eV 1602 176 462 (63) x 10-19 J Elementary charge e 1602 176 462 (63) x 10-19 C Gas constant R 8314 472 (15) JKmol Gravitational constant G 6673 (10) x 10-11 Nm2kg2 Neutron mass mn 1674 927 16 (13) x 10-27 kg
1008 664 915 78 (55) u 939565 330 (38) MeVc2
neutron mass 1675 x 10-27 kg Nuclear magneton
2n p
em 5050 783 17 (20) x 10-27 JT
permeability constant μ0 1257 x 10-6 Hm permittivity constant ε0 8854 x 10-12 Fm Plancks constant h
2h
6626 068 76 (52) x 10-34 Js 1054 571 596 (82) x 10-34 Js
Planckrsquos constant h ħ
6626 x 10-34 Js 1055 x 10-34 Js
Proton mass mp 1672621 58 (13) x 10-27 kg 1007 276 466 88 (13) u 938271 998 (38) MeVc2
proton mass 1673 x 10-27 kg Rydberg constant RH 1097 373 156 854 9 (83) x 107 m-1 Speed of light in vacuum c 2997 92458 x 108 ms (exact) speed of light c 300 x 108 ms a
These constants are the values recommended in 1998 by CODATA based on a least-squares adjustment of data from different measurements For a more complete list see P J Mohr and B N Taylor Rev Mod Phys 72351 2000 b The numbers in parentheses for the values above represent the uncertainties of the last two digits
NOTE The ones in red are the ones from Appendix 13 The ones in yellow are the ones that are from the Serway book table but that were already on the Appendix 13 table
42
INDEX
Absolute zero 9
Adiabatic 17
Adiabatic exponent 18
Adiabatic process 17
Alpha decay 46
Amplitude 4
Angle of reflection 25
Angular frequency 3
Angular magnification 30
Angular momentum atomic 43
Angular size 29
Archimedes Principle 1
Atomic spectra 43
Atoms 43
Avogadrorsquos number 10
Beats 8
Bernoullirsquos Equation 2
Beta decay 46
Bifocals 29
Brewsterrsquos angle 34
British Thermal Unit Btu 11
Buoyancy 2
calorie 11
Calorie 11
Camera 28
Carnot cycle 21
Carnot efficiency 22
Celsius scale 9
Ciliary muscle 29
Coefficient of performance 21
Compton effect 39
Constructive interference 7
Continuity equation of 2
Contraction length 35
Convection 14
Curved mirrors 27
Decay constant 46
Decay rate radioactive 46
Decibel scale 5
Degrees of freedom 16
Density 1
Destructive interference 7
Diffraction single slit 33
Diffraction grating 33
Dilation time 35
Disintegration energy 47
Dispersion of light 26
Doppler effect 6
Double slit interference 31
Efficiency 20
Electron interference 39
Emissivity 15
Energy nuclear decay 47
Energy relativistic 36
43
Engineering work 20
Entropy 22
Entropy ideal gas 24
Equipartition of energy 18
Equivalence principle 36
Exclusion principle 44
Expansion thermal 9
Expansion coefficient linear 9
Eye 29
Fahrenheit scale 9
Far point 29
Farsightedness 29
First Law of Thermodynamics 12
Fission 48
Flux volume 2
Free expansion 23
Fringes 31
Fusion 48
Fusion heat of 11
Gamma relativistic 35
Gamma decay 46
General relativity 36
Half-life 46
Heat 11
Heat capacity 11
Heat conduction 13
Heat engine 20
Heat of fusion 11
Heat of vaporization 11
Heat pump 20
Heats of transformation 11
Hydrostatics 1
Ideal Gas Law 10
Images real and virtual 27
Intensity sound 5
Interference 7
Interference two-slit 31
Internal energy 12
Internal energy degrees of freedom 16
Irreversible process 19 23
Isotopes 45
Joule 11
Kelvin scale 9
Kinetic energy relativity 36
Kinetic theory 16
Latent heat 11
Length contraction 35
Linear expansion coefficient 9
Linear polarization 34
Linear superposition 7
Longitudinal wave 3
Loudness 5
Magnification lateral 27
Magnifying glass 30
Malusrsquos law 34
Microscope 30
Momentum relativistic 36
Muscle ciliary 29
44
Musical instruments 8
Musical scale 8
Near point 29
Nearsightedness 29
Nonlinear 7
Nuclear energy 48
Nuclear properties 45
Nuclear stability 45
Octave 8
Optical resolution 33
Orbital quantum number ℓ 43
Orbitals 43
Particle in a Box 41
Particles are waves 39
Pascalrsquos Principle 1
Period 3
Photoelectric effect 38
Photons 38
Pitch 8
Polarization 34
Power sound 5
Presbyopia 29
Pressure 1
Principal quantum number n 43
Principle of equivalence 36
Principle of linear superposition 7
Probability amplitude ψ 41
Processes thermodynamic 13
R-value 14
Radiation thermal 14
Radioactive decay 46
Radioactive decay rate 46
Ray tracing 27
Rayleighrsquos criterion 33
Reading glasses 29
Real image 27
Refraction 25
Refrigerator 20
Relativistic gamma 35
Relativity principles 35
Resolved for light sources 33
Rest energy 36
Reversible process 19 22
Rope wave speed 4
Second Law of Thermodynamics 19
Shock waves 6
Simple magnifier 30
Simultaneity 35
Single slit diffraction 33
Snellrsquos law 25
Sound speed 5
Specific heat 11
Spin Angular momentum 44
Standing waves 7
Stefanrsquos law 14
Strong force 45
Telescope 30
Temperature 9
45
Temperature Scales 9
Thermal conductivity 14
Thermal energy 12
Thermal expansion 9
Thin film interference 31
Thin lenses 28
Time dilation 35
Tone musical 8
Total internal reflection 25
Transverse wave 3
Traveling Waves 3
Two-slit interference 31
Uncertainty principle 40
Vaporization heat of 11
Velocity addition relativity 36
Virtual image 27
Volume flux 2
Wave function 41
Wave Function ψ and Probability 41
Wave speed 3
Wave-particle duality 39
Wavelength 3
Wavenumber 3
Wien Displacement Law 16
Work 11
2
heating process in just the right way its pressure could remain fixed In isobaric
processes both Q and W are non-zero (Work is eacy = ndashP(Vf ndash Vi)
Isovolumetric process The volume is held fixed ΔV = 0 The work done by the gas is
then zero and so we have ΔEint = Q
Isothermal process The temperature is held fixed ΔT = 0 There is no change in Eint
so Q = ndashW for isothermal processes For an ideal gas PV = nRT so PV = constant in
an isothermal process The work done on the gas is then given by
lnf
i
Vi
Vf
VW PdV nRT
V
Heat Conduction
The quantity P is defined to be the rate at which heat flows through an object and is a
power having units of watts It is analogous to electric current which is the rate at
which charge flows through an object If the flow of heat through a slab of length L
and cross-sectional area A is steady in time then P is given by the equation
h cT TdQkA
dt L
P =
where Th and Tc are the (hot and cold) temperatures of the two ends of the slab The
heat flows of course because of this temperature difference The quantity k is called
the thermal conductivity and is a constant that is characteristic of the material It is
analogous to the electrical conductivity h cT TL is sometimes called the temperature
gradient and is written dTd dTdx
R-Values
It is common to have the heat-conducting properties of materials described by their R-
values especially for insulating materials like fiberglass batting The connection
between k and R is R = Lk where L is the material thickness In this country R-values
always have units of 2ft F hourBtu
Polyurethane foam 6 (1 in thick) Air space 1 (35 in thick) Fiber glass batting 11
(35 in thick) Wood about 1 (1 in thick) Glass 09 (0125 in thick)
The heat flow rate through a slab of area A is given by
3
h cT TA
R
P
in units of Btuhour Note that A must be in square feet and the temperatures must be
in degrees Fahrenheit
Convection
Convection is the transfer of thermal energy by flow of material For instance a home
furnace doesnrsquot heat a house by waiting for the heat from the burner to slowly conduct
throughout the house instead it quickly pumps warm air to all of the rooms
Generally convection is a much faster way to transfer heat than conduction
Radiation
Electromagnetic radiation can also transfer heat When you warm yourself near a
campfire which has burned itself down into a bed of glowing embers you are
receiving radiant heat from the infrared portion of the electromagnetic spectrum The
rate at which an object emits radiant heat is given by Stefanrsquos law
4AeTP
where P is the radiated power in watts σ is a constant
8 2 45696 10 W m K
A is the surface area of the object in m2 and T is the temperature in kelvins The
constant e is called the emissivity and it varies from substance to substance A perfect
absorber (think black velvet) has e = 1 while a perfect reflector (think mirror) has e =
0 Hence black objects radiate very well while shiny ones do not Also an object that
is hotter than its surroundings radiates more energy than it absorbs whereas an object
that is cooler than its surroundings absorbs more energy than it radiates
Terminology
Transfer variables vs state variables
Energy transfer by heat as well as work done depends on the initial final and
intermediate states of the system They are transfer variables But their sum (Q + W =
4
Eint) is a state variable
Figure 205
5
Serway Chapter 21
Kinetic Theory
The ideal gas law works for all atoms and molecules at low pressure It is rather
amazing that it does Kinetic theory explains why The properties of an ideal gas can
be understood by thinking of it as N rapidly moving particles of mass m As these
particles collide with the container walls momentum is imparted to the walls which
we call the force of gas pressure In this picture the pressure is related to the average
of the square of the particle velocity 2v by
22 1( )
3 2
NP mv
V
Using the ideal gas law we obtain the average translational kinetic energy per
molecule
21 3
2 2 Bmv k T
The rms speed is then given by
2rms
3 3Bk T RTv v
m M
where M is the molecular mass in kgmol
Degrees of Freedom
Roughly speaking a degree of freedom is a way in which a molecule can store energy
For instance since there are three different directions in space along which a molecule
can move there are three degrees of freedom for the translational kinetic energy
There are also three different axes of rotation about which a polyatomic molecule can
spin so we say there are three degrees of freedom for the rotational kinetic energy
There are even degrees of freedom associated with the various ways in which a
molecule can vibrate and with the different energy levels in which the electrons of
the molecule can exist
Internal Energy and Degrees of Freedom The internal energy of an ideal gas made
up of molecules with J degrees of freedom is given by
int 2 2 B
J JE nRT Nk T
6
Heat Capacities of an Ideal Gas The heat capacity of a gas is described by means of
molar heat capacities CV and CP These are the heat capacities per mole and the
subscript V on CV means that the volume is being held constant while for CP the
pressure is held constant For example to raise the temperature of n moles of a gas
whose pressure is held constant by 10 K we would have to supply an amount of heat Q
= nCP (10) K
Molar Specific Heat of an Ideal Gas at Constant Volume
VQ nC T
3monatomic
2VC R
5diatomic
2VC R
5polyatomic
2VC
Real gases deviate from these formulas because in addition to the translational and ro-
tational degrees of freedom they also have vibrational and electronic degrees of
freedom These are unimportant at low temperatures due to quantum mechanical
effects but become increasingly important at higher temperatures The rough rule is
No of degrees of freedom
2VC R
Molar Specific Heat of an Ideal Gas at Constant Pressure
PQ nC T
P VC C R
The internal energy of an ideal gas depends only on the temperature
int VE nC T
Adiabatic Processes in an Ideal Gas
7
An adiabatic process is one in which no heat is exchanged between the system and the
environment When an ideal gas expands or contracts adiabatically not only does its
pressure change as expected from the ideal gas law but its temperature changes as
well Under these conditions the final pressure Pf can be computed from the initial
pressure Pi and from the final and initial volumes Vf and Vi by
or constantf f i iP V PV PV
where γ = CPCV The quantity γ is called the adiabatic exponent Note that this
doesnrsquot mean that the ideal gas law no longer holds it does and in fact it can be
combined with the adiabatic law for pressure given above to obtain the adiabatic law
for temperatures
1 constantTV
Compressions in sound waves are adiabatic because they happen too rapidly for any
appreciable amount of heat to flow This is why the adiabatic exponent appears in the
formula for the speed of sound in an ideal gas
RTv
M
Note that v depends only on T and not on P Because it depends only on the
temperature the speed of sound is the same in Provo as at sea level in spite of the
lower pressure here due to the difference in elevation
Equipartition of Energy
Every kind of molecule has a certain number of degrees of freedom which are
independent ways in which it can store energy Each such degree of freedom has
associated with it ndash on average ndash an energy of 12 Bk T per molecule (or 1
2 RT per mole)
(Note since a molecule has so many possible degrees of freedom it would seem that
there should be a lot of 12 sBk T to spread around But because energy is quantized
some of these degrees of freedom are not ldquoactiverdquo until the temperature becomes high
enough that 12 Bk T is as big as the lowest quantum of energy
8
Serway Chapter 22
Second Law of Thermodynamics
There are several equivalent forms of this important law
Kelvin It is not possible to change heat completely into work with no other change
taking place Or in other words there are no perfect heat engines
Clausius It is not possible for heat to flow from one body to another body at a higher
temperature with no other change taking place Or in other words there are no
perfect refrigerators
Entropy In any thermodynamic process that proceeds from one equilibrium state to
another the entropy of the system + environment either remains unchanged or
increases The total entropy never decreases This law is a bit of an oddity among the
laws of physics because it is not absolute Things are forbidden by the second law not
because it is impossible for them to happen but because it is extremely unlikely for
them to happen (See below for more information about entropy)
Reversible and Irreversible Processes
A reversible process is one which occurs so slowly that it is in thermal equilibrium (or
very nearly so) at all times A hallmark of such processes is that a motion picture of
them looks perfectly normal whether run forward or backward Imagine for instance
the slow expansion of a gas at constant temperature in a cylinder whose volume is being
increased by a slowly moving piston Run the movie backwards and what do you see
You see the slow compression of a gas at constant temperature which looks perfectly
normal
An irreversible process is one which occurs in such a way that thermal equilibrium is
not maintained throughout the process The mark of this kind of process is that a motion
picture of it looks very odd when run backward Imagine the sudden expansion of a gas
into a previously evacuated chamber because a hole was punched in the wall between a
pressurized chamber and the evacuated one Run the movie backward and what do you
see You see the gas in the soon-to-be-evacuated chamber gather itself together and
stream through a tiny hole into a chamber in which there is already plenty of gas If you
have ever seen this happen get in touch with the support group for those who have
witnessed the spontaneous combustion of large mammals by calling 1-800-PYROCOW
9
Heat Engines
Because of the vexing difference in sign between work done on and system and work done
by a system we will invent a new work variable Weng Heat engines do work and so the net
W for these engines is negative But in engineering applications hidden minus signs are
regarded as evil so for heat engines we donrsquot talk about W instead we talk about its
magnitude engW W So for heat engines the first law is
int engE Q W
But for heat pumps and refrigerators work is done on the system so we use the usual work
W when we talk about these systems
A heat engine is a machine that absorbs heat converts part of it to work and exhausts the
rest The heat must be absorbed at high temperature and exhausted at low temperature If the
absorbed heat is Qh the exhausted heat is Qc and the work done by the engine is Weng then
eng h cW Q Q
and the efficiency of the engine is defined to be
eng
h
We
Q
A perfect engine would convert the heat hQ completely into work Weng giving an effi-
ciency of e = 1 Energy conservation alone allows a perfect engine but the second law
requires e lt 1
Refrigerators and Heat Pumps
A refrigerator is a machine that absorbs heat at low temperature and exhausts it at high
temperature the ldquobackwardsrdquo heat transfer being driven by the work done on the machine
by some source of power A heat pump is a machine that either works like a refrigerator
keeping a place cold by transferring heat from this cold place to a higher temperature
environment (cooling mode like an air conditioner) or it functions as a heater
transferring heat into a warm place from a cooler one (heating mode like a window unit
that heats a house by extracting thermal energy from the cold outdoors) The coefficient
of performance of a refrigerator or of a heat pump in cooling mode is defined to be
COP cooling modecQ
W
10
For a heat pump in heating mode the coefficient of performance is
COP heating modehQ
W
Note that we donrsquot have to use the engineering work here because in these systems
work is done on the system and W is naturally positive
A perfect refrigerator would take in heat Qc and exhaust the same amount of heat without
doing any work giving an infinite coefficient of performance Energy conservation alone
allows a perfect refrigerator but the second law requires COP lt infin A perfect heat pump
in heating mode would transfer Qh into the house without doing any work and so would
also have an infinite coefficient of performance The second law forbids this too A good
coefficient of performance for a real device would be around 5 or 6
Carnot Cycle
The most efficient of all possible engines is one that uses the Carnot cycle This cycle
employs an ideal gas has no friction and operates very slowly so that the gas can be in
thermal equilibrium at all parts of the cycle This means of course that it canrsquot
possibly be built and even if it could be built it would not run fast enough to be useful
Nevertheless this cycle is very important because it gives an upper bound on the
efficiency of real engines There cannot possibly be an engine that is more efficient
than one based on the Carnot cycle This cycle consists of the following four steps
1 The ideal gas absorbs heat Qh at constant temperature Th while the gas increases its
volume The reason that heat is absorbed is that expansion tends to cool the gas but
thermal contact with the environment at Th keeps the temperature high by heat
conduction into the ideal gas
2 The ideal gas further increases its volume by an adiabatic expansion This expansion
causes the gas to cool so at the end of this part of the cycle the gas is at temperature Tc
3 The gas exhausts heat Qc at constant temperature Tc while the gas decreases in
volume The reason that heat is exhausted is that compression tends to heat the gas
but thermal contact with the environment at Tc keeps the temperature low by heat
conduction out of the ideal gas
4 The gas is adiabatically compressed back to its original volume (the volume it started
with in step 1) This compression heats the gas from Tc up to Th
11
The efficiency of a Carnot engine is given by the very simple formula
1 cC
h
Te
T
where the temperatures must in be Kelvin No real engine can be more efficient than
this
The coefficient of performance of a Carnot refrigerator or heat pump in cooling mode
is given by
COP cooling mode cC
h c
T
T T
and no real refrigerator can have a coefficient of performance greater than this
The coefficient of performance of a Carnot heat pump in heating mode is
COP heating mode hC
h c
T
T T
So why donrsquot we just use these wonderful Carnot engines and have perfect efficiency
To make the reversible steps in the cycle really reversible they would have to occur
infinitely slowly So the price you pay for making a perfect engine is that it takes
forever to get it to do any work
Entropy
The entropy of a system is defined in terms of its molecular makeup and measures
roughly the disorder of the system If the system is packed into a very small volume
then it is quite ordered and the entropy will be low If it occupies a large volume the
entropy is high (To see what this has to do with disorder note that socks in a drawer
occupy a small volume while socks on the bed in the corner by the door and
hanging from the chandelier occupy a large volume) If the system is very cold then
the molecules hardly move and may even reach out to each other and form a crystal
This is a highly ordered state and therefore has low entropy If the system is very hot
with rapidly speeding molecules crashing into the container walls and bouncing off
each other things are disordered and the entropy is high
It is possible to calculate the entropy of a system in terms of its macroscopic thermody-
namic properties ie pressure volume temperature number of moles etc The key to this
calculation is the concept of a reversible process A reversible process is one that is
12
carried out without friction and so slowly that the process can be reversed at any stage by
making an infinitesimal change in the environment of the system The slow expansion of the
gas at Th in the Carnot cycle with tiny amounts of heat being transferred to the environment
is reversible If we turned around and began slowly to compress the gas would just slowly
exhaust heat to the environment in the exact reverse way that it absorbed it during expansion
Most processes however are irreversible For example if a gas-filled box were suddenly
increased in size so that the particles were free to wander into the void created by the sudden
expansion then the gas would eventually fill the new volume uniformly at the same
temperature as before the expansion (The temperature is unchanged in this imaginary
process because the kinetic energy of the molecules would be unaffected by such an
instantaneous expansion of the container walls) This imaginary but highly thought-
stimulating process is called a free expansion and it is impossible to reverse it During the
expansion we didnrsquot push on any of the molecules so reversing this process would mean
making them go back into their original volume without pushing on them they simply will
not cooperate to this extent Another way to see that just pushing them back where they came
from does not reverse the free expansion is to think about what would happen if we just
compressed either adiabatically or isothermally An adiabatic compression back to the
original volume would heat the gas above its original temperature and an isothermal
compression would require that heat be exhausted to the environment But the free expansion
involved neither temperature changes nor heat exchanges so neither of these two processes
is the reverse of the free expansion It is simply impossible to reverse this rapid expansion in
a way that takes us back to the initial state
It is possible to calculate the change of entropy for both reversible and irreversible processes
Letrsquos consider a reversible process first In a reversible process the entropy change is given
by the formula
dQS
T
where dQ is the amount of heat added to the system during a small step of the process
The total energy change during the process may then simply be calculated by integration
f f
f i i i
dQS S S dS
T
13
(Just as in the case of energy we are mainly interested in differences rather than in
absolute magnitudes)
This integration method does not work for irreversible processes For instance in the
rapid free expansion discussed above no heat is added to the system but its disorder
obviously goes up We need to find some other way to calculate the entropy The key
is the fact that the entropy of a system depends only on its current state and not at all
on how it arrived there So to calculate the entropy change in an irreversible process
first find out what the initial and final conditions of the process are Then invent a
reversible process that takes the system from the initial state to the final state Since
the entropy depends only on the state of system and not on the process the entropy
change for the reversible process is the same as that for the irreversible process
Entropy is thus like pressure or temperature if the pressure changes from Pl to to P2
then the pressure difference is P2 - Pl regardless of how the pressure change was
made This seems obvious for pressure but not for entropy only because you arent
comfortable with entropy yet
Entropy of an Ideal Gas
For n moles of an ideal gas the difference in entropy between a state with temperature
T and volume V and some standard state with temperature To and volume Vo is given
by
ln lno o Vo o
T VS T V S T V nC nR
T V
Entropy in a Phase Change
Calculating entropy change in a phase change like melting or boiling is easy It is
QT Where Q is in the latent heat for example the flows in to cause the melting and
T is the temperature of the phase change There is NO integral to do
14
Serway Chapter 35
Angle of Reflection
If plane waves are incident on a reflecting surface with the propagation direction of
the waves making angle θ1 with the normal direction to the surface then the reflected
angle θ1prime relative to the surface normal is simply
1 1
ie the incident angles and reflected angles are the same
Refraction
If plane waves traveling through medium 1 are incident on a plane interface between
medium 1 and medium 2 then the angle of incidence of the incoming wave θ1 and
the angle of refraction of the transmitted wave θ2 are related by Snellrsquos law
2 2
1 1
sin
sin
v
v
where v1 and v2 are the wave speeds in medium 1 and medium 2 The angle of
incidence and the angle of refraction are both measured between the wave propagation
direction and the normal to the interface In terms of indices of refraction in the case
of light waves Snellrsquos law takes the more familiar form
1 1 2 2sin sinn n
where ni = cvi
Total Internal Reflection
If a wave is incident from a medium of low wave speed into a medium of high wave
speed the law of refraction requires that the angle of refraction be greater than the
angle of incidence If the angle of refraction is required to be greater than 90deg then no
refracted wave can exist and total internal reflection occurs The critical incident
angle θc beyond which total internal reflection occurs is given by
2
1
sin c
n
n
15
Dispersion of Light
In addition to the speed of light varying from material to material it also varies with
wavelength within each material This means that the index of refraction is generally a
function of wavelength
c
nv
Since the wave speed is not constant such a medium is dispersive meaning in this
context that refraction actually disperses white light into its various colors because
Snellrsquos law gives a different angle for each wavelength In most materials the
variation with wavelength is quite small but this small effect is responsible for some
of the most spectacular color effects we ever see including rainbows a flashing
crystal chandelier and the colored fire of a diamond solitaire by candlelight
16
Serway Chapter 36
Real and Virtual Images
When light rays are focused at a certain plane producing an image if a sheet of white
paper is placed there we call the image a real image The images produced by film
projectors and overhead projectors are examples of real images
When light rays appear to come from a certain location but no image is produced
when a screen is placed there we say that there is a virtual image at that location For
instance when you look in a mirror it appears that someone is behind the mirror but
a screen placed back there in the dark would show nothing Your image in the mirror
is a virtual image
Ray Tracing
There are lots of rules about how to find the images in optical systems but the best
way to keep things straight is to learn how to draw the principal rays for curved
mirrors and lenses The rays for convex and concave mirrors are shown in Fig 3615
The rays for converging and diverging thin lenses are shown in Fig 3627 You should
memorize the principal rays and know how to use them to locate images
Curved Mirrors
The focal length of a curved mirror with radius of curvature R is given by
2
Rf
The relation between object distance p image distance q and focal length f is
1 1 1
p q f
If the mirror is a diverging mirror f should be negative and if q should turn out to be
negative the image is virtual
Lateral Magnification
The lateral magnification in an optical system is defined by the ratio of the image size
to the object size
17
Image height
Object height
qM
p
for curved mirrors
As usual there are sign conventions here too but this formula just gives the
magnitude It is better to keep keep track of upright images versus inverted images by
means of ray diagrams rather than by memorizing sign conventions
Thin Lenses
The focal length of a thin lens is related to the radii of curvature of the two faces R1
and R2 of the lens by
1 2
1 1 11n
f R R
Note that this formula differs from Eq (3611) in the text by not having a minus sign
between the two R-terms We like this form better because for a simple converging
lens like a magnifying glass we just use positive values of R for both surfaces If one
of the faces is concave producing divergence use a negative value for R And if a
surface is flat use R = infin If the face is flat the radius is infinite
The relation between the image object and focal distances for a thin lens is the same
as that for a curved mirror
1 1 1
q p f
Use a negative focal length if the lens is diverging
The lateral magnification for a thin lens is the same as for a curved mirror
Image height
Object height
qM
p
Camera
The lens system in a camera projects a real image of an object onto the film (or CCD
array in a digital camera) The position of the image is adjusted to be on the film by
moving the lens into or out of the camera
18
Eye
The eye is like a camera in that a real image is formed on the retina Unlike a camera
the image position is adjusted by changing the focal length of the lens This is done by
the ciliary muscle which squeezes the lens changing its shape
Near Point The near point is the closest distance from the eye for which the lens can
focus an image on the retina It is usually 18-25 cm for young persons
Far Point The far point is the greatest distance from the eye for which the lens can
focus an image on the retina For a person with normal vision the far point is at
infinity
Nearsightedness A person is nearsighted if their far point is at some finite distance
less than infinity This condition can be corrected with a lens that takes an object at
infinity and produces a virtual image at the personrsquos far point
Farsightedness and Presbyopia A person is farsighted if their near point is too far
away for comfortable near work like reading or knitting This can be corrected by a
lens which takes an object at a normal near point distance of 18-25 cm and produces a
virtual image at the personrsquos natural near point Presbyopia involves a similar
problem which nearly all people experience as they age The ciliary muscle becomes
too weak and the lens becomes too stiff to allow the eye to provide for both near and
far vision The solution for this problem is either reading glasses or bifocal lenses
Reading classes are just weak magnifying glasses mounted on eyeglass frames
Bifocal lenses are split into upper and lower halves The lower half is a lens which
gives the proper correction for near work and the upper half is a different lens for
proper focusing at infinity
Angular Size
When an object is brought closer to the eye it appears to be larger because the image
on the retina is larger The size of this image is directly proportional to the objectrsquos
angular size which is the angle subtended by the object measured from the center of
the lens of the eye In optical instruments which are to be used with the eye the
angular size of the final image is whatrsquos important because it determines how large
the image will appear to the viewer
19
Simple Magnifier
A simple magnifier is a single converging lens or magnifying glass It takes an object
closer to the eye than a normal near point and produces a virtual image at or beyond
this near point The angular magnification is defined to be the ratio of the angular size
when viewed through the lens to the angular size of the object when viewed at the
normal near point (without aid of the lens)
Microscope
This instrument has two lenses (1) The objective is near the object being viewed and
produces a greatly magnified real image (2) The eyepiece is a simple magnifier which
the viewer uses to closely examine the image from (1)
Telescope
This instrument also has two lenses (1) The objective at the front of the telescope
takes light from a distant object and produces a real inverted image (which is rather
small) near its focal point (2) This small real image is then examined by the eyepiece
functioning as a simple magnifier to produce a virtual image with a larger angular
size
20
Serway Chapter 37
Two-Slit Interference
If light is incident on two closely spaced narrow slits a pattern of light and dark
stripes is produced beyond the slits The bright stripes or fringes are caused by
constructive interference of the two waves coming from the slits Constructive
interference occurs whenever two waves arrive at a location in phase with each other
This occurs when the distance x1 from slit 1 to a point P on the screen and the
distance x2 from slit 2 to point P differ by in integral number of wavelengths
1 2 where 0 1 2x x m m
where λ is the wavelength of the light When x1 and x2 are much larger than the slit
spacing d this condition reduces to
sind m
where θ is the angle between the direction of the incident light and the direction of the
light arriving at the screen
Thin Films
When light is partially reflected and partially transmitted by a thin film of transparent
material it is possible to have interference between the wave reflected from the front
of the film and light reflected from the back of the film (The colored reflections from
the thin film of oil on the water in a rain-soaked parking lot are an example of this
effect) It is difficult to write down formulas that will work in all cases so we will
just review the important principles here
1 If the two reflected waves are in phase with each other the film has enhanced
reflection (constructive interference) but if the two reflected waves are out of phase
with each other reflection is diminished (destructive interference) Phase shifts occur
due to reflection and due to the extra path length through the film of the wave
reflected from the back of the film
2 The phase change due to reflection is determined by the difference in index of
refraction between the two media involved in the reflection If the wave is incident
21
from a medium with a low index of refraction into a medium with a high index of
refraction a phase change of 180deg occurs and the reflected wave is inverted If
incident from high to low no phase shift occurs and the reflected wave is non-
inverted
3 The extra path length through the film of the wave reflected from the back of the
film is equal to 2t where t is the thickness of the film (The incident light is assumed
to be normal to the surface of the film) The number of wavelengths contained in the
extra path length is equal to 2tλn where λn = λn is the wavelength of the light in the
film
4 Rules for reflection from thin films
If one ray is inverted and the other is not then we have
12 constructive
2 nt m
2 destructivent m
If either both rays are inverted or both are non-inverted we have
2 constructivent m
12 destructive
2 nt m
22
Serway Chapter 38
Diffraction Grating
A diffraction grating is simply a fancy version of two-slit interference with the two
slits replaced by thousands of slits Just as in the two-slit case the bright fringes
occur at angles given by
sin md
but in this case d the distance between neighboring slits is made to be very small
The effect of having many slits instead of two is to make each bright fringe highly
localized with wide dark regions between neighboring maxima
Single Slit Diffraction
When light passes through an opening in an opaque screen an interference pattern is
produced beyond the opening To understand why we may replace the single opening
by many small coherent sources of light These many sources interfere with each
other producing a pattern known as a diffraction pattern (Note that many authors do
not distinguish between interference and diffraction treating them as interchangeable
terms) If the opening is a slit of width a then the diffraction pattern far from the slit
will have a bright central maximum with a succession of minima and weaker maxima
on either side The angle between the incident direction and the minima is given by
sin ma
where m = plusmn1 plusmn2
If the opening is circular with diameter D the angle between the incident direction
and the first minimum is given by
sin 122D
Optical Resolution and Rayleighrsquos Criterion
Two point sources can just be resolved (distinguished from each other) if the peak of
the diffraction image of the first source overlies the first minimum of the diffraction
image of the second source For circular holes of the kind usually encountered in
23
optical devices this condition is approximately satisfied when the angular separation
between the two sources as viewed from the optical instrument is greater than or equal
to the critical angle
min 122D
where λ is the wavelength of the light and where D is the diameter of the aperture in
the instrument
Polarization
We say that an electromagnetic wave is polarized if its electric field vector doesnrsquot
change direction in a random fashion The simplest kind of polarization is linear
polarization in which the electric field vector oscillates back and forth along the
same axis in space Polarized light can be produced from normal unpolarized light by
selective absorption (as in Polaroid sunglasses) by passing light through crystals that
have different indices of refraction for different polarizations (double refraction) by
scattering (the blue sky is polarized) and by reflection (glare) Polarization by
reflection occurs when light reflects from a shiny insulating (non-metallic) surface
The amount of polarization is greatest for reflection at Brewsterrsquos angle
2
1
tan P
n
n
where θP is the incidence angle of light from medium 1 onto medium 2 and where n1
and n2 are the indices of refraction for the media (In our everyday experience n1 = 1
since the light comes in through the air and n2 is the index of refraction of the shiny
insulating material producing the glare eg water glass plastic paint etc)
Malusrsquos Law
The intensity of transmitted polarized light through a perfect polarizer is related to the
incident intensity of polarized light by Malusrsquos law
2cosoI I
where Io is the intensity of the incident light and where θ is the angle between the
electric field vector in the incident wave and the transmission axis the polarizer
24
Serway Chapter 39
Principles of Relativity
All of the weirdness of relativity flows from two simple principles
(1) The laws of physics must be the same in all inertial (non-accelerating) reference
frames
(2) The speed of light in vacuum has the same value in all inertial reference frames
Note that (1) seems reasonable but (2) is very odd It says that if two space ships are
approaching each other at nearly the speed of light and a laser pulse is shot from ship
1 toward ship 2 then when the people on ship 2 measure the speed of the pulse as it
goes by it is moving at 3 times 108 ms the same as if the ships were stationary
Simultaneity
If observer 1 sees two events in her own frame as simultaneous at two different
locations a moving observer 2 will see these two events happening at different times
Relativistic gamma
The factor γ (gamma) appears regularly in the formulas of relativity
2
2
1
1 vc
where v is the relative speed between two inertial frames
Time Dilation
If observer 1 sees two events at the same location in space separated by time Δtp in his
own frame then observer 2 moving at speed v relative to observer 1 will see these
two events separated by a longer time Δt
pt t
Moving clocks run slow
Length Contraction
If observer 1 measures the length of an object along the x-direction at some instant of
time in her frame to be Lp then observer 2 moving in the x-direction at speed v will
measure the length L of the object to be shorter
pLL
25
Moving meter sticks are short
Velocity Addition
If a particle is moving at velocity vac relative to frame c and if frame c is moving at
speed vcb with respect to frame b (with both velocities directed along the same line)
what would be the velocity of the particle in frame b In classical physics the answer is
ab ac cbv v v
but if any of these velocities are an appreciable fraction of the speed of light the
answer changes
21ac cb
abac cb
v Vv
v v c
In using this equation make sure that vac has the same sign in the numerator and in the
denominator and that vbc also has the same sign in both places Note that this equation
is different (and in our opinion easier to understand) than the equations in the text
Momentum and Energy
The momentum and total energy of a particle moving at velocity v (its speed is v v )
are given by
22
2 2
22
1 1 vvc c
m mcm E mc
vp v
Rest Energy
When a particle is at rest (v = 0 and γ = 1) then E = mc2 is called the rest energy It
includes the potential energy of the particle so that if we change the potential energy
of a particle we change its rest mass
Kinetic Energy The kinetic energy is the difference between the total energy and the
rest energy
2 2K mc mc
For v ltlt c this reduces to K = mv22
General Relativity
Einsteinrsquos principle of equivalence in the theory of general relativity is In the vicinity
of a point a gravitational field is equivalent to an accelerated frame of reference in
26
the absence of gravitational effects For example there is no difference whatever in
the physics experienced by an astronaut on earth and one who is in interstellar space
far away from any stars or planets but who has just turned on her spaceshiprsquos engines
and is accelerating at 98 ms2
27
Serway Chapter 40
Photons
We now know that even though light behaves like a wave it is better described as
consisting of small packets of energy called photons The energy of a photon is related
to its frequency by
E hf
where h is Planckrsquos constant
346626 10 J sh
Photons also carry momentum given by
hf hp
c
Since the amount of energy in each photon is fixed the difference between dim light
and intense light (of the same wavelength) is that dim light consists of fewer photons
than intense light
Photoelectric Effect
In the photoelectric effect light shining onto a metal surface gives the electrons in the
metal enough energy to escape and be detected It requires a certain amount of energy
(called the work function typically = 3 ndash 5 eV) just to kick the electron out of the
metal so the light must deliver at least this much energy to an electron to produce the
effect It is observed that red light no matter how intense never produces electrons
But ultraviolet light even if quite dim will eject electrons from the metal Since red
light consists of 2 eV photons and ultraviolet light has photons with energies around
4-6 eV the photon idea explains the behavior of the photoelectric effect The
maximum energy that an ejected electron can have is
maxK hf
where f is the photon frequency The minimum frequency that light can have and cause
electrons to be ejected is called the cutoff energy and is found by setting Kmax = 0
cf h
28
Compton Effect
When high-frequency light interacts with free electrons the scattered light does not
have the same wavelength as the incident light contrary to what classical
electromagnetism would predict But the photon picture in which photons have
momentum and energy allows us to treat scattering as a collision between the photon
and the electron In this collision the electron and the scattered photon both have a
different momentum and energy than they did before And since p = hλ for a photon
if the momentum is different the wavelength will also be different
after before 1 cose
h
m c
where the angle θ is the angle between the incoming photon direction and the
direction of the scattered photon
Particles are Waves
Since photons behave like particles sometimes it is not surprising that elementary
particles can behave like waves sometimes The wavelength of a particle with
momentum p is given by
h
p
Wave-particle Duality
Both photons and elementary particles have a dual nature sometimes they behave like
particles and sometimes they behave like waves It is hard for us to comprehend the
nature of such an object by making mental pictures but experiments definitively show
that this is the case Since we have no direct experience with photons and elementary
particles (because their energies are so much smaller than the energies of the everyday
objects we encounter) it is perhaps not surprising that we have a hard time forming a
mental picture of how they behave
Electron Interference
Since an electron is both a particle and a wave just like a photon it should be able to
produce an interference pattern This is observed If an electron beam is shot at two
closely-spaced slits and if the electrons that pass through the slits are detected
downstream it is found that there are some locations where electrons are never detected
29
and others where lots of electrons are detected The pattern is exactly the same as the
one observed for light waves provided that we use the electron wavelength λ = hp in
place of the wavelength of light This pattern is observed even though each electron is
detected as a single dot on the screen Only after many such dots are collected does the
pattern emerge And if we try to understand how this effect could possibly work by
looking closely at each slit to see which one the electron came through the pattern
disappears the act of measurement destroys the interference This means that each
single electron somehow comes through both slits (as a wave would)
Uncertainty Principle
In classical physics we always imagine that the positions and momenta of moving
particles have definite values It might be hard to measure them but surely at each
instant of time a particle should be precisely located at some point in space and have a
similarly precise momentum This turns out not to be true Instead both position and
momentum are required to be uncertain with their uncertainties Δx and Δp satisfying
the Heisenberg uncertainty relation
2x p
So if the particle were known to be precisely at some particular location (so that Δx =
0) we couldnrsquot know anything about its momentum (Δp = infin) and if its momentum
were exactly known we couldnrsquot know its position
There is a similar relation involving the particlersquos energy E and the time interval Δt
over which this energy is measured
2E t
This means that energy is not actually conserved if we are considering very small time
intervals and this brief non-conservation of energy has been observed
30
Serway Chapter 41
Wave Function ψ and Probability
Quantum mechanics does not predict exactly what an electron or a photon will do Instead
it specifies the wave function or probability amplitude ψ of an electron or a photon
This wave function is a complex-valued function of space and time whose squared
magnitude is the probability density P for finding a particle at a particular place in
space at a certain time
2P
where is the complex conjugate of ψ
For example the wave function of an electron with perfectly specified momentum p
would have a wavelength given by
h
p
and its probability amplitude would be proportional to
2i x ipxe e
The corresponding probability density would then be
21ipx ip ipxP e e e
which means that the electron is equally probable to be anywhere along the x axis
This is in accord with the uncertainty principle since we specified the momentum
precisely we canrsquot have any idea about the position of the electron
As another example you have probably seen ldquofuzzy ballrdquo drawings of electron
orbitals in a chemistry book These fuzzy balls are meant to indicate the distribution
of the probability density 2
P in the orbital
Particle in a Box
A simple example in which we can calculate the wave function is the case of a particle of
mass m constrained to be inside a 1-dimensional box between x = 0 and x = L In this
case the general wave function is a linear superposition of wave functions ψn of the form
sin niE tn
n xx t A e
L
31
where A is a positive constant where n = 1 2 3 and where the energy associated
with each of the quantum states ψn is given by
22
28n
hE n
mL
This wave function is zero at x = 0 and x = L which means that the particle will never
be found at the walls of the box The wave function has maximum values in the
interior and at these places the particle is most likely to be found and it also has
places where it is zero and at these places the particle will also never be found as
expressed by the formula
2sinn x
PL
The particle in the box is interfering with itself producing a probability interference
pattern across the box just like the interference patterns we studied with light and
sound
32
Serway Chapter 42
Atoms
Many people picture an atom as a miniature solar system where electrons orbit around
a massive nucleus at the center This picture is misleading because in atoms the wave
nature of electrons dominates The electrons form 3-dimensional standing waves
(called orbitals) centered at the nucleus And if you ask what is it thatrsquos waving the
answer is ψ the probability amplitude (see Chapter 41)
Standing waves on a string can occur only for certain frequencies (the fundamental
and higher harmonics) Similarly atomic orbitals occur only for certain energies For
the hydrogen atom the energies of the orbitals have the a particularly simple form
2
1136 eVnE
n
where n = 1 2 3 is called the principal quantum number For other atoms the
determination of the orbital energies requires numerical calculation by computers
Atomic Spectra
If an electron is somehow given extra extra energy (we say that it is excited) so that it
occupies a higher orbital it will eventually ldquofallrdquo back down to a lower orbital Each
time an electron falls to a lower orbital it loses the difference in energy between the
two orbitals in the form of a photon Since the orbital energies are discrete so are the
energy differences and so are the wavelengths of the emitted light The entire set of
these discrete wavelengths is called the atomic spectrum and it is unique to each type
of atom For hydrogen the spectrum can be simply written as
H 2 2
1 1 1
f i
Rn n
where RH = 10973732 times 107 m-1 For other atoms the spectrum cannot be expressed as
a formula
Orbital Angular Momentum Electrons in atoms also have quantized values of
angular momentum The orbital quantum number ℓ specifies the value of this
quantized angular momentum through the formula
1L
33
If we want to know the value of the angular momentum along some direction in space
say the z direction the answer is not L but rather
zL m
where mℓ is another quantum number which runs from
1 1m
This quantum number is important when an atom sits in a magnetic field
Spin Angular Momentum It has been found experimentally that electrons and other
charged particles also carry internal angular momentum which we call spin
Electrons have an intrinsic spin angular momentum s along a specified axis that is
extremely quantized it can only take on 2 possible values
1
2zs s
We thus say that electrons have spin s = 12 or that electrons are rdquospin one-half parti-
cles Since s is an angular momentum it obeys the same rule as that for orbital angular
momentum namely that if its value along some axis is s then its total magnitude is
given by
31
2S s s
I know this seems weird but quantum mechanics is weird The only excuse for this
bizarre way of looking at the world is that it predicts what happens in experiments
Exclusion Principle The answer to the question of how many electrons (or any other
spin one-half particle) can be in one particular quantum state was discovered by
Wolfgang Pauli and is called the exclusion principle
ldquoNo two electrons can ever be in the same quantum state therefore no two electrons
in the same atom can have the same set of quantum numbersrdquo
This is the reason that we have atoms with different properties instead of every atom
simple having all of its electrons in the ground state All of the variety we see around
us in the world is the result of chemical differences and these differences would not
exist unless electrons obeyed this important principle The entire structure of the
periodic table (see pages 1377-1379 in Serway) is an expression of this principle
34
Serway Chapter 44
Nuclear Properties
The nucleus is composed of protons with charge +e and mass mp = 1007226 u and of
neutrons with zero charge and mass mn = 1008665 u where u is one atomic mass unit
-271 u=1660540 10 kg
The atomic number Z counts the number of protons in a nucleus while the neutron
number N counts the number of neutrons The mass number A is the sum of the two
A N Z
Protons and nuetrons have about the same mass and some times called baryons (heavy
ones) or nucleons (partcles in the nucleus) A is sometimes called the baryon
number The chemical elements are distinguished by Z ie hydrogen has Z = 1 iron
has Z = 26 uranium has Z = 92 etc But for a given Z there might be several nuclei
with different numbers of neutrons These nuclear siblings are called isotopes
These numbers are used to label nuclei according to the pattern
5626 ie FeA
Z X
denotes the isotope of iron with N = 56 ndash 26 = 30 neutrons
The nucleus is roughly spherical with a radius given approximately by
1 3 150 0where 12 10 mr r A r
Nuclear Stability
Because the positively charged protons electrically repel each other with an enormous
force at distances as small as 10-15 m there must be some really strong force that that
overcomes electrical repulsion to hold protons and neutrons together This force is
called with some lack of imagination the strong force It is a very short-range force
(it only acts over a distance of about 2 x 10-15 m) and attracts protons to protons
neutrons to neutrons and neutrons to protons But in spite of this strong nuclear force
the coulomb repulsion of the protons is still present so anything that might keep the
protons from being right next to each other would help keep the nucleus from
35
exploding This role is played by the neutrons and for nuclei with Z le 20 the stable
nuclei roughly have N = Z
For Z gt 20 the coulomb repulsion force is more powerful and more neutrons are
needed to dilute it up to about N = 15Z around Z = 80 For Z gt 83 no amount of
neutrons can help and these nuclei are unstable (radioactive) There is an ldquoisland of
(relative) stabilityrdquo around Z=90-92 (thorium and uranium respectively) These
elements have one or two isotopes with half-lifes of billions of years so there are
substantial amounts of such elements on Earth This fact makes it possible to have
practical fission devices
Radioactive Decay
There are three types of radioactive decay
Alpha decay the nucleus kicks out a helium nucleus (N = 2 Z = 2 A = 4) So if the
original nucleus is called X and the new nucleus is called Y then the decay would look
like this
4 42 2X Y+ HeA A
Z Z
Beta decay the nucleus either kicks out an electron (endash) or its positively-charged
antimatter twin the positron (e+) plus either an electron neutrino v or an electron
anti-neutrino v
1X Y+e electron decayA AZ Z v
1X Y+e positron decayA AZ Z v
A neutrino is a particle with no charge hardly any mass (much less than the electron
mass) and interacts so weakly with matter that most neutrinos upon encountering the
planet earth just pass right through it as if it werenrsquot there
Gamma decay the nucleons in the nucleus X are in an excited energy state X
(perhaps as a result of having undergone alpha or beta decay) and they drop down to a
lower energy state shedding the energy as a high frequency photon
X XA AZ Z
36
This process is exactly analogous to the way that the electrons in atoms emit photons
Decay Rate and Half Life
There is no way to predict exactly when an unstable or excited nucleus will decay but
there is an average rate at which this decay occurs called the decay constant λ The
meaning of this constant is that if there are a large number N of nuclei in a sample
then the number of decays per second that will be observed (called the decay rate R)
is R = λN In mathematical language
dMR N
dt
This simple differential equation has for its solution
0tN t N e
where N0 is the number of nuclei in the sample at time t = 0
The half-life is the time it takes for half of the nuclei in the sample to decay and is
related to the decay constant by
1 2
ln 2 0693T
Disintegration Energy
When a nucleus decays it is making a transition to an overall state of lower energy
which means according to Einsteinrsquos famous formula E = mc2 that the sum of the
masses after the decay must be less than the mass before with the lost mass appearing
as kinetic energy among the decay products For example in alpha decay this kinetic
energy called the disintegration energy Q is given by
2X YQ M M M c
37
Serway Chapter 45
Nuclear energy
There are two ways to extract energy from the nucleus fission and fusion For nuclei
with Z greater than 26 breaking the nucleus apart into pieces leads to a lower
2mc energy than the original nucleus so energy can be extracted by fission For
nuclei with Z less than 26 a lower energy is achieved by combining nuclei so energy
can be extracted by fusion
Fission
Since neutrons have no charge they are not repelled from nuclei as protons are for
this reason their behavior is the key to understanding how fission works
Fast neutrons mostly bounce off other nuclei with each collision slowing the neutron
down This elastic energy loss is most effective if the other nuclei have low mass (like
hydrogen) and these materials are called moderators because of their ability to slow
down fast neutrons
The reason that slowing neutrons is important is that slow neutrons are much more
likely to be absorbed by a nucleus which then leads to nuclear reactions of various
kinds For a few very large nuclei like uranium-235 and some plutonium isotopes
absorption of a slow neutron causes the nucleus to split into two large fragments plus
2 or 3 fast neutrons (energy is released in the process too) Thus one slow neutron
can produce energy plus 2 or 3 more neutrons which if moderated from fast to slow
can split 2 or 3 more nuclei leading to a runaway chain reaction If the reaction is
allowed to proceed unchecked it produces a large explosion If a neutron absorbing
material is added to the mix (like the cadmium in reactor control rods) it is possible to
keep the reaction under control and to extract the released energy as heat to drive
steam turbines and produce electricity
Fusion
Fusion involves mashing two nuclei together and since they are both charged and repel
each other this reaction is much harder to make go The nuclei must have enough energy to
overcome the coulomb repulsion which is why this reaction requires a high temperature
(hundreds of millions of degrees K) like that in the sun (which is burning hydrogen to
helium via fusion) or like that in the center of the fission explosion that is used to detonate
38
a hydrogen bomb
This reaction is of interest for power production in spite of this difficult temperature
requirement because of the abundance of fusion fuel on the planet There are about
012 g of deuterium ( 21H ) in every gallon of water on earth and it only costs about 4
cents to extract it The fusion energy available from this minuscule amount of
deuterium would run a 1000-Megawatt power plant for 10 seconds Water is so
abundant on earth that if fusion were to work we would have an essentially
inexhaustible source of energy
So why donrsquot we have fusion power plants Well the fuel is cheap but the match is
incredibly expensive The only way we know to control this difficult high-temperature
reaction is with large and expensive pieces of equipment involving either large
magnetic fields and complex high-power electromagnetic antennas or with gigantic
(football-fieldsized) laser facilities involving more than a hundred of the highest-
energy lasers ever built Power plants based on these current methods for controlling
fusion are unattractive to the fiscally-minded people who run the electric power
industry Hopefully better designs will be discovered as experiments continue
39
c actinium 68 Er erbium 101 Md mendelevium 104 Rf rutherfordium l aluminum 63 Eu europium 80 Hg mercury 62 Sm samarium m americium 100 Fm fermium 42 Mo molybdenum 21 Sc scandium b antimony 9 F fluorine 60 Nd neodymium 106 Sg seaborgium r argon 87 Fr francium 10 Ne neon 34 Se selenium s arsenic 64 Gd gadolinium 93 Np neptunium 14 Si silicon t astatine 31 Ga gallium 28 Ni nickel 47 Ag silver a barium 32 Ge germanium 41 Nb niobium 11 Na sodium k berkelium 79 Au gold 7 N nitrogen 38 Sr strontium e beryllium 72 Hf hafnium 102 No nobelium 16 S sulfur i bismuth 108 Hs hassium 76 Os osmium 73 Ta tantalum h bohrium 2 He helium 8 O oxygen 43 Tc technetium
boron 67 Ho holmium 46 Pd palladium 52 Te tellurium r bromine 1 H hydrogen 15 P phosphorus 65 Tb terbium d cadmium 49 In indium 78 Pt platinum 81 Tl thallium a calcium 53 I iodine 94 Pu plutonium 90 Th thorium f californium 77 Ir iridium 84 Po polonium 69 Tm thulium
carbon 26 Fe iron 19 K potassium 50 Sn tin e cerium 36 Kr krypton 59 Pr praseodymium 22 Ti titanium s cesium 57 La lanthanum 61 Pm promethium 74 W tungsten l chlorine 103 Lr lawrencium 91 Pa protactinium 92 U uranium r chromium 82 Pb lead 88 Ra radium 23 V vanadium o cobalt 3 Li lithium 86 Rn radon 54 Xe xenon u copper 71 Lu lutetium 75 Re rhenium 70 Yb ytterbium m curium 12 Mg magnesium 45 Rh rhodium 39 Y yttrium b dubnium 25 Mn manganese 37 Rb rubidium 30 Zn zinc y dysprosium 109 Mt meitnerium 44 Ru ruthenium 40 Zr zirconium s einsteinium
40
UNITS The SI units are given in parentheses Other commonly used units are given in terms of the SI units Symbols conform with the recommendations of the American National Standards Institute (ANSI) and the American Institute of Physics (AIP) Commonly used multiples of SI units
prefix symbol factor tera T 1012 giga G 109
mega M 106 kilo k 103
centi c 10-2 milli m 10-3
micro μ 10-6 nano n 10-9 pico p 10-12
femto f 10-15
length meter (m) angstrom 1 Ǻ = 10-10 m inch 1 in = 254 x 10-2 m foot 1 ft = 03048 m mile 1 mi = 1609 m
mass kilogram (kg) atomic mass unit 1 u = 1661 x 10-27 kgslug 1 slug = 1459 kg
force newton (N = kg bull ms2) dyne 1 dyn = 10-5 N pound 1 lb = 4448 N
pressure Pascal (Pa = kgm bull s2) atmosphere 1 atm = 1013 x 105 Pa poundssquare inch 1 psi = 6895 Pa cm of mercury 1 cm Hg = 1333 Pa bar 1 bar = 1000 x 105 Pa torr 1 torr = 1333 Pa
time second (s) minute 1 min = 60 s hour 1 h =3600 s
frequency hertz (Hz = s-1) radianssecond 1 rads = 12π Hz
energy joule (J = kg bull m2s2) erg 1 erg = 10-7 J electron volt 1 eV = 1602 x 10-19 J calorie 1 cal = 4187 J kilowatt-hour 1 kWh = 36 x 106 J British thermal unit 1 Btu = 1055 J
power watt (W = kg bull m2s3) horsepower 1 hp = 7457 W
charge coulomb (C = A bull s)
electric potential volt (V = kg bull m2 s3 bull A)
current ampere (A)
resistance ohm (Ω = kg bull m2s3 bull A2)
capacitance farad (F = s4 bull A2kg bull m2)
magnetic field tesla (T = kgs2 bull A) gauss 1 G = 10-4 T
magnetic flux weber (Wb = kg m2s2 bull A) maxwell 1 Mx = 10-8 Wb
magnetic inductance henry (H = kg bull m2s2 bullA2)
temperature kelvin (K) degrees Celsius 0degC = 27315 K
angle radian (rad) degree 1 deg = π180 rad revolution 1 rev = 2π rad
41
Some Physical Constantsa Quantity Symbol Valueb Atomic mass unit μ 1660 538 73 (13) x 10-27 kg
931494 013 (37) MeVc2 Avogadros number NA 6022 141 99 (47) x 1023 particlesmol Avogadrorsquos number 6022 x 1023 mol Bohr magneton
2B e
em 9274 008 99 (37) x 10-24 JT
Bohr magneton μB 927 x 10-24 JT Bohr radius
0
2
2e em e k
a 5291 772 083 (19) x 10-11 m
Boltzmanns constant B A
RNk 1380 650 3 (24) x 10-23 JK
Boltzmannrsquos constant kB 1380 x 10-23 JK Compton wavelength
C eh
m c 2426 310 215 (18) x 10-12 m
Deuteron mass md 3343 583 09 (26) x 10-27 kg 2013 553 212 71 (35) u
electron charge e 1602 x 10-19 C Electron mass me 9109 381 88 (72) x 10-31 kg
5485 799 110 (12) x 10-4 u 0510 998 902 (21) MeVc2
electron mass 911 x 10-31 kg Electron volt eV 1602 176 462 (63) x 10-19 J Elementary charge e 1602 176 462 (63) x 10-19 C Gas constant R 8314 472 (15) JKmol Gravitational constant G 6673 (10) x 10-11 Nm2kg2 Neutron mass mn 1674 927 16 (13) x 10-27 kg
1008 664 915 78 (55) u 939565 330 (38) MeVc2
neutron mass 1675 x 10-27 kg Nuclear magneton
2n p
em 5050 783 17 (20) x 10-27 JT
permeability constant μ0 1257 x 10-6 Hm permittivity constant ε0 8854 x 10-12 Fm Plancks constant h
2h
6626 068 76 (52) x 10-34 Js 1054 571 596 (82) x 10-34 Js
Planckrsquos constant h ħ
6626 x 10-34 Js 1055 x 10-34 Js
Proton mass mp 1672621 58 (13) x 10-27 kg 1007 276 466 88 (13) u 938271 998 (38) MeVc2
proton mass 1673 x 10-27 kg Rydberg constant RH 1097 373 156 854 9 (83) x 107 m-1 Speed of light in vacuum c 2997 92458 x 108 ms (exact) speed of light c 300 x 108 ms a
These constants are the values recommended in 1998 by CODATA based on a least-squares adjustment of data from different measurements For a more complete list see P J Mohr and B N Taylor Rev Mod Phys 72351 2000 b The numbers in parentheses for the values above represent the uncertainties of the last two digits
NOTE The ones in red are the ones from Appendix 13 The ones in yellow are the ones that are from the Serway book table but that were already on the Appendix 13 table
42
INDEX
Absolute zero 9
Adiabatic 17
Adiabatic exponent 18
Adiabatic process 17
Alpha decay 46
Amplitude 4
Angle of reflection 25
Angular frequency 3
Angular magnification 30
Angular momentum atomic 43
Angular size 29
Archimedes Principle 1
Atomic spectra 43
Atoms 43
Avogadrorsquos number 10
Beats 8
Bernoullirsquos Equation 2
Beta decay 46
Bifocals 29
Brewsterrsquos angle 34
British Thermal Unit Btu 11
Buoyancy 2
calorie 11
Calorie 11
Camera 28
Carnot cycle 21
Carnot efficiency 22
Celsius scale 9
Ciliary muscle 29
Coefficient of performance 21
Compton effect 39
Constructive interference 7
Continuity equation of 2
Contraction length 35
Convection 14
Curved mirrors 27
Decay constant 46
Decay rate radioactive 46
Decibel scale 5
Degrees of freedom 16
Density 1
Destructive interference 7
Diffraction single slit 33
Diffraction grating 33
Dilation time 35
Disintegration energy 47
Dispersion of light 26
Doppler effect 6
Double slit interference 31
Efficiency 20
Electron interference 39
Emissivity 15
Energy nuclear decay 47
Energy relativistic 36
43
Engineering work 20
Entropy 22
Entropy ideal gas 24
Equipartition of energy 18
Equivalence principle 36
Exclusion principle 44
Expansion thermal 9
Expansion coefficient linear 9
Eye 29
Fahrenheit scale 9
Far point 29
Farsightedness 29
First Law of Thermodynamics 12
Fission 48
Flux volume 2
Free expansion 23
Fringes 31
Fusion 48
Fusion heat of 11
Gamma relativistic 35
Gamma decay 46
General relativity 36
Half-life 46
Heat 11
Heat capacity 11
Heat conduction 13
Heat engine 20
Heat of fusion 11
Heat of vaporization 11
Heat pump 20
Heats of transformation 11
Hydrostatics 1
Ideal Gas Law 10
Images real and virtual 27
Intensity sound 5
Interference 7
Interference two-slit 31
Internal energy 12
Internal energy degrees of freedom 16
Irreversible process 19 23
Isotopes 45
Joule 11
Kelvin scale 9
Kinetic energy relativity 36
Kinetic theory 16
Latent heat 11
Length contraction 35
Linear expansion coefficient 9
Linear polarization 34
Linear superposition 7
Longitudinal wave 3
Loudness 5
Magnification lateral 27
Magnifying glass 30
Malusrsquos law 34
Microscope 30
Momentum relativistic 36
Muscle ciliary 29
44
Musical instruments 8
Musical scale 8
Near point 29
Nearsightedness 29
Nonlinear 7
Nuclear energy 48
Nuclear properties 45
Nuclear stability 45
Octave 8
Optical resolution 33
Orbital quantum number ℓ 43
Orbitals 43
Particle in a Box 41
Particles are waves 39
Pascalrsquos Principle 1
Period 3
Photoelectric effect 38
Photons 38
Pitch 8
Polarization 34
Power sound 5
Presbyopia 29
Pressure 1
Principal quantum number n 43
Principle of equivalence 36
Principle of linear superposition 7
Probability amplitude ψ 41
Processes thermodynamic 13
R-value 14
Radiation thermal 14
Radioactive decay 46
Radioactive decay rate 46
Ray tracing 27
Rayleighrsquos criterion 33
Reading glasses 29
Real image 27
Refraction 25
Refrigerator 20
Relativistic gamma 35
Relativity principles 35
Resolved for light sources 33
Rest energy 36
Reversible process 19 22
Rope wave speed 4
Second Law of Thermodynamics 19
Shock waves 6
Simple magnifier 30
Simultaneity 35
Single slit diffraction 33
Snellrsquos law 25
Sound speed 5
Specific heat 11
Spin Angular momentum 44
Standing waves 7
Stefanrsquos law 14
Strong force 45
Telescope 30
Temperature 9
45
Temperature Scales 9
Thermal conductivity 14
Thermal energy 12
Thermal expansion 9
Thin film interference 31
Thin lenses 28
Time dilation 35
Tone musical 8
Total internal reflection 25
Transverse wave 3
Traveling Waves 3
Two-slit interference 31
Uncertainty principle 40
Vaporization heat of 11
Velocity addition relativity 36
Virtual image 27
Volume flux 2
Wave function 41
Wave Function ψ and Probability 41
Wave speed 3
Wave-particle duality 39
Wavelength 3
Wavenumber 3
Wien Displacement Law 16
Work 11
3
h cT TA
R
P
in units of Btuhour Note that A must be in square feet and the temperatures must be
in degrees Fahrenheit
Convection
Convection is the transfer of thermal energy by flow of material For instance a home
furnace doesnrsquot heat a house by waiting for the heat from the burner to slowly conduct
throughout the house instead it quickly pumps warm air to all of the rooms
Generally convection is a much faster way to transfer heat than conduction
Radiation
Electromagnetic radiation can also transfer heat When you warm yourself near a
campfire which has burned itself down into a bed of glowing embers you are
receiving radiant heat from the infrared portion of the electromagnetic spectrum The
rate at which an object emits radiant heat is given by Stefanrsquos law
4AeTP
where P is the radiated power in watts σ is a constant
8 2 45696 10 W m K
A is the surface area of the object in m2 and T is the temperature in kelvins The
constant e is called the emissivity and it varies from substance to substance A perfect
absorber (think black velvet) has e = 1 while a perfect reflector (think mirror) has e =
0 Hence black objects radiate very well while shiny ones do not Also an object that
is hotter than its surroundings radiates more energy than it absorbs whereas an object
that is cooler than its surroundings absorbs more energy than it radiates
Terminology
Transfer variables vs state variables
Energy transfer by heat as well as work done depends on the initial final and
intermediate states of the system They are transfer variables But their sum (Q + W =
4
Eint) is a state variable
Figure 205
5
Serway Chapter 21
Kinetic Theory
The ideal gas law works for all atoms and molecules at low pressure It is rather
amazing that it does Kinetic theory explains why The properties of an ideal gas can
be understood by thinking of it as N rapidly moving particles of mass m As these
particles collide with the container walls momentum is imparted to the walls which
we call the force of gas pressure In this picture the pressure is related to the average
of the square of the particle velocity 2v by
22 1( )
3 2
NP mv
V
Using the ideal gas law we obtain the average translational kinetic energy per
molecule
21 3
2 2 Bmv k T
The rms speed is then given by
2rms
3 3Bk T RTv v
m M
where M is the molecular mass in kgmol
Degrees of Freedom
Roughly speaking a degree of freedom is a way in which a molecule can store energy
For instance since there are three different directions in space along which a molecule
can move there are three degrees of freedom for the translational kinetic energy
There are also three different axes of rotation about which a polyatomic molecule can
spin so we say there are three degrees of freedom for the rotational kinetic energy
There are even degrees of freedom associated with the various ways in which a
molecule can vibrate and with the different energy levels in which the electrons of
the molecule can exist
Internal Energy and Degrees of Freedom The internal energy of an ideal gas made
up of molecules with J degrees of freedom is given by
int 2 2 B
J JE nRT Nk T
6
Heat Capacities of an Ideal Gas The heat capacity of a gas is described by means of
molar heat capacities CV and CP These are the heat capacities per mole and the
subscript V on CV means that the volume is being held constant while for CP the
pressure is held constant For example to raise the temperature of n moles of a gas
whose pressure is held constant by 10 K we would have to supply an amount of heat Q
= nCP (10) K
Molar Specific Heat of an Ideal Gas at Constant Volume
VQ nC T
3monatomic
2VC R
5diatomic
2VC R
5polyatomic
2VC
Real gases deviate from these formulas because in addition to the translational and ro-
tational degrees of freedom they also have vibrational and electronic degrees of
freedom These are unimportant at low temperatures due to quantum mechanical
effects but become increasingly important at higher temperatures The rough rule is
No of degrees of freedom
2VC R
Molar Specific Heat of an Ideal Gas at Constant Pressure
PQ nC T
P VC C R
The internal energy of an ideal gas depends only on the temperature
int VE nC T
Adiabatic Processes in an Ideal Gas
7
An adiabatic process is one in which no heat is exchanged between the system and the
environment When an ideal gas expands or contracts adiabatically not only does its
pressure change as expected from the ideal gas law but its temperature changes as
well Under these conditions the final pressure Pf can be computed from the initial
pressure Pi and from the final and initial volumes Vf and Vi by
or constantf f i iP V PV PV
where γ = CPCV The quantity γ is called the adiabatic exponent Note that this
doesnrsquot mean that the ideal gas law no longer holds it does and in fact it can be
combined with the adiabatic law for pressure given above to obtain the adiabatic law
for temperatures
1 constantTV
Compressions in sound waves are adiabatic because they happen too rapidly for any
appreciable amount of heat to flow This is why the adiabatic exponent appears in the
formula for the speed of sound in an ideal gas
RTv
M
Note that v depends only on T and not on P Because it depends only on the
temperature the speed of sound is the same in Provo as at sea level in spite of the
lower pressure here due to the difference in elevation
Equipartition of Energy
Every kind of molecule has a certain number of degrees of freedom which are
independent ways in which it can store energy Each such degree of freedom has
associated with it ndash on average ndash an energy of 12 Bk T per molecule (or 1
2 RT per mole)
(Note since a molecule has so many possible degrees of freedom it would seem that
there should be a lot of 12 sBk T to spread around But because energy is quantized
some of these degrees of freedom are not ldquoactiverdquo until the temperature becomes high
enough that 12 Bk T is as big as the lowest quantum of energy
8
Serway Chapter 22
Second Law of Thermodynamics
There are several equivalent forms of this important law
Kelvin It is not possible to change heat completely into work with no other change
taking place Or in other words there are no perfect heat engines
Clausius It is not possible for heat to flow from one body to another body at a higher
temperature with no other change taking place Or in other words there are no
perfect refrigerators
Entropy In any thermodynamic process that proceeds from one equilibrium state to
another the entropy of the system + environment either remains unchanged or
increases The total entropy never decreases This law is a bit of an oddity among the
laws of physics because it is not absolute Things are forbidden by the second law not
because it is impossible for them to happen but because it is extremely unlikely for
them to happen (See below for more information about entropy)
Reversible and Irreversible Processes
A reversible process is one which occurs so slowly that it is in thermal equilibrium (or
very nearly so) at all times A hallmark of such processes is that a motion picture of
them looks perfectly normal whether run forward or backward Imagine for instance
the slow expansion of a gas at constant temperature in a cylinder whose volume is being
increased by a slowly moving piston Run the movie backwards and what do you see
You see the slow compression of a gas at constant temperature which looks perfectly
normal
An irreversible process is one which occurs in such a way that thermal equilibrium is
not maintained throughout the process The mark of this kind of process is that a motion
picture of it looks very odd when run backward Imagine the sudden expansion of a gas
into a previously evacuated chamber because a hole was punched in the wall between a
pressurized chamber and the evacuated one Run the movie backward and what do you
see You see the gas in the soon-to-be-evacuated chamber gather itself together and
stream through a tiny hole into a chamber in which there is already plenty of gas If you
have ever seen this happen get in touch with the support group for those who have
witnessed the spontaneous combustion of large mammals by calling 1-800-PYROCOW
9
Heat Engines
Because of the vexing difference in sign between work done on and system and work done
by a system we will invent a new work variable Weng Heat engines do work and so the net
W for these engines is negative But in engineering applications hidden minus signs are
regarded as evil so for heat engines we donrsquot talk about W instead we talk about its
magnitude engW W So for heat engines the first law is
int engE Q W
But for heat pumps and refrigerators work is done on the system so we use the usual work
W when we talk about these systems
A heat engine is a machine that absorbs heat converts part of it to work and exhausts the
rest The heat must be absorbed at high temperature and exhausted at low temperature If the
absorbed heat is Qh the exhausted heat is Qc and the work done by the engine is Weng then
eng h cW Q Q
and the efficiency of the engine is defined to be
eng
h
We
Q
A perfect engine would convert the heat hQ completely into work Weng giving an effi-
ciency of e = 1 Energy conservation alone allows a perfect engine but the second law
requires e lt 1
Refrigerators and Heat Pumps
A refrigerator is a machine that absorbs heat at low temperature and exhausts it at high
temperature the ldquobackwardsrdquo heat transfer being driven by the work done on the machine
by some source of power A heat pump is a machine that either works like a refrigerator
keeping a place cold by transferring heat from this cold place to a higher temperature
environment (cooling mode like an air conditioner) or it functions as a heater
transferring heat into a warm place from a cooler one (heating mode like a window unit
that heats a house by extracting thermal energy from the cold outdoors) The coefficient
of performance of a refrigerator or of a heat pump in cooling mode is defined to be
COP cooling modecQ
W
10
For a heat pump in heating mode the coefficient of performance is
COP heating modehQ
W
Note that we donrsquot have to use the engineering work here because in these systems
work is done on the system and W is naturally positive
A perfect refrigerator would take in heat Qc and exhaust the same amount of heat without
doing any work giving an infinite coefficient of performance Energy conservation alone
allows a perfect refrigerator but the second law requires COP lt infin A perfect heat pump
in heating mode would transfer Qh into the house without doing any work and so would
also have an infinite coefficient of performance The second law forbids this too A good
coefficient of performance for a real device would be around 5 or 6
Carnot Cycle
The most efficient of all possible engines is one that uses the Carnot cycle This cycle
employs an ideal gas has no friction and operates very slowly so that the gas can be in
thermal equilibrium at all parts of the cycle This means of course that it canrsquot
possibly be built and even if it could be built it would not run fast enough to be useful
Nevertheless this cycle is very important because it gives an upper bound on the
efficiency of real engines There cannot possibly be an engine that is more efficient
than one based on the Carnot cycle This cycle consists of the following four steps
1 The ideal gas absorbs heat Qh at constant temperature Th while the gas increases its
volume The reason that heat is absorbed is that expansion tends to cool the gas but
thermal contact with the environment at Th keeps the temperature high by heat
conduction into the ideal gas
2 The ideal gas further increases its volume by an adiabatic expansion This expansion
causes the gas to cool so at the end of this part of the cycle the gas is at temperature Tc
3 The gas exhausts heat Qc at constant temperature Tc while the gas decreases in
volume The reason that heat is exhausted is that compression tends to heat the gas
but thermal contact with the environment at Tc keeps the temperature low by heat
conduction out of the ideal gas
4 The gas is adiabatically compressed back to its original volume (the volume it started
with in step 1) This compression heats the gas from Tc up to Th
11
The efficiency of a Carnot engine is given by the very simple formula
1 cC
h
Te
T
where the temperatures must in be Kelvin No real engine can be more efficient than
this
The coefficient of performance of a Carnot refrigerator or heat pump in cooling mode
is given by
COP cooling mode cC
h c
T
T T
and no real refrigerator can have a coefficient of performance greater than this
The coefficient of performance of a Carnot heat pump in heating mode is
COP heating mode hC
h c
T
T T
So why donrsquot we just use these wonderful Carnot engines and have perfect efficiency
To make the reversible steps in the cycle really reversible they would have to occur
infinitely slowly So the price you pay for making a perfect engine is that it takes
forever to get it to do any work
Entropy
The entropy of a system is defined in terms of its molecular makeup and measures
roughly the disorder of the system If the system is packed into a very small volume
then it is quite ordered and the entropy will be low If it occupies a large volume the
entropy is high (To see what this has to do with disorder note that socks in a drawer
occupy a small volume while socks on the bed in the corner by the door and
hanging from the chandelier occupy a large volume) If the system is very cold then
the molecules hardly move and may even reach out to each other and form a crystal
This is a highly ordered state and therefore has low entropy If the system is very hot
with rapidly speeding molecules crashing into the container walls and bouncing off
each other things are disordered and the entropy is high
It is possible to calculate the entropy of a system in terms of its macroscopic thermody-
namic properties ie pressure volume temperature number of moles etc The key to this
calculation is the concept of a reversible process A reversible process is one that is
12
carried out without friction and so slowly that the process can be reversed at any stage by
making an infinitesimal change in the environment of the system The slow expansion of the
gas at Th in the Carnot cycle with tiny amounts of heat being transferred to the environment
is reversible If we turned around and began slowly to compress the gas would just slowly
exhaust heat to the environment in the exact reverse way that it absorbed it during expansion
Most processes however are irreversible For example if a gas-filled box were suddenly
increased in size so that the particles were free to wander into the void created by the sudden
expansion then the gas would eventually fill the new volume uniformly at the same
temperature as before the expansion (The temperature is unchanged in this imaginary
process because the kinetic energy of the molecules would be unaffected by such an
instantaneous expansion of the container walls) This imaginary but highly thought-
stimulating process is called a free expansion and it is impossible to reverse it During the
expansion we didnrsquot push on any of the molecules so reversing this process would mean
making them go back into their original volume without pushing on them they simply will
not cooperate to this extent Another way to see that just pushing them back where they came
from does not reverse the free expansion is to think about what would happen if we just
compressed either adiabatically or isothermally An adiabatic compression back to the
original volume would heat the gas above its original temperature and an isothermal
compression would require that heat be exhausted to the environment But the free expansion
involved neither temperature changes nor heat exchanges so neither of these two processes
is the reverse of the free expansion It is simply impossible to reverse this rapid expansion in
a way that takes us back to the initial state
It is possible to calculate the change of entropy for both reversible and irreversible processes
Letrsquos consider a reversible process first In a reversible process the entropy change is given
by the formula
dQS
T
where dQ is the amount of heat added to the system during a small step of the process
The total energy change during the process may then simply be calculated by integration
f f
f i i i
dQS S S dS
T
13
(Just as in the case of energy we are mainly interested in differences rather than in
absolute magnitudes)
This integration method does not work for irreversible processes For instance in the
rapid free expansion discussed above no heat is added to the system but its disorder
obviously goes up We need to find some other way to calculate the entropy The key
is the fact that the entropy of a system depends only on its current state and not at all
on how it arrived there So to calculate the entropy change in an irreversible process
first find out what the initial and final conditions of the process are Then invent a
reversible process that takes the system from the initial state to the final state Since
the entropy depends only on the state of system and not on the process the entropy
change for the reversible process is the same as that for the irreversible process
Entropy is thus like pressure or temperature if the pressure changes from Pl to to P2
then the pressure difference is P2 - Pl regardless of how the pressure change was
made This seems obvious for pressure but not for entropy only because you arent
comfortable with entropy yet
Entropy of an Ideal Gas
For n moles of an ideal gas the difference in entropy between a state with temperature
T and volume V and some standard state with temperature To and volume Vo is given
by
ln lno o Vo o
T VS T V S T V nC nR
T V
Entropy in a Phase Change
Calculating entropy change in a phase change like melting or boiling is easy It is
QT Where Q is in the latent heat for example the flows in to cause the melting and
T is the temperature of the phase change There is NO integral to do
14
Serway Chapter 35
Angle of Reflection
If plane waves are incident on a reflecting surface with the propagation direction of
the waves making angle θ1 with the normal direction to the surface then the reflected
angle θ1prime relative to the surface normal is simply
1 1
ie the incident angles and reflected angles are the same
Refraction
If plane waves traveling through medium 1 are incident on a plane interface between
medium 1 and medium 2 then the angle of incidence of the incoming wave θ1 and
the angle of refraction of the transmitted wave θ2 are related by Snellrsquos law
2 2
1 1
sin
sin
v
v
where v1 and v2 are the wave speeds in medium 1 and medium 2 The angle of
incidence and the angle of refraction are both measured between the wave propagation
direction and the normal to the interface In terms of indices of refraction in the case
of light waves Snellrsquos law takes the more familiar form
1 1 2 2sin sinn n
where ni = cvi
Total Internal Reflection
If a wave is incident from a medium of low wave speed into a medium of high wave
speed the law of refraction requires that the angle of refraction be greater than the
angle of incidence If the angle of refraction is required to be greater than 90deg then no
refracted wave can exist and total internal reflection occurs The critical incident
angle θc beyond which total internal reflection occurs is given by
2
1
sin c
n
n
15
Dispersion of Light
In addition to the speed of light varying from material to material it also varies with
wavelength within each material This means that the index of refraction is generally a
function of wavelength
c
nv
Since the wave speed is not constant such a medium is dispersive meaning in this
context that refraction actually disperses white light into its various colors because
Snellrsquos law gives a different angle for each wavelength In most materials the
variation with wavelength is quite small but this small effect is responsible for some
of the most spectacular color effects we ever see including rainbows a flashing
crystal chandelier and the colored fire of a diamond solitaire by candlelight
16
Serway Chapter 36
Real and Virtual Images
When light rays are focused at a certain plane producing an image if a sheet of white
paper is placed there we call the image a real image The images produced by film
projectors and overhead projectors are examples of real images
When light rays appear to come from a certain location but no image is produced
when a screen is placed there we say that there is a virtual image at that location For
instance when you look in a mirror it appears that someone is behind the mirror but
a screen placed back there in the dark would show nothing Your image in the mirror
is a virtual image
Ray Tracing
There are lots of rules about how to find the images in optical systems but the best
way to keep things straight is to learn how to draw the principal rays for curved
mirrors and lenses The rays for convex and concave mirrors are shown in Fig 3615
The rays for converging and diverging thin lenses are shown in Fig 3627 You should
memorize the principal rays and know how to use them to locate images
Curved Mirrors
The focal length of a curved mirror with radius of curvature R is given by
2
Rf
The relation between object distance p image distance q and focal length f is
1 1 1
p q f
If the mirror is a diverging mirror f should be negative and if q should turn out to be
negative the image is virtual
Lateral Magnification
The lateral magnification in an optical system is defined by the ratio of the image size
to the object size
17
Image height
Object height
qM
p
for curved mirrors
As usual there are sign conventions here too but this formula just gives the
magnitude It is better to keep keep track of upright images versus inverted images by
means of ray diagrams rather than by memorizing sign conventions
Thin Lenses
The focal length of a thin lens is related to the radii of curvature of the two faces R1
and R2 of the lens by
1 2
1 1 11n
f R R
Note that this formula differs from Eq (3611) in the text by not having a minus sign
between the two R-terms We like this form better because for a simple converging
lens like a magnifying glass we just use positive values of R for both surfaces If one
of the faces is concave producing divergence use a negative value for R And if a
surface is flat use R = infin If the face is flat the radius is infinite
The relation between the image object and focal distances for a thin lens is the same
as that for a curved mirror
1 1 1
q p f
Use a negative focal length if the lens is diverging
The lateral magnification for a thin lens is the same as for a curved mirror
Image height
Object height
qM
p
Camera
The lens system in a camera projects a real image of an object onto the film (or CCD
array in a digital camera) The position of the image is adjusted to be on the film by
moving the lens into or out of the camera
18
Eye
The eye is like a camera in that a real image is formed on the retina Unlike a camera
the image position is adjusted by changing the focal length of the lens This is done by
the ciliary muscle which squeezes the lens changing its shape
Near Point The near point is the closest distance from the eye for which the lens can
focus an image on the retina It is usually 18-25 cm for young persons
Far Point The far point is the greatest distance from the eye for which the lens can
focus an image on the retina For a person with normal vision the far point is at
infinity
Nearsightedness A person is nearsighted if their far point is at some finite distance
less than infinity This condition can be corrected with a lens that takes an object at
infinity and produces a virtual image at the personrsquos far point
Farsightedness and Presbyopia A person is farsighted if their near point is too far
away for comfortable near work like reading or knitting This can be corrected by a
lens which takes an object at a normal near point distance of 18-25 cm and produces a
virtual image at the personrsquos natural near point Presbyopia involves a similar
problem which nearly all people experience as they age The ciliary muscle becomes
too weak and the lens becomes too stiff to allow the eye to provide for both near and
far vision The solution for this problem is either reading glasses or bifocal lenses
Reading classes are just weak magnifying glasses mounted on eyeglass frames
Bifocal lenses are split into upper and lower halves The lower half is a lens which
gives the proper correction for near work and the upper half is a different lens for
proper focusing at infinity
Angular Size
When an object is brought closer to the eye it appears to be larger because the image
on the retina is larger The size of this image is directly proportional to the objectrsquos
angular size which is the angle subtended by the object measured from the center of
the lens of the eye In optical instruments which are to be used with the eye the
angular size of the final image is whatrsquos important because it determines how large
the image will appear to the viewer
19
Simple Magnifier
A simple magnifier is a single converging lens or magnifying glass It takes an object
closer to the eye than a normal near point and produces a virtual image at or beyond
this near point The angular magnification is defined to be the ratio of the angular size
when viewed through the lens to the angular size of the object when viewed at the
normal near point (without aid of the lens)
Microscope
This instrument has two lenses (1) The objective is near the object being viewed and
produces a greatly magnified real image (2) The eyepiece is a simple magnifier which
the viewer uses to closely examine the image from (1)
Telescope
This instrument also has two lenses (1) The objective at the front of the telescope
takes light from a distant object and produces a real inverted image (which is rather
small) near its focal point (2) This small real image is then examined by the eyepiece
functioning as a simple magnifier to produce a virtual image with a larger angular
size
20
Serway Chapter 37
Two-Slit Interference
If light is incident on two closely spaced narrow slits a pattern of light and dark
stripes is produced beyond the slits The bright stripes or fringes are caused by
constructive interference of the two waves coming from the slits Constructive
interference occurs whenever two waves arrive at a location in phase with each other
This occurs when the distance x1 from slit 1 to a point P on the screen and the
distance x2 from slit 2 to point P differ by in integral number of wavelengths
1 2 where 0 1 2x x m m
where λ is the wavelength of the light When x1 and x2 are much larger than the slit
spacing d this condition reduces to
sind m
where θ is the angle between the direction of the incident light and the direction of the
light arriving at the screen
Thin Films
When light is partially reflected and partially transmitted by a thin film of transparent
material it is possible to have interference between the wave reflected from the front
of the film and light reflected from the back of the film (The colored reflections from
the thin film of oil on the water in a rain-soaked parking lot are an example of this
effect) It is difficult to write down formulas that will work in all cases so we will
just review the important principles here
1 If the two reflected waves are in phase with each other the film has enhanced
reflection (constructive interference) but if the two reflected waves are out of phase
with each other reflection is diminished (destructive interference) Phase shifts occur
due to reflection and due to the extra path length through the film of the wave
reflected from the back of the film
2 The phase change due to reflection is determined by the difference in index of
refraction between the two media involved in the reflection If the wave is incident
21
from a medium with a low index of refraction into a medium with a high index of
refraction a phase change of 180deg occurs and the reflected wave is inverted If
incident from high to low no phase shift occurs and the reflected wave is non-
inverted
3 The extra path length through the film of the wave reflected from the back of the
film is equal to 2t where t is the thickness of the film (The incident light is assumed
to be normal to the surface of the film) The number of wavelengths contained in the
extra path length is equal to 2tλn where λn = λn is the wavelength of the light in the
film
4 Rules for reflection from thin films
If one ray is inverted and the other is not then we have
12 constructive
2 nt m
2 destructivent m
If either both rays are inverted or both are non-inverted we have
2 constructivent m
12 destructive
2 nt m
22
Serway Chapter 38
Diffraction Grating
A diffraction grating is simply a fancy version of two-slit interference with the two
slits replaced by thousands of slits Just as in the two-slit case the bright fringes
occur at angles given by
sin md
but in this case d the distance between neighboring slits is made to be very small
The effect of having many slits instead of two is to make each bright fringe highly
localized with wide dark regions between neighboring maxima
Single Slit Diffraction
When light passes through an opening in an opaque screen an interference pattern is
produced beyond the opening To understand why we may replace the single opening
by many small coherent sources of light These many sources interfere with each
other producing a pattern known as a diffraction pattern (Note that many authors do
not distinguish between interference and diffraction treating them as interchangeable
terms) If the opening is a slit of width a then the diffraction pattern far from the slit
will have a bright central maximum with a succession of minima and weaker maxima
on either side The angle between the incident direction and the minima is given by
sin ma
where m = plusmn1 plusmn2
If the opening is circular with diameter D the angle between the incident direction
and the first minimum is given by
sin 122D
Optical Resolution and Rayleighrsquos Criterion
Two point sources can just be resolved (distinguished from each other) if the peak of
the diffraction image of the first source overlies the first minimum of the diffraction
image of the second source For circular holes of the kind usually encountered in
23
optical devices this condition is approximately satisfied when the angular separation
between the two sources as viewed from the optical instrument is greater than or equal
to the critical angle
min 122D
where λ is the wavelength of the light and where D is the diameter of the aperture in
the instrument
Polarization
We say that an electromagnetic wave is polarized if its electric field vector doesnrsquot
change direction in a random fashion The simplest kind of polarization is linear
polarization in which the electric field vector oscillates back and forth along the
same axis in space Polarized light can be produced from normal unpolarized light by
selective absorption (as in Polaroid sunglasses) by passing light through crystals that
have different indices of refraction for different polarizations (double refraction) by
scattering (the blue sky is polarized) and by reflection (glare) Polarization by
reflection occurs when light reflects from a shiny insulating (non-metallic) surface
The amount of polarization is greatest for reflection at Brewsterrsquos angle
2
1
tan P
n
n
where θP is the incidence angle of light from medium 1 onto medium 2 and where n1
and n2 are the indices of refraction for the media (In our everyday experience n1 = 1
since the light comes in through the air and n2 is the index of refraction of the shiny
insulating material producing the glare eg water glass plastic paint etc)
Malusrsquos Law
The intensity of transmitted polarized light through a perfect polarizer is related to the
incident intensity of polarized light by Malusrsquos law
2cosoI I
where Io is the intensity of the incident light and where θ is the angle between the
electric field vector in the incident wave and the transmission axis the polarizer
24
Serway Chapter 39
Principles of Relativity
All of the weirdness of relativity flows from two simple principles
(1) The laws of physics must be the same in all inertial (non-accelerating) reference
frames
(2) The speed of light in vacuum has the same value in all inertial reference frames
Note that (1) seems reasonable but (2) is very odd It says that if two space ships are
approaching each other at nearly the speed of light and a laser pulse is shot from ship
1 toward ship 2 then when the people on ship 2 measure the speed of the pulse as it
goes by it is moving at 3 times 108 ms the same as if the ships were stationary
Simultaneity
If observer 1 sees two events in her own frame as simultaneous at two different
locations a moving observer 2 will see these two events happening at different times
Relativistic gamma
The factor γ (gamma) appears regularly in the formulas of relativity
2
2
1
1 vc
where v is the relative speed between two inertial frames
Time Dilation
If observer 1 sees two events at the same location in space separated by time Δtp in his
own frame then observer 2 moving at speed v relative to observer 1 will see these
two events separated by a longer time Δt
pt t
Moving clocks run slow
Length Contraction
If observer 1 measures the length of an object along the x-direction at some instant of
time in her frame to be Lp then observer 2 moving in the x-direction at speed v will
measure the length L of the object to be shorter
pLL
25
Moving meter sticks are short
Velocity Addition
If a particle is moving at velocity vac relative to frame c and if frame c is moving at
speed vcb with respect to frame b (with both velocities directed along the same line)
what would be the velocity of the particle in frame b In classical physics the answer is
ab ac cbv v v
but if any of these velocities are an appreciable fraction of the speed of light the
answer changes
21ac cb
abac cb
v Vv
v v c
In using this equation make sure that vac has the same sign in the numerator and in the
denominator and that vbc also has the same sign in both places Note that this equation
is different (and in our opinion easier to understand) than the equations in the text
Momentum and Energy
The momentum and total energy of a particle moving at velocity v (its speed is v v )
are given by
22
2 2
22
1 1 vvc c
m mcm E mc
vp v
Rest Energy
When a particle is at rest (v = 0 and γ = 1) then E = mc2 is called the rest energy It
includes the potential energy of the particle so that if we change the potential energy
of a particle we change its rest mass
Kinetic Energy The kinetic energy is the difference between the total energy and the
rest energy
2 2K mc mc
For v ltlt c this reduces to K = mv22
General Relativity
Einsteinrsquos principle of equivalence in the theory of general relativity is In the vicinity
of a point a gravitational field is equivalent to an accelerated frame of reference in
26
the absence of gravitational effects For example there is no difference whatever in
the physics experienced by an astronaut on earth and one who is in interstellar space
far away from any stars or planets but who has just turned on her spaceshiprsquos engines
and is accelerating at 98 ms2
27
Serway Chapter 40
Photons
We now know that even though light behaves like a wave it is better described as
consisting of small packets of energy called photons The energy of a photon is related
to its frequency by
E hf
where h is Planckrsquos constant
346626 10 J sh
Photons also carry momentum given by
hf hp
c
Since the amount of energy in each photon is fixed the difference between dim light
and intense light (of the same wavelength) is that dim light consists of fewer photons
than intense light
Photoelectric Effect
In the photoelectric effect light shining onto a metal surface gives the electrons in the
metal enough energy to escape and be detected It requires a certain amount of energy
(called the work function typically = 3 ndash 5 eV) just to kick the electron out of the
metal so the light must deliver at least this much energy to an electron to produce the
effect It is observed that red light no matter how intense never produces electrons
But ultraviolet light even if quite dim will eject electrons from the metal Since red
light consists of 2 eV photons and ultraviolet light has photons with energies around
4-6 eV the photon idea explains the behavior of the photoelectric effect The
maximum energy that an ejected electron can have is
maxK hf
where f is the photon frequency The minimum frequency that light can have and cause
electrons to be ejected is called the cutoff energy and is found by setting Kmax = 0
cf h
28
Compton Effect
When high-frequency light interacts with free electrons the scattered light does not
have the same wavelength as the incident light contrary to what classical
electromagnetism would predict But the photon picture in which photons have
momentum and energy allows us to treat scattering as a collision between the photon
and the electron In this collision the electron and the scattered photon both have a
different momentum and energy than they did before And since p = hλ for a photon
if the momentum is different the wavelength will also be different
after before 1 cose
h
m c
where the angle θ is the angle between the incoming photon direction and the
direction of the scattered photon
Particles are Waves
Since photons behave like particles sometimes it is not surprising that elementary
particles can behave like waves sometimes The wavelength of a particle with
momentum p is given by
h
p
Wave-particle Duality
Both photons and elementary particles have a dual nature sometimes they behave like
particles and sometimes they behave like waves It is hard for us to comprehend the
nature of such an object by making mental pictures but experiments definitively show
that this is the case Since we have no direct experience with photons and elementary
particles (because their energies are so much smaller than the energies of the everyday
objects we encounter) it is perhaps not surprising that we have a hard time forming a
mental picture of how they behave
Electron Interference
Since an electron is both a particle and a wave just like a photon it should be able to
produce an interference pattern This is observed If an electron beam is shot at two
closely-spaced slits and if the electrons that pass through the slits are detected
downstream it is found that there are some locations where electrons are never detected
29
and others where lots of electrons are detected The pattern is exactly the same as the
one observed for light waves provided that we use the electron wavelength λ = hp in
place of the wavelength of light This pattern is observed even though each electron is
detected as a single dot on the screen Only after many such dots are collected does the
pattern emerge And if we try to understand how this effect could possibly work by
looking closely at each slit to see which one the electron came through the pattern
disappears the act of measurement destroys the interference This means that each
single electron somehow comes through both slits (as a wave would)
Uncertainty Principle
In classical physics we always imagine that the positions and momenta of moving
particles have definite values It might be hard to measure them but surely at each
instant of time a particle should be precisely located at some point in space and have a
similarly precise momentum This turns out not to be true Instead both position and
momentum are required to be uncertain with their uncertainties Δx and Δp satisfying
the Heisenberg uncertainty relation
2x p
So if the particle were known to be precisely at some particular location (so that Δx =
0) we couldnrsquot know anything about its momentum (Δp = infin) and if its momentum
were exactly known we couldnrsquot know its position
There is a similar relation involving the particlersquos energy E and the time interval Δt
over which this energy is measured
2E t
This means that energy is not actually conserved if we are considering very small time
intervals and this brief non-conservation of energy has been observed
30
Serway Chapter 41
Wave Function ψ and Probability
Quantum mechanics does not predict exactly what an electron or a photon will do Instead
it specifies the wave function or probability amplitude ψ of an electron or a photon
This wave function is a complex-valued function of space and time whose squared
magnitude is the probability density P for finding a particle at a particular place in
space at a certain time
2P
where is the complex conjugate of ψ
For example the wave function of an electron with perfectly specified momentum p
would have a wavelength given by
h
p
and its probability amplitude would be proportional to
2i x ipxe e
The corresponding probability density would then be
21ipx ip ipxP e e e
which means that the electron is equally probable to be anywhere along the x axis
This is in accord with the uncertainty principle since we specified the momentum
precisely we canrsquot have any idea about the position of the electron
As another example you have probably seen ldquofuzzy ballrdquo drawings of electron
orbitals in a chemistry book These fuzzy balls are meant to indicate the distribution
of the probability density 2
P in the orbital
Particle in a Box
A simple example in which we can calculate the wave function is the case of a particle of
mass m constrained to be inside a 1-dimensional box between x = 0 and x = L In this
case the general wave function is a linear superposition of wave functions ψn of the form
sin niE tn
n xx t A e
L
31
where A is a positive constant where n = 1 2 3 and where the energy associated
with each of the quantum states ψn is given by
22
28n
hE n
mL
This wave function is zero at x = 0 and x = L which means that the particle will never
be found at the walls of the box The wave function has maximum values in the
interior and at these places the particle is most likely to be found and it also has
places where it is zero and at these places the particle will also never be found as
expressed by the formula
2sinn x
PL
The particle in the box is interfering with itself producing a probability interference
pattern across the box just like the interference patterns we studied with light and
sound
32
Serway Chapter 42
Atoms
Many people picture an atom as a miniature solar system where electrons orbit around
a massive nucleus at the center This picture is misleading because in atoms the wave
nature of electrons dominates The electrons form 3-dimensional standing waves
(called orbitals) centered at the nucleus And if you ask what is it thatrsquos waving the
answer is ψ the probability amplitude (see Chapter 41)
Standing waves on a string can occur only for certain frequencies (the fundamental
and higher harmonics) Similarly atomic orbitals occur only for certain energies For
the hydrogen atom the energies of the orbitals have the a particularly simple form
2
1136 eVnE
n
where n = 1 2 3 is called the principal quantum number For other atoms the
determination of the orbital energies requires numerical calculation by computers
Atomic Spectra
If an electron is somehow given extra extra energy (we say that it is excited) so that it
occupies a higher orbital it will eventually ldquofallrdquo back down to a lower orbital Each
time an electron falls to a lower orbital it loses the difference in energy between the
two orbitals in the form of a photon Since the orbital energies are discrete so are the
energy differences and so are the wavelengths of the emitted light The entire set of
these discrete wavelengths is called the atomic spectrum and it is unique to each type
of atom For hydrogen the spectrum can be simply written as
H 2 2
1 1 1
f i
Rn n
where RH = 10973732 times 107 m-1 For other atoms the spectrum cannot be expressed as
a formula
Orbital Angular Momentum Electrons in atoms also have quantized values of
angular momentum The orbital quantum number ℓ specifies the value of this
quantized angular momentum through the formula
1L
33
If we want to know the value of the angular momentum along some direction in space
say the z direction the answer is not L but rather
zL m
where mℓ is another quantum number which runs from
1 1m
This quantum number is important when an atom sits in a magnetic field
Spin Angular Momentum It has been found experimentally that electrons and other
charged particles also carry internal angular momentum which we call spin
Electrons have an intrinsic spin angular momentum s along a specified axis that is
extremely quantized it can only take on 2 possible values
1
2zs s
We thus say that electrons have spin s = 12 or that electrons are rdquospin one-half parti-
cles Since s is an angular momentum it obeys the same rule as that for orbital angular
momentum namely that if its value along some axis is s then its total magnitude is
given by
31
2S s s
I know this seems weird but quantum mechanics is weird The only excuse for this
bizarre way of looking at the world is that it predicts what happens in experiments
Exclusion Principle The answer to the question of how many electrons (or any other
spin one-half particle) can be in one particular quantum state was discovered by
Wolfgang Pauli and is called the exclusion principle
ldquoNo two electrons can ever be in the same quantum state therefore no two electrons
in the same atom can have the same set of quantum numbersrdquo
This is the reason that we have atoms with different properties instead of every atom
simple having all of its electrons in the ground state All of the variety we see around
us in the world is the result of chemical differences and these differences would not
exist unless electrons obeyed this important principle The entire structure of the
periodic table (see pages 1377-1379 in Serway) is an expression of this principle
34
Serway Chapter 44
Nuclear Properties
The nucleus is composed of protons with charge +e and mass mp = 1007226 u and of
neutrons with zero charge and mass mn = 1008665 u where u is one atomic mass unit
-271 u=1660540 10 kg
The atomic number Z counts the number of protons in a nucleus while the neutron
number N counts the number of neutrons The mass number A is the sum of the two
A N Z
Protons and nuetrons have about the same mass and some times called baryons (heavy
ones) or nucleons (partcles in the nucleus) A is sometimes called the baryon
number The chemical elements are distinguished by Z ie hydrogen has Z = 1 iron
has Z = 26 uranium has Z = 92 etc But for a given Z there might be several nuclei
with different numbers of neutrons These nuclear siblings are called isotopes
These numbers are used to label nuclei according to the pattern
5626 ie FeA
Z X
denotes the isotope of iron with N = 56 ndash 26 = 30 neutrons
The nucleus is roughly spherical with a radius given approximately by
1 3 150 0where 12 10 mr r A r
Nuclear Stability
Because the positively charged protons electrically repel each other with an enormous
force at distances as small as 10-15 m there must be some really strong force that that
overcomes electrical repulsion to hold protons and neutrons together This force is
called with some lack of imagination the strong force It is a very short-range force
(it only acts over a distance of about 2 x 10-15 m) and attracts protons to protons
neutrons to neutrons and neutrons to protons But in spite of this strong nuclear force
the coulomb repulsion of the protons is still present so anything that might keep the
protons from being right next to each other would help keep the nucleus from
35
exploding This role is played by the neutrons and for nuclei with Z le 20 the stable
nuclei roughly have N = Z
For Z gt 20 the coulomb repulsion force is more powerful and more neutrons are
needed to dilute it up to about N = 15Z around Z = 80 For Z gt 83 no amount of
neutrons can help and these nuclei are unstable (radioactive) There is an ldquoisland of
(relative) stabilityrdquo around Z=90-92 (thorium and uranium respectively) These
elements have one or two isotopes with half-lifes of billions of years so there are
substantial amounts of such elements on Earth This fact makes it possible to have
practical fission devices
Radioactive Decay
There are three types of radioactive decay
Alpha decay the nucleus kicks out a helium nucleus (N = 2 Z = 2 A = 4) So if the
original nucleus is called X and the new nucleus is called Y then the decay would look
like this
4 42 2X Y+ HeA A
Z Z
Beta decay the nucleus either kicks out an electron (endash) or its positively-charged
antimatter twin the positron (e+) plus either an electron neutrino v or an electron
anti-neutrino v
1X Y+e electron decayA AZ Z v
1X Y+e positron decayA AZ Z v
A neutrino is a particle with no charge hardly any mass (much less than the electron
mass) and interacts so weakly with matter that most neutrinos upon encountering the
planet earth just pass right through it as if it werenrsquot there
Gamma decay the nucleons in the nucleus X are in an excited energy state X
(perhaps as a result of having undergone alpha or beta decay) and they drop down to a
lower energy state shedding the energy as a high frequency photon
X XA AZ Z
36
This process is exactly analogous to the way that the electrons in atoms emit photons
Decay Rate and Half Life
There is no way to predict exactly when an unstable or excited nucleus will decay but
there is an average rate at which this decay occurs called the decay constant λ The
meaning of this constant is that if there are a large number N of nuclei in a sample
then the number of decays per second that will be observed (called the decay rate R)
is R = λN In mathematical language
dMR N
dt
This simple differential equation has for its solution
0tN t N e
where N0 is the number of nuclei in the sample at time t = 0
The half-life is the time it takes for half of the nuclei in the sample to decay and is
related to the decay constant by
1 2
ln 2 0693T
Disintegration Energy
When a nucleus decays it is making a transition to an overall state of lower energy
which means according to Einsteinrsquos famous formula E = mc2 that the sum of the
masses after the decay must be less than the mass before with the lost mass appearing
as kinetic energy among the decay products For example in alpha decay this kinetic
energy called the disintegration energy Q is given by
2X YQ M M M c
37
Serway Chapter 45
Nuclear energy
There are two ways to extract energy from the nucleus fission and fusion For nuclei
with Z greater than 26 breaking the nucleus apart into pieces leads to a lower
2mc energy than the original nucleus so energy can be extracted by fission For
nuclei with Z less than 26 a lower energy is achieved by combining nuclei so energy
can be extracted by fusion
Fission
Since neutrons have no charge they are not repelled from nuclei as protons are for
this reason their behavior is the key to understanding how fission works
Fast neutrons mostly bounce off other nuclei with each collision slowing the neutron
down This elastic energy loss is most effective if the other nuclei have low mass (like
hydrogen) and these materials are called moderators because of their ability to slow
down fast neutrons
The reason that slowing neutrons is important is that slow neutrons are much more
likely to be absorbed by a nucleus which then leads to nuclear reactions of various
kinds For a few very large nuclei like uranium-235 and some plutonium isotopes
absorption of a slow neutron causes the nucleus to split into two large fragments plus
2 or 3 fast neutrons (energy is released in the process too) Thus one slow neutron
can produce energy plus 2 or 3 more neutrons which if moderated from fast to slow
can split 2 or 3 more nuclei leading to a runaway chain reaction If the reaction is
allowed to proceed unchecked it produces a large explosion If a neutron absorbing
material is added to the mix (like the cadmium in reactor control rods) it is possible to
keep the reaction under control and to extract the released energy as heat to drive
steam turbines and produce electricity
Fusion
Fusion involves mashing two nuclei together and since they are both charged and repel
each other this reaction is much harder to make go The nuclei must have enough energy to
overcome the coulomb repulsion which is why this reaction requires a high temperature
(hundreds of millions of degrees K) like that in the sun (which is burning hydrogen to
helium via fusion) or like that in the center of the fission explosion that is used to detonate
38
a hydrogen bomb
This reaction is of interest for power production in spite of this difficult temperature
requirement because of the abundance of fusion fuel on the planet There are about
012 g of deuterium ( 21H ) in every gallon of water on earth and it only costs about 4
cents to extract it The fusion energy available from this minuscule amount of
deuterium would run a 1000-Megawatt power plant for 10 seconds Water is so
abundant on earth that if fusion were to work we would have an essentially
inexhaustible source of energy
So why donrsquot we have fusion power plants Well the fuel is cheap but the match is
incredibly expensive The only way we know to control this difficult high-temperature
reaction is with large and expensive pieces of equipment involving either large
magnetic fields and complex high-power electromagnetic antennas or with gigantic
(football-fieldsized) laser facilities involving more than a hundred of the highest-
energy lasers ever built Power plants based on these current methods for controlling
fusion are unattractive to the fiscally-minded people who run the electric power
industry Hopefully better designs will be discovered as experiments continue
39
c actinium 68 Er erbium 101 Md mendelevium 104 Rf rutherfordium l aluminum 63 Eu europium 80 Hg mercury 62 Sm samarium m americium 100 Fm fermium 42 Mo molybdenum 21 Sc scandium b antimony 9 F fluorine 60 Nd neodymium 106 Sg seaborgium r argon 87 Fr francium 10 Ne neon 34 Se selenium s arsenic 64 Gd gadolinium 93 Np neptunium 14 Si silicon t astatine 31 Ga gallium 28 Ni nickel 47 Ag silver a barium 32 Ge germanium 41 Nb niobium 11 Na sodium k berkelium 79 Au gold 7 N nitrogen 38 Sr strontium e beryllium 72 Hf hafnium 102 No nobelium 16 S sulfur i bismuth 108 Hs hassium 76 Os osmium 73 Ta tantalum h bohrium 2 He helium 8 O oxygen 43 Tc technetium
boron 67 Ho holmium 46 Pd palladium 52 Te tellurium r bromine 1 H hydrogen 15 P phosphorus 65 Tb terbium d cadmium 49 In indium 78 Pt platinum 81 Tl thallium a calcium 53 I iodine 94 Pu plutonium 90 Th thorium f californium 77 Ir iridium 84 Po polonium 69 Tm thulium
carbon 26 Fe iron 19 K potassium 50 Sn tin e cerium 36 Kr krypton 59 Pr praseodymium 22 Ti titanium s cesium 57 La lanthanum 61 Pm promethium 74 W tungsten l chlorine 103 Lr lawrencium 91 Pa protactinium 92 U uranium r chromium 82 Pb lead 88 Ra radium 23 V vanadium o cobalt 3 Li lithium 86 Rn radon 54 Xe xenon u copper 71 Lu lutetium 75 Re rhenium 70 Yb ytterbium m curium 12 Mg magnesium 45 Rh rhodium 39 Y yttrium b dubnium 25 Mn manganese 37 Rb rubidium 30 Zn zinc y dysprosium 109 Mt meitnerium 44 Ru ruthenium 40 Zr zirconium s einsteinium
40
UNITS The SI units are given in parentheses Other commonly used units are given in terms of the SI units Symbols conform with the recommendations of the American National Standards Institute (ANSI) and the American Institute of Physics (AIP) Commonly used multiples of SI units
prefix symbol factor tera T 1012 giga G 109
mega M 106 kilo k 103
centi c 10-2 milli m 10-3
micro μ 10-6 nano n 10-9 pico p 10-12
femto f 10-15
length meter (m) angstrom 1 Ǻ = 10-10 m inch 1 in = 254 x 10-2 m foot 1 ft = 03048 m mile 1 mi = 1609 m
mass kilogram (kg) atomic mass unit 1 u = 1661 x 10-27 kgslug 1 slug = 1459 kg
force newton (N = kg bull ms2) dyne 1 dyn = 10-5 N pound 1 lb = 4448 N
pressure Pascal (Pa = kgm bull s2) atmosphere 1 atm = 1013 x 105 Pa poundssquare inch 1 psi = 6895 Pa cm of mercury 1 cm Hg = 1333 Pa bar 1 bar = 1000 x 105 Pa torr 1 torr = 1333 Pa
time second (s) minute 1 min = 60 s hour 1 h =3600 s
frequency hertz (Hz = s-1) radianssecond 1 rads = 12π Hz
energy joule (J = kg bull m2s2) erg 1 erg = 10-7 J electron volt 1 eV = 1602 x 10-19 J calorie 1 cal = 4187 J kilowatt-hour 1 kWh = 36 x 106 J British thermal unit 1 Btu = 1055 J
power watt (W = kg bull m2s3) horsepower 1 hp = 7457 W
charge coulomb (C = A bull s)
electric potential volt (V = kg bull m2 s3 bull A)
current ampere (A)
resistance ohm (Ω = kg bull m2s3 bull A2)
capacitance farad (F = s4 bull A2kg bull m2)
magnetic field tesla (T = kgs2 bull A) gauss 1 G = 10-4 T
magnetic flux weber (Wb = kg m2s2 bull A) maxwell 1 Mx = 10-8 Wb
magnetic inductance henry (H = kg bull m2s2 bullA2)
temperature kelvin (K) degrees Celsius 0degC = 27315 K
angle radian (rad) degree 1 deg = π180 rad revolution 1 rev = 2π rad
41
Some Physical Constantsa Quantity Symbol Valueb Atomic mass unit μ 1660 538 73 (13) x 10-27 kg
931494 013 (37) MeVc2 Avogadros number NA 6022 141 99 (47) x 1023 particlesmol Avogadrorsquos number 6022 x 1023 mol Bohr magneton
2B e
em 9274 008 99 (37) x 10-24 JT
Bohr magneton μB 927 x 10-24 JT Bohr radius
0
2
2e em e k
a 5291 772 083 (19) x 10-11 m
Boltzmanns constant B A
RNk 1380 650 3 (24) x 10-23 JK
Boltzmannrsquos constant kB 1380 x 10-23 JK Compton wavelength
C eh
m c 2426 310 215 (18) x 10-12 m
Deuteron mass md 3343 583 09 (26) x 10-27 kg 2013 553 212 71 (35) u
electron charge e 1602 x 10-19 C Electron mass me 9109 381 88 (72) x 10-31 kg
5485 799 110 (12) x 10-4 u 0510 998 902 (21) MeVc2
electron mass 911 x 10-31 kg Electron volt eV 1602 176 462 (63) x 10-19 J Elementary charge e 1602 176 462 (63) x 10-19 C Gas constant R 8314 472 (15) JKmol Gravitational constant G 6673 (10) x 10-11 Nm2kg2 Neutron mass mn 1674 927 16 (13) x 10-27 kg
1008 664 915 78 (55) u 939565 330 (38) MeVc2
neutron mass 1675 x 10-27 kg Nuclear magneton
2n p
em 5050 783 17 (20) x 10-27 JT
permeability constant μ0 1257 x 10-6 Hm permittivity constant ε0 8854 x 10-12 Fm Plancks constant h
2h
6626 068 76 (52) x 10-34 Js 1054 571 596 (82) x 10-34 Js
Planckrsquos constant h ħ
6626 x 10-34 Js 1055 x 10-34 Js
Proton mass mp 1672621 58 (13) x 10-27 kg 1007 276 466 88 (13) u 938271 998 (38) MeVc2
proton mass 1673 x 10-27 kg Rydberg constant RH 1097 373 156 854 9 (83) x 107 m-1 Speed of light in vacuum c 2997 92458 x 108 ms (exact) speed of light c 300 x 108 ms a
These constants are the values recommended in 1998 by CODATA based on a least-squares adjustment of data from different measurements For a more complete list see P J Mohr and B N Taylor Rev Mod Phys 72351 2000 b The numbers in parentheses for the values above represent the uncertainties of the last two digits
NOTE The ones in red are the ones from Appendix 13 The ones in yellow are the ones that are from the Serway book table but that were already on the Appendix 13 table
42
INDEX
Absolute zero 9
Adiabatic 17
Adiabatic exponent 18
Adiabatic process 17
Alpha decay 46
Amplitude 4
Angle of reflection 25
Angular frequency 3
Angular magnification 30
Angular momentum atomic 43
Angular size 29
Archimedes Principle 1
Atomic spectra 43
Atoms 43
Avogadrorsquos number 10
Beats 8
Bernoullirsquos Equation 2
Beta decay 46
Bifocals 29
Brewsterrsquos angle 34
British Thermal Unit Btu 11
Buoyancy 2
calorie 11
Calorie 11
Camera 28
Carnot cycle 21
Carnot efficiency 22
Celsius scale 9
Ciliary muscle 29
Coefficient of performance 21
Compton effect 39
Constructive interference 7
Continuity equation of 2
Contraction length 35
Convection 14
Curved mirrors 27
Decay constant 46
Decay rate radioactive 46
Decibel scale 5
Degrees of freedom 16
Density 1
Destructive interference 7
Diffraction single slit 33
Diffraction grating 33
Dilation time 35
Disintegration energy 47
Dispersion of light 26
Doppler effect 6
Double slit interference 31
Efficiency 20
Electron interference 39
Emissivity 15
Energy nuclear decay 47
Energy relativistic 36
43
Engineering work 20
Entropy 22
Entropy ideal gas 24
Equipartition of energy 18
Equivalence principle 36
Exclusion principle 44
Expansion thermal 9
Expansion coefficient linear 9
Eye 29
Fahrenheit scale 9
Far point 29
Farsightedness 29
First Law of Thermodynamics 12
Fission 48
Flux volume 2
Free expansion 23
Fringes 31
Fusion 48
Fusion heat of 11
Gamma relativistic 35
Gamma decay 46
General relativity 36
Half-life 46
Heat 11
Heat capacity 11
Heat conduction 13
Heat engine 20
Heat of fusion 11
Heat of vaporization 11
Heat pump 20
Heats of transformation 11
Hydrostatics 1
Ideal Gas Law 10
Images real and virtual 27
Intensity sound 5
Interference 7
Interference two-slit 31
Internal energy 12
Internal energy degrees of freedom 16
Irreversible process 19 23
Isotopes 45
Joule 11
Kelvin scale 9
Kinetic energy relativity 36
Kinetic theory 16
Latent heat 11
Length contraction 35
Linear expansion coefficient 9
Linear polarization 34
Linear superposition 7
Longitudinal wave 3
Loudness 5
Magnification lateral 27
Magnifying glass 30
Malusrsquos law 34
Microscope 30
Momentum relativistic 36
Muscle ciliary 29
44
Musical instruments 8
Musical scale 8
Near point 29
Nearsightedness 29
Nonlinear 7
Nuclear energy 48
Nuclear properties 45
Nuclear stability 45
Octave 8
Optical resolution 33
Orbital quantum number ℓ 43
Orbitals 43
Particle in a Box 41
Particles are waves 39
Pascalrsquos Principle 1
Period 3
Photoelectric effect 38
Photons 38
Pitch 8
Polarization 34
Power sound 5
Presbyopia 29
Pressure 1
Principal quantum number n 43
Principle of equivalence 36
Principle of linear superposition 7
Probability amplitude ψ 41
Processes thermodynamic 13
R-value 14
Radiation thermal 14
Radioactive decay 46
Radioactive decay rate 46
Ray tracing 27
Rayleighrsquos criterion 33
Reading glasses 29
Real image 27
Refraction 25
Refrigerator 20
Relativistic gamma 35
Relativity principles 35
Resolved for light sources 33
Rest energy 36
Reversible process 19 22
Rope wave speed 4
Second Law of Thermodynamics 19
Shock waves 6
Simple magnifier 30
Simultaneity 35
Single slit diffraction 33
Snellrsquos law 25
Sound speed 5
Specific heat 11
Spin Angular momentum 44
Standing waves 7
Stefanrsquos law 14
Strong force 45
Telescope 30
Temperature 9
45
Temperature Scales 9
Thermal conductivity 14
Thermal energy 12
Thermal expansion 9
Thin film interference 31
Thin lenses 28
Time dilation 35
Tone musical 8
Total internal reflection 25
Transverse wave 3
Traveling Waves 3
Two-slit interference 31
Uncertainty principle 40
Vaporization heat of 11
Velocity addition relativity 36
Virtual image 27
Volume flux 2
Wave function 41
Wave Function ψ and Probability 41
Wave speed 3
Wave-particle duality 39
Wavelength 3
Wavenumber 3
Wien Displacement Law 16
Work 11
4
Eint) is a state variable
Figure 205
5
Serway Chapter 21
Kinetic Theory
The ideal gas law works for all atoms and molecules at low pressure It is rather
amazing that it does Kinetic theory explains why The properties of an ideal gas can
be understood by thinking of it as N rapidly moving particles of mass m As these
particles collide with the container walls momentum is imparted to the walls which
we call the force of gas pressure In this picture the pressure is related to the average
of the square of the particle velocity 2v by
22 1( )
3 2
NP mv
V
Using the ideal gas law we obtain the average translational kinetic energy per
molecule
21 3
2 2 Bmv k T
The rms speed is then given by
2rms
3 3Bk T RTv v
m M
where M is the molecular mass in kgmol
Degrees of Freedom
Roughly speaking a degree of freedom is a way in which a molecule can store energy
For instance since there are three different directions in space along which a molecule
can move there are three degrees of freedom for the translational kinetic energy
There are also three different axes of rotation about which a polyatomic molecule can
spin so we say there are three degrees of freedom for the rotational kinetic energy
There are even degrees of freedom associated with the various ways in which a
molecule can vibrate and with the different energy levels in which the electrons of
the molecule can exist
Internal Energy and Degrees of Freedom The internal energy of an ideal gas made
up of molecules with J degrees of freedom is given by
int 2 2 B
J JE nRT Nk T
6
Heat Capacities of an Ideal Gas The heat capacity of a gas is described by means of
molar heat capacities CV and CP These are the heat capacities per mole and the
subscript V on CV means that the volume is being held constant while for CP the
pressure is held constant For example to raise the temperature of n moles of a gas
whose pressure is held constant by 10 K we would have to supply an amount of heat Q
= nCP (10) K
Molar Specific Heat of an Ideal Gas at Constant Volume
VQ nC T
3monatomic
2VC R
5diatomic
2VC R
5polyatomic
2VC
Real gases deviate from these formulas because in addition to the translational and ro-
tational degrees of freedom they also have vibrational and electronic degrees of
freedom These are unimportant at low temperatures due to quantum mechanical
effects but become increasingly important at higher temperatures The rough rule is
No of degrees of freedom
2VC R
Molar Specific Heat of an Ideal Gas at Constant Pressure
PQ nC T
P VC C R
The internal energy of an ideal gas depends only on the temperature
int VE nC T
Adiabatic Processes in an Ideal Gas
7
An adiabatic process is one in which no heat is exchanged between the system and the
environment When an ideal gas expands or contracts adiabatically not only does its
pressure change as expected from the ideal gas law but its temperature changes as
well Under these conditions the final pressure Pf can be computed from the initial
pressure Pi and from the final and initial volumes Vf and Vi by
or constantf f i iP V PV PV
where γ = CPCV The quantity γ is called the adiabatic exponent Note that this
doesnrsquot mean that the ideal gas law no longer holds it does and in fact it can be
combined with the adiabatic law for pressure given above to obtain the adiabatic law
for temperatures
1 constantTV
Compressions in sound waves are adiabatic because they happen too rapidly for any
appreciable amount of heat to flow This is why the adiabatic exponent appears in the
formula for the speed of sound in an ideal gas
RTv
M
Note that v depends only on T and not on P Because it depends only on the
temperature the speed of sound is the same in Provo as at sea level in spite of the
lower pressure here due to the difference in elevation
Equipartition of Energy
Every kind of molecule has a certain number of degrees of freedom which are
independent ways in which it can store energy Each such degree of freedom has
associated with it ndash on average ndash an energy of 12 Bk T per molecule (or 1
2 RT per mole)
(Note since a molecule has so many possible degrees of freedom it would seem that
there should be a lot of 12 sBk T to spread around But because energy is quantized
some of these degrees of freedom are not ldquoactiverdquo until the temperature becomes high
enough that 12 Bk T is as big as the lowest quantum of energy
8
Serway Chapter 22
Second Law of Thermodynamics
There are several equivalent forms of this important law
Kelvin It is not possible to change heat completely into work with no other change
taking place Or in other words there are no perfect heat engines
Clausius It is not possible for heat to flow from one body to another body at a higher
temperature with no other change taking place Or in other words there are no
perfect refrigerators
Entropy In any thermodynamic process that proceeds from one equilibrium state to
another the entropy of the system + environment either remains unchanged or
increases The total entropy never decreases This law is a bit of an oddity among the
laws of physics because it is not absolute Things are forbidden by the second law not
because it is impossible for them to happen but because it is extremely unlikely for
them to happen (See below for more information about entropy)
Reversible and Irreversible Processes
A reversible process is one which occurs so slowly that it is in thermal equilibrium (or
very nearly so) at all times A hallmark of such processes is that a motion picture of
them looks perfectly normal whether run forward or backward Imagine for instance
the slow expansion of a gas at constant temperature in a cylinder whose volume is being
increased by a slowly moving piston Run the movie backwards and what do you see
You see the slow compression of a gas at constant temperature which looks perfectly
normal
An irreversible process is one which occurs in such a way that thermal equilibrium is
not maintained throughout the process The mark of this kind of process is that a motion
picture of it looks very odd when run backward Imagine the sudden expansion of a gas
into a previously evacuated chamber because a hole was punched in the wall between a
pressurized chamber and the evacuated one Run the movie backward and what do you
see You see the gas in the soon-to-be-evacuated chamber gather itself together and
stream through a tiny hole into a chamber in which there is already plenty of gas If you
have ever seen this happen get in touch with the support group for those who have
witnessed the spontaneous combustion of large mammals by calling 1-800-PYROCOW
9
Heat Engines
Because of the vexing difference in sign between work done on and system and work done
by a system we will invent a new work variable Weng Heat engines do work and so the net
W for these engines is negative But in engineering applications hidden minus signs are
regarded as evil so for heat engines we donrsquot talk about W instead we talk about its
magnitude engW W So for heat engines the first law is
int engE Q W
But for heat pumps and refrigerators work is done on the system so we use the usual work
W when we talk about these systems
A heat engine is a machine that absorbs heat converts part of it to work and exhausts the
rest The heat must be absorbed at high temperature and exhausted at low temperature If the
absorbed heat is Qh the exhausted heat is Qc and the work done by the engine is Weng then
eng h cW Q Q
and the efficiency of the engine is defined to be
eng
h
We
Q
A perfect engine would convert the heat hQ completely into work Weng giving an effi-
ciency of e = 1 Energy conservation alone allows a perfect engine but the second law
requires e lt 1
Refrigerators and Heat Pumps
A refrigerator is a machine that absorbs heat at low temperature and exhausts it at high
temperature the ldquobackwardsrdquo heat transfer being driven by the work done on the machine
by some source of power A heat pump is a machine that either works like a refrigerator
keeping a place cold by transferring heat from this cold place to a higher temperature
environment (cooling mode like an air conditioner) or it functions as a heater
transferring heat into a warm place from a cooler one (heating mode like a window unit
that heats a house by extracting thermal energy from the cold outdoors) The coefficient
of performance of a refrigerator or of a heat pump in cooling mode is defined to be
COP cooling modecQ
W
10
For a heat pump in heating mode the coefficient of performance is
COP heating modehQ
W
Note that we donrsquot have to use the engineering work here because in these systems
work is done on the system and W is naturally positive
A perfect refrigerator would take in heat Qc and exhaust the same amount of heat without
doing any work giving an infinite coefficient of performance Energy conservation alone
allows a perfect refrigerator but the second law requires COP lt infin A perfect heat pump
in heating mode would transfer Qh into the house without doing any work and so would
also have an infinite coefficient of performance The second law forbids this too A good
coefficient of performance for a real device would be around 5 or 6
Carnot Cycle
The most efficient of all possible engines is one that uses the Carnot cycle This cycle
employs an ideal gas has no friction and operates very slowly so that the gas can be in
thermal equilibrium at all parts of the cycle This means of course that it canrsquot
possibly be built and even if it could be built it would not run fast enough to be useful
Nevertheless this cycle is very important because it gives an upper bound on the
efficiency of real engines There cannot possibly be an engine that is more efficient
than one based on the Carnot cycle This cycle consists of the following four steps
1 The ideal gas absorbs heat Qh at constant temperature Th while the gas increases its
volume The reason that heat is absorbed is that expansion tends to cool the gas but
thermal contact with the environment at Th keeps the temperature high by heat
conduction into the ideal gas
2 The ideal gas further increases its volume by an adiabatic expansion This expansion
causes the gas to cool so at the end of this part of the cycle the gas is at temperature Tc
3 The gas exhausts heat Qc at constant temperature Tc while the gas decreases in
volume The reason that heat is exhausted is that compression tends to heat the gas
but thermal contact with the environment at Tc keeps the temperature low by heat
conduction out of the ideal gas
4 The gas is adiabatically compressed back to its original volume (the volume it started
with in step 1) This compression heats the gas from Tc up to Th
11
The efficiency of a Carnot engine is given by the very simple formula
1 cC
h
Te
T
where the temperatures must in be Kelvin No real engine can be more efficient than
this
The coefficient of performance of a Carnot refrigerator or heat pump in cooling mode
is given by
COP cooling mode cC
h c
T
T T
and no real refrigerator can have a coefficient of performance greater than this
The coefficient of performance of a Carnot heat pump in heating mode is
COP heating mode hC
h c
T
T T
So why donrsquot we just use these wonderful Carnot engines and have perfect efficiency
To make the reversible steps in the cycle really reversible they would have to occur
infinitely slowly So the price you pay for making a perfect engine is that it takes
forever to get it to do any work
Entropy
The entropy of a system is defined in terms of its molecular makeup and measures
roughly the disorder of the system If the system is packed into a very small volume
then it is quite ordered and the entropy will be low If it occupies a large volume the
entropy is high (To see what this has to do with disorder note that socks in a drawer
occupy a small volume while socks on the bed in the corner by the door and
hanging from the chandelier occupy a large volume) If the system is very cold then
the molecules hardly move and may even reach out to each other and form a crystal
This is a highly ordered state and therefore has low entropy If the system is very hot
with rapidly speeding molecules crashing into the container walls and bouncing off
each other things are disordered and the entropy is high
It is possible to calculate the entropy of a system in terms of its macroscopic thermody-
namic properties ie pressure volume temperature number of moles etc The key to this
calculation is the concept of a reversible process A reversible process is one that is
12
carried out without friction and so slowly that the process can be reversed at any stage by
making an infinitesimal change in the environment of the system The slow expansion of the
gas at Th in the Carnot cycle with tiny amounts of heat being transferred to the environment
is reversible If we turned around and began slowly to compress the gas would just slowly
exhaust heat to the environment in the exact reverse way that it absorbed it during expansion
Most processes however are irreversible For example if a gas-filled box were suddenly
increased in size so that the particles were free to wander into the void created by the sudden
expansion then the gas would eventually fill the new volume uniformly at the same
temperature as before the expansion (The temperature is unchanged in this imaginary
process because the kinetic energy of the molecules would be unaffected by such an
instantaneous expansion of the container walls) This imaginary but highly thought-
stimulating process is called a free expansion and it is impossible to reverse it During the
expansion we didnrsquot push on any of the molecules so reversing this process would mean
making them go back into their original volume without pushing on them they simply will
not cooperate to this extent Another way to see that just pushing them back where they came
from does not reverse the free expansion is to think about what would happen if we just
compressed either adiabatically or isothermally An adiabatic compression back to the
original volume would heat the gas above its original temperature and an isothermal
compression would require that heat be exhausted to the environment But the free expansion
involved neither temperature changes nor heat exchanges so neither of these two processes
is the reverse of the free expansion It is simply impossible to reverse this rapid expansion in
a way that takes us back to the initial state
It is possible to calculate the change of entropy for both reversible and irreversible processes
Letrsquos consider a reversible process first In a reversible process the entropy change is given
by the formula
dQS
T
where dQ is the amount of heat added to the system during a small step of the process
The total energy change during the process may then simply be calculated by integration
f f
f i i i
dQS S S dS
T
13
(Just as in the case of energy we are mainly interested in differences rather than in
absolute magnitudes)
This integration method does not work for irreversible processes For instance in the
rapid free expansion discussed above no heat is added to the system but its disorder
obviously goes up We need to find some other way to calculate the entropy The key
is the fact that the entropy of a system depends only on its current state and not at all
on how it arrived there So to calculate the entropy change in an irreversible process
first find out what the initial and final conditions of the process are Then invent a
reversible process that takes the system from the initial state to the final state Since
the entropy depends only on the state of system and not on the process the entropy
change for the reversible process is the same as that for the irreversible process
Entropy is thus like pressure or temperature if the pressure changes from Pl to to P2
then the pressure difference is P2 - Pl regardless of how the pressure change was
made This seems obvious for pressure but not for entropy only because you arent
comfortable with entropy yet
Entropy of an Ideal Gas
For n moles of an ideal gas the difference in entropy between a state with temperature
T and volume V and some standard state with temperature To and volume Vo is given
by
ln lno o Vo o
T VS T V S T V nC nR
T V
Entropy in a Phase Change
Calculating entropy change in a phase change like melting or boiling is easy It is
QT Where Q is in the latent heat for example the flows in to cause the melting and
T is the temperature of the phase change There is NO integral to do
14
Serway Chapter 35
Angle of Reflection
If plane waves are incident on a reflecting surface with the propagation direction of
the waves making angle θ1 with the normal direction to the surface then the reflected
angle θ1prime relative to the surface normal is simply
1 1
ie the incident angles and reflected angles are the same
Refraction
If plane waves traveling through medium 1 are incident on a plane interface between
medium 1 and medium 2 then the angle of incidence of the incoming wave θ1 and
the angle of refraction of the transmitted wave θ2 are related by Snellrsquos law
2 2
1 1
sin
sin
v
v
where v1 and v2 are the wave speeds in medium 1 and medium 2 The angle of
incidence and the angle of refraction are both measured between the wave propagation
direction and the normal to the interface In terms of indices of refraction in the case
of light waves Snellrsquos law takes the more familiar form
1 1 2 2sin sinn n
where ni = cvi
Total Internal Reflection
If a wave is incident from a medium of low wave speed into a medium of high wave
speed the law of refraction requires that the angle of refraction be greater than the
angle of incidence If the angle of refraction is required to be greater than 90deg then no
refracted wave can exist and total internal reflection occurs The critical incident
angle θc beyond which total internal reflection occurs is given by
2
1
sin c
n
n
15
Dispersion of Light
In addition to the speed of light varying from material to material it also varies with
wavelength within each material This means that the index of refraction is generally a
function of wavelength
c
nv
Since the wave speed is not constant such a medium is dispersive meaning in this
context that refraction actually disperses white light into its various colors because
Snellrsquos law gives a different angle for each wavelength In most materials the
variation with wavelength is quite small but this small effect is responsible for some
of the most spectacular color effects we ever see including rainbows a flashing
crystal chandelier and the colored fire of a diamond solitaire by candlelight
16
Serway Chapter 36
Real and Virtual Images
When light rays are focused at a certain plane producing an image if a sheet of white
paper is placed there we call the image a real image The images produced by film
projectors and overhead projectors are examples of real images
When light rays appear to come from a certain location but no image is produced
when a screen is placed there we say that there is a virtual image at that location For
instance when you look in a mirror it appears that someone is behind the mirror but
a screen placed back there in the dark would show nothing Your image in the mirror
is a virtual image
Ray Tracing
There are lots of rules about how to find the images in optical systems but the best
way to keep things straight is to learn how to draw the principal rays for curved
mirrors and lenses The rays for convex and concave mirrors are shown in Fig 3615
The rays for converging and diverging thin lenses are shown in Fig 3627 You should
memorize the principal rays and know how to use them to locate images
Curved Mirrors
The focal length of a curved mirror with radius of curvature R is given by
2
Rf
The relation between object distance p image distance q and focal length f is
1 1 1
p q f
If the mirror is a diverging mirror f should be negative and if q should turn out to be
negative the image is virtual
Lateral Magnification
The lateral magnification in an optical system is defined by the ratio of the image size
to the object size
17
Image height
Object height
qM
p
for curved mirrors
As usual there are sign conventions here too but this formula just gives the
magnitude It is better to keep keep track of upright images versus inverted images by
means of ray diagrams rather than by memorizing sign conventions
Thin Lenses
The focal length of a thin lens is related to the radii of curvature of the two faces R1
and R2 of the lens by
1 2
1 1 11n
f R R
Note that this formula differs from Eq (3611) in the text by not having a minus sign
between the two R-terms We like this form better because for a simple converging
lens like a magnifying glass we just use positive values of R for both surfaces If one
of the faces is concave producing divergence use a negative value for R And if a
surface is flat use R = infin If the face is flat the radius is infinite
The relation between the image object and focal distances for a thin lens is the same
as that for a curved mirror
1 1 1
q p f
Use a negative focal length if the lens is diverging
The lateral magnification for a thin lens is the same as for a curved mirror
Image height
Object height
qM
p
Camera
The lens system in a camera projects a real image of an object onto the film (or CCD
array in a digital camera) The position of the image is adjusted to be on the film by
moving the lens into or out of the camera
18
Eye
The eye is like a camera in that a real image is formed on the retina Unlike a camera
the image position is adjusted by changing the focal length of the lens This is done by
the ciliary muscle which squeezes the lens changing its shape
Near Point The near point is the closest distance from the eye for which the lens can
focus an image on the retina It is usually 18-25 cm for young persons
Far Point The far point is the greatest distance from the eye for which the lens can
focus an image on the retina For a person with normal vision the far point is at
infinity
Nearsightedness A person is nearsighted if their far point is at some finite distance
less than infinity This condition can be corrected with a lens that takes an object at
infinity and produces a virtual image at the personrsquos far point
Farsightedness and Presbyopia A person is farsighted if their near point is too far
away for comfortable near work like reading or knitting This can be corrected by a
lens which takes an object at a normal near point distance of 18-25 cm and produces a
virtual image at the personrsquos natural near point Presbyopia involves a similar
problem which nearly all people experience as they age The ciliary muscle becomes
too weak and the lens becomes too stiff to allow the eye to provide for both near and
far vision The solution for this problem is either reading glasses or bifocal lenses
Reading classes are just weak magnifying glasses mounted on eyeglass frames
Bifocal lenses are split into upper and lower halves The lower half is a lens which
gives the proper correction for near work and the upper half is a different lens for
proper focusing at infinity
Angular Size
When an object is brought closer to the eye it appears to be larger because the image
on the retina is larger The size of this image is directly proportional to the objectrsquos
angular size which is the angle subtended by the object measured from the center of
the lens of the eye In optical instruments which are to be used with the eye the
angular size of the final image is whatrsquos important because it determines how large
the image will appear to the viewer
19
Simple Magnifier
A simple magnifier is a single converging lens or magnifying glass It takes an object
closer to the eye than a normal near point and produces a virtual image at or beyond
this near point The angular magnification is defined to be the ratio of the angular size
when viewed through the lens to the angular size of the object when viewed at the
normal near point (without aid of the lens)
Microscope
This instrument has two lenses (1) The objective is near the object being viewed and
produces a greatly magnified real image (2) The eyepiece is a simple magnifier which
the viewer uses to closely examine the image from (1)
Telescope
This instrument also has two lenses (1) The objective at the front of the telescope
takes light from a distant object and produces a real inverted image (which is rather
small) near its focal point (2) This small real image is then examined by the eyepiece
functioning as a simple magnifier to produce a virtual image with a larger angular
size
20
Serway Chapter 37
Two-Slit Interference
If light is incident on two closely spaced narrow slits a pattern of light and dark
stripes is produced beyond the slits The bright stripes or fringes are caused by
constructive interference of the two waves coming from the slits Constructive
interference occurs whenever two waves arrive at a location in phase with each other
This occurs when the distance x1 from slit 1 to a point P on the screen and the
distance x2 from slit 2 to point P differ by in integral number of wavelengths
1 2 where 0 1 2x x m m
where λ is the wavelength of the light When x1 and x2 are much larger than the slit
spacing d this condition reduces to
sind m
where θ is the angle between the direction of the incident light and the direction of the
light arriving at the screen
Thin Films
When light is partially reflected and partially transmitted by a thin film of transparent
material it is possible to have interference between the wave reflected from the front
of the film and light reflected from the back of the film (The colored reflections from
the thin film of oil on the water in a rain-soaked parking lot are an example of this
effect) It is difficult to write down formulas that will work in all cases so we will
just review the important principles here
1 If the two reflected waves are in phase with each other the film has enhanced
reflection (constructive interference) but if the two reflected waves are out of phase
with each other reflection is diminished (destructive interference) Phase shifts occur
due to reflection and due to the extra path length through the film of the wave
reflected from the back of the film
2 The phase change due to reflection is determined by the difference in index of
refraction between the two media involved in the reflection If the wave is incident
21
from a medium with a low index of refraction into a medium with a high index of
refraction a phase change of 180deg occurs and the reflected wave is inverted If
incident from high to low no phase shift occurs and the reflected wave is non-
inverted
3 The extra path length through the film of the wave reflected from the back of the
film is equal to 2t where t is the thickness of the film (The incident light is assumed
to be normal to the surface of the film) The number of wavelengths contained in the
extra path length is equal to 2tλn where λn = λn is the wavelength of the light in the
film
4 Rules for reflection from thin films
If one ray is inverted and the other is not then we have
12 constructive
2 nt m
2 destructivent m
If either both rays are inverted or both are non-inverted we have
2 constructivent m
12 destructive
2 nt m
22
Serway Chapter 38
Diffraction Grating
A diffraction grating is simply a fancy version of two-slit interference with the two
slits replaced by thousands of slits Just as in the two-slit case the bright fringes
occur at angles given by
sin md
but in this case d the distance between neighboring slits is made to be very small
The effect of having many slits instead of two is to make each bright fringe highly
localized with wide dark regions between neighboring maxima
Single Slit Diffraction
When light passes through an opening in an opaque screen an interference pattern is
produced beyond the opening To understand why we may replace the single opening
by many small coherent sources of light These many sources interfere with each
other producing a pattern known as a diffraction pattern (Note that many authors do
not distinguish between interference and diffraction treating them as interchangeable
terms) If the opening is a slit of width a then the diffraction pattern far from the slit
will have a bright central maximum with a succession of minima and weaker maxima
on either side The angle between the incident direction and the minima is given by
sin ma
where m = plusmn1 plusmn2
If the opening is circular with diameter D the angle between the incident direction
and the first minimum is given by
sin 122D
Optical Resolution and Rayleighrsquos Criterion
Two point sources can just be resolved (distinguished from each other) if the peak of
the diffraction image of the first source overlies the first minimum of the diffraction
image of the second source For circular holes of the kind usually encountered in
23
optical devices this condition is approximately satisfied when the angular separation
between the two sources as viewed from the optical instrument is greater than or equal
to the critical angle
min 122D
where λ is the wavelength of the light and where D is the diameter of the aperture in
the instrument
Polarization
We say that an electromagnetic wave is polarized if its electric field vector doesnrsquot
change direction in a random fashion The simplest kind of polarization is linear
polarization in which the electric field vector oscillates back and forth along the
same axis in space Polarized light can be produced from normal unpolarized light by
selective absorption (as in Polaroid sunglasses) by passing light through crystals that
have different indices of refraction for different polarizations (double refraction) by
scattering (the blue sky is polarized) and by reflection (glare) Polarization by
reflection occurs when light reflects from a shiny insulating (non-metallic) surface
The amount of polarization is greatest for reflection at Brewsterrsquos angle
2
1
tan P
n
n
where θP is the incidence angle of light from medium 1 onto medium 2 and where n1
and n2 are the indices of refraction for the media (In our everyday experience n1 = 1
since the light comes in through the air and n2 is the index of refraction of the shiny
insulating material producing the glare eg water glass plastic paint etc)
Malusrsquos Law
The intensity of transmitted polarized light through a perfect polarizer is related to the
incident intensity of polarized light by Malusrsquos law
2cosoI I
where Io is the intensity of the incident light and where θ is the angle between the
electric field vector in the incident wave and the transmission axis the polarizer
24
Serway Chapter 39
Principles of Relativity
All of the weirdness of relativity flows from two simple principles
(1) The laws of physics must be the same in all inertial (non-accelerating) reference
frames
(2) The speed of light in vacuum has the same value in all inertial reference frames
Note that (1) seems reasonable but (2) is very odd It says that if two space ships are
approaching each other at nearly the speed of light and a laser pulse is shot from ship
1 toward ship 2 then when the people on ship 2 measure the speed of the pulse as it
goes by it is moving at 3 times 108 ms the same as if the ships were stationary
Simultaneity
If observer 1 sees two events in her own frame as simultaneous at two different
locations a moving observer 2 will see these two events happening at different times
Relativistic gamma
The factor γ (gamma) appears regularly in the formulas of relativity
2
2
1
1 vc
where v is the relative speed between two inertial frames
Time Dilation
If observer 1 sees two events at the same location in space separated by time Δtp in his
own frame then observer 2 moving at speed v relative to observer 1 will see these
two events separated by a longer time Δt
pt t
Moving clocks run slow
Length Contraction
If observer 1 measures the length of an object along the x-direction at some instant of
time in her frame to be Lp then observer 2 moving in the x-direction at speed v will
measure the length L of the object to be shorter
pLL
25
Moving meter sticks are short
Velocity Addition
If a particle is moving at velocity vac relative to frame c and if frame c is moving at
speed vcb with respect to frame b (with both velocities directed along the same line)
what would be the velocity of the particle in frame b In classical physics the answer is
ab ac cbv v v
but if any of these velocities are an appreciable fraction of the speed of light the
answer changes
21ac cb
abac cb
v Vv
v v c
In using this equation make sure that vac has the same sign in the numerator and in the
denominator and that vbc also has the same sign in both places Note that this equation
is different (and in our opinion easier to understand) than the equations in the text
Momentum and Energy
The momentum and total energy of a particle moving at velocity v (its speed is v v )
are given by
22
2 2
22
1 1 vvc c
m mcm E mc
vp v
Rest Energy
When a particle is at rest (v = 0 and γ = 1) then E = mc2 is called the rest energy It
includes the potential energy of the particle so that if we change the potential energy
of a particle we change its rest mass
Kinetic Energy The kinetic energy is the difference between the total energy and the
rest energy
2 2K mc mc
For v ltlt c this reduces to K = mv22
General Relativity
Einsteinrsquos principle of equivalence in the theory of general relativity is In the vicinity
of a point a gravitational field is equivalent to an accelerated frame of reference in
26
the absence of gravitational effects For example there is no difference whatever in
the physics experienced by an astronaut on earth and one who is in interstellar space
far away from any stars or planets but who has just turned on her spaceshiprsquos engines
and is accelerating at 98 ms2
27
Serway Chapter 40
Photons
We now know that even though light behaves like a wave it is better described as
consisting of small packets of energy called photons The energy of a photon is related
to its frequency by
E hf
where h is Planckrsquos constant
346626 10 J sh
Photons also carry momentum given by
hf hp
c
Since the amount of energy in each photon is fixed the difference between dim light
and intense light (of the same wavelength) is that dim light consists of fewer photons
than intense light
Photoelectric Effect
In the photoelectric effect light shining onto a metal surface gives the electrons in the
metal enough energy to escape and be detected It requires a certain amount of energy
(called the work function typically = 3 ndash 5 eV) just to kick the electron out of the
metal so the light must deliver at least this much energy to an electron to produce the
effect It is observed that red light no matter how intense never produces electrons
But ultraviolet light even if quite dim will eject electrons from the metal Since red
light consists of 2 eV photons and ultraviolet light has photons with energies around
4-6 eV the photon idea explains the behavior of the photoelectric effect The
maximum energy that an ejected electron can have is
maxK hf
where f is the photon frequency The minimum frequency that light can have and cause
electrons to be ejected is called the cutoff energy and is found by setting Kmax = 0
cf h
28
Compton Effect
When high-frequency light interacts with free electrons the scattered light does not
have the same wavelength as the incident light contrary to what classical
electromagnetism would predict But the photon picture in which photons have
momentum and energy allows us to treat scattering as a collision between the photon
and the electron In this collision the electron and the scattered photon both have a
different momentum and energy than they did before And since p = hλ for a photon
if the momentum is different the wavelength will also be different
after before 1 cose
h
m c
where the angle θ is the angle between the incoming photon direction and the
direction of the scattered photon
Particles are Waves
Since photons behave like particles sometimes it is not surprising that elementary
particles can behave like waves sometimes The wavelength of a particle with
momentum p is given by
h
p
Wave-particle Duality
Both photons and elementary particles have a dual nature sometimes they behave like
particles and sometimes they behave like waves It is hard for us to comprehend the
nature of such an object by making mental pictures but experiments definitively show
that this is the case Since we have no direct experience with photons and elementary
particles (because their energies are so much smaller than the energies of the everyday
objects we encounter) it is perhaps not surprising that we have a hard time forming a
mental picture of how they behave
Electron Interference
Since an electron is both a particle and a wave just like a photon it should be able to
produce an interference pattern This is observed If an electron beam is shot at two
closely-spaced slits and if the electrons that pass through the slits are detected
downstream it is found that there are some locations where electrons are never detected
29
and others where lots of electrons are detected The pattern is exactly the same as the
one observed for light waves provided that we use the electron wavelength λ = hp in
place of the wavelength of light This pattern is observed even though each electron is
detected as a single dot on the screen Only after many such dots are collected does the
pattern emerge And if we try to understand how this effect could possibly work by
looking closely at each slit to see which one the electron came through the pattern
disappears the act of measurement destroys the interference This means that each
single electron somehow comes through both slits (as a wave would)
Uncertainty Principle
In classical physics we always imagine that the positions and momenta of moving
particles have definite values It might be hard to measure them but surely at each
instant of time a particle should be precisely located at some point in space and have a
similarly precise momentum This turns out not to be true Instead both position and
momentum are required to be uncertain with their uncertainties Δx and Δp satisfying
the Heisenberg uncertainty relation
2x p
So if the particle were known to be precisely at some particular location (so that Δx =
0) we couldnrsquot know anything about its momentum (Δp = infin) and if its momentum
were exactly known we couldnrsquot know its position
There is a similar relation involving the particlersquos energy E and the time interval Δt
over which this energy is measured
2E t
This means that energy is not actually conserved if we are considering very small time
intervals and this brief non-conservation of energy has been observed
30
Serway Chapter 41
Wave Function ψ and Probability
Quantum mechanics does not predict exactly what an electron or a photon will do Instead
it specifies the wave function or probability amplitude ψ of an electron or a photon
This wave function is a complex-valued function of space and time whose squared
magnitude is the probability density P for finding a particle at a particular place in
space at a certain time
2P
where is the complex conjugate of ψ
For example the wave function of an electron with perfectly specified momentum p
would have a wavelength given by
h
p
and its probability amplitude would be proportional to
2i x ipxe e
The corresponding probability density would then be
21ipx ip ipxP e e e
which means that the electron is equally probable to be anywhere along the x axis
This is in accord with the uncertainty principle since we specified the momentum
precisely we canrsquot have any idea about the position of the electron
As another example you have probably seen ldquofuzzy ballrdquo drawings of electron
orbitals in a chemistry book These fuzzy balls are meant to indicate the distribution
of the probability density 2
P in the orbital
Particle in a Box
A simple example in which we can calculate the wave function is the case of a particle of
mass m constrained to be inside a 1-dimensional box between x = 0 and x = L In this
case the general wave function is a linear superposition of wave functions ψn of the form
sin niE tn
n xx t A e
L
31
where A is a positive constant where n = 1 2 3 and where the energy associated
with each of the quantum states ψn is given by
22
28n
hE n
mL
This wave function is zero at x = 0 and x = L which means that the particle will never
be found at the walls of the box The wave function has maximum values in the
interior and at these places the particle is most likely to be found and it also has
places where it is zero and at these places the particle will also never be found as
expressed by the formula
2sinn x
PL
The particle in the box is interfering with itself producing a probability interference
pattern across the box just like the interference patterns we studied with light and
sound
32
Serway Chapter 42
Atoms
Many people picture an atom as a miniature solar system where electrons orbit around
a massive nucleus at the center This picture is misleading because in atoms the wave
nature of electrons dominates The electrons form 3-dimensional standing waves
(called orbitals) centered at the nucleus And if you ask what is it thatrsquos waving the
answer is ψ the probability amplitude (see Chapter 41)
Standing waves on a string can occur only for certain frequencies (the fundamental
and higher harmonics) Similarly atomic orbitals occur only for certain energies For
the hydrogen atom the energies of the orbitals have the a particularly simple form
2
1136 eVnE
n
where n = 1 2 3 is called the principal quantum number For other atoms the
determination of the orbital energies requires numerical calculation by computers
Atomic Spectra
If an electron is somehow given extra extra energy (we say that it is excited) so that it
occupies a higher orbital it will eventually ldquofallrdquo back down to a lower orbital Each
time an electron falls to a lower orbital it loses the difference in energy between the
two orbitals in the form of a photon Since the orbital energies are discrete so are the
energy differences and so are the wavelengths of the emitted light The entire set of
these discrete wavelengths is called the atomic spectrum and it is unique to each type
of atom For hydrogen the spectrum can be simply written as
H 2 2
1 1 1
f i
Rn n
where RH = 10973732 times 107 m-1 For other atoms the spectrum cannot be expressed as
a formula
Orbital Angular Momentum Electrons in atoms also have quantized values of
angular momentum The orbital quantum number ℓ specifies the value of this
quantized angular momentum through the formula
1L
33
If we want to know the value of the angular momentum along some direction in space
say the z direction the answer is not L but rather
zL m
where mℓ is another quantum number which runs from
1 1m
This quantum number is important when an atom sits in a magnetic field
Spin Angular Momentum It has been found experimentally that electrons and other
charged particles also carry internal angular momentum which we call spin
Electrons have an intrinsic spin angular momentum s along a specified axis that is
extremely quantized it can only take on 2 possible values
1
2zs s
We thus say that electrons have spin s = 12 or that electrons are rdquospin one-half parti-
cles Since s is an angular momentum it obeys the same rule as that for orbital angular
momentum namely that if its value along some axis is s then its total magnitude is
given by
31
2S s s
I know this seems weird but quantum mechanics is weird The only excuse for this
bizarre way of looking at the world is that it predicts what happens in experiments
Exclusion Principle The answer to the question of how many electrons (or any other
spin one-half particle) can be in one particular quantum state was discovered by
Wolfgang Pauli and is called the exclusion principle
ldquoNo two electrons can ever be in the same quantum state therefore no two electrons
in the same atom can have the same set of quantum numbersrdquo
This is the reason that we have atoms with different properties instead of every atom
simple having all of its electrons in the ground state All of the variety we see around
us in the world is the result of chemical differences and these differences would not
exist unless electrons obeyed this important principle The entire structure of the
periodic table (see pages 1377-1379 in Serway) is an expression of this principle
34
Serway Chapter 44
Nuclear Properties
The nucleus is composed of protons with charge +e and mass mp = 1007226 u and of
neutrons with zero charge and mass mn = 1008665 u where u is one atomic mass unit
-271 u=1660540 10 kg
The atomic number Z counts the number of protons in a nucleus while the neutron
number N counts the number of neutrons The mass number A is the sum of the two
A N Z
Protons and nuetrons have about the same mass and some times called baryons (heavy
ones) or nucleons (partcles in the nucleus) A is sometimes called the baryon
number The chemical elements are distinguished by Z ie hydrogen has Z = 1 iron
has Z = 26 uranium has Z = 92 etc But for a given Z there might be several nuclei
with different numbers of neutrons These nuclear siblings are called isotopes
These numbers are used to label nuclei according to the pattern
5626 ie FeA
Z X
denotes the isotope of iron with N = 56 ndash 26 = 30 neutrons
The nucleus is roughly spherical with a radius given approximately by
1 3 150 0where 12 10 mr r A r
Nuclear Stability
Because the positively charged protons electrically repel each other with an enormous
force at distances as small as 10-15 m there must be some really strong force that that
overcomes electrical repulsion to hold protons and neutrons together This force is
called with some lack of imagination the strong force It is a very short-range force
(it only acts over a distance of about 2 x 10-15 m) and attracts protons to protons
neutrons to neutrons and neutrons to protons But in spite of this strong nuclear force
the coulomb repulsion of the protons is still present so anything that might keep the
protons from being right next to each other would help keep the nucleus from
35
exploding This role is played by the neutrons and for nuclei with Z le 20 the stable
nuclei roughly have N = Z
For Z gt 20 the coulomb repulsion force is more powerful and more neutrons are
needed to dilute it up to about N = 15Z around Z = 80 For Z gt 83 no amount of
neutrons can help and these nuclei are unstable (radioactive) There is an ldquoisland of
(relative) stabilityrdquo around Z=90-92 (thorium and uranium respectively) These
elements have one or two isotopes with half-lifes of billions of years so there are
substantial amounts of such elements on Earth This fact makes it possible to have
practical fission devices
Radioactive Decay
There are three types of radioactive decay
Alpha decay the nucleus kicks out a helium nucleus (N = 2 Z = 2 A = 4) So if the
original nucleus is called X and the new nucleus is called Y then the decay would look
like this
4 42 2X Y+ HeA A
Z Z
Beta decay the nucleus either kicks out an electron (endash) or its positively-charged
antimatter twin the positron (e+) plus either an electron neutrino v or an electron
anti-neutrino v
1X Y+e electron decayA AZ Z v
1X Y+e positron decayA AZ Z v
A neutrino is a particle with no charge hardly any mass (much less than the electron
mass) and interacts so weakly with matter that most neutrinos upon encountering the
planet earth just pass right through it as if it werenrsquot there
Gamma decay the nucleons in the nucleus X are in an excited energy state X
(perhaps as a result of having undergone alpha or beta decay) and they drop down to a
lower energy state shedding the energy as a high frequency photon
X XA AZ Z
36
This process is exactly analogous to the way that the electrons in atoms emit photons
Decay Rate and Half Life
There is no way to predict exactly when an unstable or excited nucleus will decay but
there is an average rate at which this decay occurs called the decay constant λ The
meaning of this constant is that if there are a large number N of nuclei in a sample
then the number of decays per second that will be observed (called the decay rate R)
is R = λN In mathematical language
dMR N
dt
This simple differential equation has for its solution
0tN t N e
where N0 is the number of nuclei in the sample at time t = 0
The half-life is the time it takes for half of the nuclei in the sample to decay and is
related to the decay constant by
1 2
ln 2 0693T
Disintegration Energy
When a nucleus decays it is making a transition to an overall state of lower energy
which means according to Einsteinrsquos famous formula E = mc2 that the sum of the
masses after the decay must be less than the mass before with the lost mass appearing
as kinetic energy among the decay products For example in alpha decay this kinetic
energy called the disintegration energy Q is given by
2X YQ M M M c
37
Serway Chapter 45
Nuclear energy
There are two ways to extract energy from the nucleus fission and fusion For nuclei
with Z greater than 26 breaking the nucleus apart into pieces leads to a lower
2mc energy than the original nucleus so energy can be extracted by fission For
nuclei with Z less than 26 a lower energy is achieved by combining nuclei so energy
can be extracted by fusion
Fission
Since neutrons have no charge they are not repelled from nuclei as protons are for
this reason their behavior is the key to understanding how fission works
Fast neutrons mostly bounce off other nuclei with each collision slowing the neutron
down This elastic energy loss is most effective if the other nuclei have low mass (like
hydrogen) and these materials are called moderators because of their ability to slow
down fast neutrons
The reason that slowing neutrons is important is that slow neutrons are much more
likely to be absorbed by a nucleus which then leads to nuclear reactions of various
kinds For a few very large nuclei like uranium-235 and some plutonium isotopes
absorption of a slow neutron causes the nucleus to split into two large fragments plus
2 or 3 fast neutrons (energy is released in the process too) Thus one slow neutron
can produce energy plus 2 or 3 more neutrons which if moderated from fast to slow
can split 2 or 3 more nuclei leading to a runaway chain reaction If the reaction is
allowed to proceed unchecked it produces a large explosion If a neutron absorbing
material is added to the mix (like the cadmium in reactor control rods) it is possible to
keep the reaction under control and to extract the released energy as heat to drive
steam turbines and produce electricity
Fusion
Fusion involves mashing two nuclei together and since they are both charged and repel
each other this reaction is much harder to make go The nuclei must have enough energy to
overcome the coulomb repulsion which is why this reaction requires a high temperature
(hundreds of millions of degrees K) like that in the sun (which is burning hydrogen to
helium via fusion) or like that in the center of the fission explosion that is used to detonate
38
a hydrogen bomb
This reaction is of interest for power production in spite of this difficult temperature
requirement because of the abundance of fusion fuel on the planet There are about
012 g of deuterium ( 21H ) in every gallon of water on earth and it only costs about 4
cents to extract it The fusion energy available from this minuscule amount of
deuterium would run a 1000-Megawatt power plant for 10 seconds Water is so
abundant on earth that if fusion were to work we would have an essentially
inexhaustible source of energy
So why donrsquot we have fusion power plants Well the fuel is cheap but the match is
incredibly expensive The only way we know to control this difficult high-temperature
reaction is with large and expensive pieces of equipment involving either large
magnetic fields and complex high-power electromagnetic antennas or with gigantic
(football-fieldsized) laser facilities involving more than a hundred of the highest-
energy lasers ever built Power plants based on these current methods for controlling
fusion are unattractive to the fiscally-minded people who run the electric power
industry Hopefully better designs will be discovered as experiments continue
39
c actinium 68 Er erbium 101 Md mendelevium 104 Rf rutherfordium l aluminum 63 Eu europium 80 Hg mercury 62 Sm samarium m americium 100 Fm fermium 42 Mo molybdenum 21 Sc scandium b antimony 9 F fluorine 60 Nd neodymium 106 Sg seaborgium r argon 87 Fr francium 10 Ne neon 34 Se selenium s arsenic 64 Gd gadolinium 93 Np neptunium 14 Si silicon t astatine 31 Ga gallium 28 Ni nickel 47 Ag silver a barium 32 Ge germanium 41 Nb niobium 11 Na sodium k berkelium 79 Au gold 7 N nitrogen 38 Sr strontium e beryllium 72 Hf hafnium 102 No nobelium 16 S sulfur i bismuth 108 Hs hassium 76 Os osmium 73 Ta tantalum h bohrium 2 He helium 8 O oxygen 43 Tc technetium
boron 67 Ho holmium 46 Pd palladium 52 Te tellurium r bromine 1 H hydrogen 15 P phosphorus 65 Tb terbium d cadmium 49 In indium 78 Pt platinum 81 Tl thallium a calcium 53 I iodine 94 Pu plutonium 90 Th thorium f californium 77 Ir iridium 84 Po polonium 69 Tm thulium
carbon 26 Fe iron 19 K potassium 50 Sn tin e cerium 36 Kr krypton 59 Pr praseodymium 22 Ti titanium s cesium 57 La lanthanum 61 Pm promethium 74 W tungsten l chlorine 103 Lr lawrencium 91 Pa protactinium 92 U uranium r chromium 82 Pb lead 88 Ra radium 23 V vanadium o cobalt 3 Li lithium 86 Rn radon 54 Xe xenon u copper 71 Lu lutetium 75 Re rhenium 70 Yb ytterbium m curium 12 Mg magnesium 45 Rh rhodium 39 Y yttrium b dubnium 25 Mn manganese 37 Rb rubidium 30 Zn zinc y dysprosium 109 Mt meitnerium 44 Ru ruthenium 40 Zr zirconium s einsteinium
40
UNITS The SI units are given in parentheses Other commonly used units are given in terms of the SI units Symbols conform with the recommendations of the American National Standards Institute (ANSI) and the American Institute of Physics (AIP) Commonly used multiples of SI units
prefix symbol factor tera T 1012 giga G 109
mega M 106 kilo k 103
centi c 10-2 milli m 10-3
micro μ 10-6 nano n 10-9 pico p 10-12
femto f 10-15
length meter (m) angstrom 1 Ǻ = 10-10 m inch 1 in = 254 x 10-2 m foot 1 ft = 03048 m mile 1 mi = 1609 m
mass kilogram (kg) atomic mass unit 1 u = 1661 x 10-27 kgslug 1 slug = 1459 kg
force newton (N = kg bull ms2) dyne 1 dyn = 10-5 N pound 1 lb = 4448 N
pressure Pascal (Pa = kgm bull s2) atmosphere 1 atm = 1013 x 105 Pa poundssquare inch 1 psi = 6895 Pa cm of mercury 1 cm Hg = 1333 Pa bar 1 bar = 1000 x 105 Pa torr 1 torr = 1333 Pa
time second (s) minute 1 min = 60 s hour 1 h =3600 s
frequency hertz (Hz = s-1) radianssecond 1 rads = 12π Hz
energy joule (J = kg bull m2s2) erg 1 erg = 10-7 J electron volt 1 eV = 1602 x 10-19 J calorie 1 cal = 4187 J kilowatt-hour 1 kWh = 36 x 106 J British thermal unit 1 Btu = 1055 J
power watt (W = kg bull m2s3) horsepower 1 hp = 7457 W
charge coulomb (C = A bull s)
electric potential volt (V = kg bull m2 s3 bull A)
current ampere (A)
resistance ohm (Ω = kg bull m2s3 bull A2)
capacitance farad (F = s4 bull A2kg bull m2)
magnetic field tesla (T = kgs2 bull A) gauss 1 G = 10-4 T
magnetic flux weber (Wb = kg m2s2 bull A) maxwell 1 Mx = 10-8 Wb
magnetic inductance henry (H = kg bull m2s2 bullA2)
temperature kelvin (K) degrees Celsius 0degC = 27315 K
angle radian (rad) degree 1 deg = π180 rad revolution 1 rev = 2π rad
41
Some Physical Constantsa Quantity Symbol Valueb Atomic mass unit μ 1660 538 73 (13) x 10-27 kg
931494 013 (37) MeVc2 Avogadros number NA 6022 141 99 (47) x 1023 particlesmol Avogadrorsquos number 6022 x 1023 mol Bohr magneton
2B e
em 9274 008 99 (37) x 10-24 JT
Bohr magneton μB 927 x 10-24 JT Bohr radius
0
2
2e em e k
a 5291 772 083 (19) x 10-11 m
Boltzmanns constant B A
RNk 1380 650 3 (24) x 10-23 JK
Boltzmannrsquos constant kB 1380 x 10-23 JK Compton wavelength
C eh
m c 2426 310 215 (18) x 10-12 m
Deuteron mass md 3343 583 09 (26) x 10-27 kg 2013 553 212 71 (35) u
electron charge e 1602 x 10-19 C Electron mass me 9109 381 88 (72) x 10-31 kg
5485 799 110 (12) x 10-4 u 0510 998 902 (21) MeVc2
electron mass 911 x 10-31 kg Electron volt eV 1602 176 462 (63) x 10-19 J Elementary charge e 1602 176 462 (63) x 10-19 C Gas constant R 8314 472 (15) JKmol Gravitational constant G 6673 (10) x 10-11 Nm2kg2 Neutron mass mn 1674 927 16 (13) x 10-27 kg
1008 664 915 78 (55) u 939565 330 (38) MeVc2
neutron mass 1675 x 10-27 kg Nuclear magneton
2n p
em 5050 783 17 (20) x 10-27 JT
permeability constant μ0 1257 x 10-6 Hm permittivity constant ε0 8854 x 10-12 Fm Plancks constant h
2h
6626 068 76 (52) x 10-34 Js 1054 571 596 (82) x 10-34 Js
Planckrsquos constant h ħ
6626 x 10-34 Js 1055 x 10-34 Js
Proton mass mp 1672621 58 (13) x 10-27 kg 1007 276 466 88 (13) u 938271 998 (38) MeVc2
proton mass 1673 x 10-27 kg Rydberg constant RH 1097 373 156 854 9 (83) x 107 m-1 Speed of light in vacuum c 2997 92458 x 108 ms (exact) speed of light c 300 x 108 ms a
These constants are the values recommended in 1998 by CODATA based on a least-squares adjustment of data from different measurements For a more complete list see P J Mohr and B N Taylor Rev Mod Phys 72351 2000 b The numbers in parentheses for the values above represent the uncertainties of the last two digits
NOTE The ones in red are the ones from Appendix 13 The ones in yellow are the ones that are from the Serway book table but that were already on the Appendix 13 table
42
INDEX
Absolute zero 9
Adiabatic 17
Adiabatic exponent 18
Adiabatic process 17
Alpha decay 46
Amplitude 4
Angle of reflection 25
Angular frequency 3
Angular magnification 30
Angular momentum atomic 43
Angular size 29
Archimedes Principle 1
Atomic spectra 43
Atoms 43
Avogadrorsquos number 10
Beats 8
Bernoullirsquos Equation 2
Beta decay 46
Bifocals 29
Brewsterrsquos angle 34
British Thermal Unit Btu 11
Buoyancy 2
calorie 11
Calorie 11
Camera 28
Carnot cycle 21
Carnot efficiency 22
Celsius scale 9
Ciliary muscle 29
Coefficient of performance 21
Compton effect 39
Constructive interference 7
Continuity equation of 2
Contraction length 35
Convection 14
Curved mirrors 27
Decay constant 46
Decay rate radioactive 46
Decibel scale 5
Degrees of freedom 16
Density 1
Destructive interference 7
Diffraction single slit 33
Diffraction grating 33
Dilation time 35
Disintegration energy 47
Dispersion of light 26
Doppler effect 6
Double slit interference 31
Efficiency 20
Electron interference 39
Emissivity 15
Energy nuclear decay 47
Energy relativistic 36
43
Engineering work 20
Entropy 22
Entropy ideal gas 24
Equipartition of energy 18
Equivalence principle 36
Exclusion principle 44
Expansion thermal 9
Expansion coefficient linear 9
Eye 29
Fahrenheit scale 9
Far point 29
Farsightedness 29
First Law of Thermodynamics 12
Fission 48
Flux volume 2
Free expansion 23
Fringes 31
Fusion 48
Fusion heat of 11
Gamma relativistic 35
Gamma decay 46
General relativity 36
Half-life 46
Heat 11
Heat capacity 11
Heat conduction 13
Heat engine 20
Heat of fusion 11
Heat of vaporization 11
Heat pump 20
Heats of transformation 11
Hydrostatics 1
Ideal Gas Law 10
Images real and virtual 27
Intensity sound 5
Interference 7
Interference two-slit 31
Internal energy 12
Internal energy degrees of freedom 16
Irreversible process 19 23
Isotopes 45
Joule 11
Kelvin scale 9
Kinetic energy relativity 36
Kinetic theory 16
Latent heat 11
Length contraction 35
Linear expansion coefficient 9
Linear polarization 34
Linear superposition 7
Longitudinal wave 3
Loudness 5
Magnification lateral 27
Magnifying glass 30
Malusrsquos law 34
Microscope 30
Momentum relativistic 36
Muscle ciliary 29
44
Musical instruments 8
Musical scale 8
Near point 29
Nearsightedness 29
Nonlinear 7
Nuclear energy 48
Nuclear properties 45
Nuclear stability 45
Octave 8
Optical resolution 33
Orbital quantum number ℓ 43
Orbitals 43
Particle in a Box 41
Particles are waves 39
Pascalrsquos Principle 1
Period 3
Photoelectric effect 38
Photons 38
Pitch 8
Polarization 34
Power sound 5
Presbyopia 29
Pressure 1
Principal quantum number n 43
Principle of equivalence 36
Principle of linear superposition 7
Probability amplitude ψ 41
Processes thermodynamic 13
R-value 14
Radiation thermal 14
Radioactive decay 46
Radioactive decay rate 46
Ray tracing 27
Rayleighrsquos criterion 33
Reading glasses 29
Real image 27
Refraction 25
Refrigerator 20
Relativistic gamma 35
Relativity principles 35
Resolved for light sources 33
Rest energy 36
Reversible process 19 22
Rope wave speed 4
Second Law of Thermodynamics 19
Shock waves 6
Simple magnifier 30
Simultaneity 35
Single slit diffraction 33
Snellrsquos law 25
Sound speed 5
Specific heat 11
Spin Angular momentum 44
Standing waves 7
Stefanrsquos law 14
Strong force 45
Telescope 30
Temperature 9
45
Temperature Scales 9
Thermal conductivity 14
Thermal energy 12
Thermal expansion 9
Thin film interference 31
Thin lenses 28
Time dilation 35
Tone musical 8
Total internal reflection 25
Transverse wave 3
Traveling Waves 3
Two-slit interference 31
Uncertainty principle 40
Vaporization heat of 11
Velocity addition relativity 36
Virtual image 27
Volume flux 2
Wave function 41
Wave Function ψ and Probability 41
Wave speed 3
Wave-particle duality 39
Wavelength 3
Wavenumber 3
Wien Displacement Law 16
Work 11
5
Serway Chapter 21
Kinetic Theory
The ideal gas law works for all atoms and molecules at low pressure It is rather
amazing that it does Kinetic theory explains why The properties of an ideal gas can
be understood by thinking of it as N rapidly moving particles of mass m As these
particles collide with the container walls momentum is imparted to the walls which
we call the force of gas pressure In this picture the pressure is related to the average
of the square of the particle velocity 2v by
22 1( )
3 2
NP mv
V
Using the ideal gas law we obtain the average translational kinetic energy per
molecule
21 3
2 2 Bmv k T
The rms speed is then given by
2rms
3 3Bk T RTv v
m M
where M is the molecular mass in kgmol
Degrees of Freedom
Roughly speaking a degree of freedom is a way in which a molecule can store energy
For instance since there are three different directions in space along which a molecule
can move there are three degrees of freedom for the translational kinetic energy
There are also three different axes of rotation about which a polyatomic molecule can
spin so we say there are three degrees of freedom for the rotational kinetic energy
There are even degrees of freedom associated with the various ways in which a
molecule can vibrate and with the different energy levels in which the electrons of
the molecule can exist
Internal Energy and Degrees of Freedom The internal energy of an ideal gas made
up of molecules with J degrees of freedom is given by
int 2 2 B
J JE nRT Nk T
6
Heat Capacities of an Ideal Gas The heat capacity of a gas is described by means of
molar heat capacities CV and CP These are the heat capacities per mole and the
subscript V on CV means that the volume is being held constant while for CP the
pressure is held constant For example to raise the temperature of n moles of a gas
whose pressure is held constant by 10 K we would have to supply an amount of heat Q
= nCP (10) K
Molar Specific Heat of an Ideal Gas at Constant Volume
VQ nC T
3monatomic
2VC R
5diatomic
2VC R
5polyatomic
2VC
Real gases deviate from these formulas because in addition to the translational and ro-
tational degrees of freedom they also have vibrational and electronic degrees of
freedom These are unimportant at low temperatures due to quantum mechanical
effects but become increasingly important at higher temperatures The rough rule is
No of degrees of freedom
2VC R
Molar Specific Heat of an Ideal Gas at Constant Pressure
PQ nC T
P VC C R
The internal energy of an ideal gas depends only on the temperature
int VE nC T
Adiabatic Processes in an Ideal Gas
7
An adiabatic process is one in which no heat is exchanged between the system and the
environment When an ideal gas expands or contracts adiabatically not only does its
pressure change as expected from the ideal gas law but its temperature changes as
well Under these conditions the final pressure Pf can be computed from the initial
pressure Pi and from the final and initial volumes Vf and Vi by
or constantf f i iP V PV PV
where γ = CPCV The quantity γ is called the adiabatic exponent Note that this
doesnrsquot mean that the ideal gas law no longer holds it does and in fact it can be
combined with the adiabatic law for pressure given above to obtain the adiabatic law
for temperatures
1 constantTV
Compressions in sound waves are adiabatic because they happen too rapidly for any
appreciable amount of heat to flow This is why the adiabatic exponent appears in the
formula for the speed of sound in an ideal gas
RTv
M
Note that v depends only on T and not on P Because it depends only on the
temperature the speed of sound is the same in Provo as at sea level in spite of the
lower pressure here due to the difference in elevation
Equipartition of Energy
Every kind of molecule has a certain number of degrees of freedom which are
independent ways in which it can store energy Each such degree of freedom has
associated with it ndash on average ndash an energy of 12 Bk T per molecule (or 1
2 RT per mole)
(Note since a molecule has so many possible degrees of freedom it would seem that
there should be a lot of 12 sBk T to spread around But because energy is quantized
some of these degrees of freedom are not ldquoactiverdquo until the temperature becomes high
enough that 12 Bk T is as big as the lowest quantum of energy
8
Serway Chapter 22
Second Law of Thermodynamics
There are several equivalent forms of this important law
Kelvin It is not possible to change heat completely into work with no other change
taking place Or in other words there are no perfect heat engines
Clausius It is not possible for heat to flow from one body to another body at a higher
temperature with no other change taking place Or in other words there are no
perfect refrigerators
Entropy In any thermodynamic process that proceeds from one equilibrium state to
another the entropy of the system + environment either remains unchanged or
increases The total entropy never decreases This law is a bit of an oddity among the
laws of physics because it is not absolute Things are forbidden by the second law not
because it is impossible for them to happen but because it is extremely unlikely for
them to happen (See below for more information about entropy)
Reversible and Irreversible Processes
A reversible process is one which occurs so slowly that it is in thermal equilibrium (or
very nearly so) at all times A hallmark of such processes is that a motion picture of
them looks perfectly normal whether run forward or backward Imagine for instance
the slow expansion of a gas at constant temperature in a cylinder whose volume is being
increased by a slowly moving piston Run the movie backwards and what do you see
You see the slow compression of a gas at constant temperature which looks perfectly
normal
An irreversible process is one which occurs in such a way that thermal equilibrium is
not maintained throughout the process The mark of this kind of process is that a motion
picture of it looks very odd when run backward Imagine the sudden expansion of a gas
into a previously evacuated chamber because a hole was punched in the wall between a
pressurized chamber and the evacuated one Run the movie backward and what do you
see You see the gas in the soon-to-be-evacuated chamber gather itself together and
stream through a tiny hole into a chamber in which there is already plenty of gas If you
have ever seen this happen get in touch with the support group for those who have
witnessed the spontaneous combustion of large mammals by calling 1-800-PYROCOW
9
Heat Engines
Because of the vexing difference in sign between work done on and system and work done
by a system we will invent a new work variable Weng Heat engines do work and so the net
W for these engines is negative But in engineering applications hidden minus signs are
regarded as evil so for heat engines we donrsquot talk about W instead we talk about its
magnitude engW W So for heat engines the first law is
int engE Q W
But for heat pumps and refrigerators work is done on the system so we use the usual work
W when we talk about these systems
A heat engine is a machine that absorbs heat converts part of it to work and exhausts the
rest The heat must be absorbed at high temperature and exhausted at low temperature If the
absorbed heat is Qh the exhausted heat is Qc and the work done by the engine is Weng then
eng h cW Q Q
and the efficiency of the engine is defined to be
eng
h
We
Q
A perfect engine would convert the heat hQ completely into work Weng giving an effi-
ciency of e = 1 Energy conservation alone allows a perfect engine but the second law
requires e lt 1
Refrigerators and Heat Pumps
A refrigerator is a machine that absorbs heat at low temperature and exhausts it at high
temperature the ldquobackwardsrdquo heat transfer being driven by the work done on the machine
by some source of power A heat pump is a machine that either works like a refrigerator
keeping a place cold by transferring heat from this cold place to a higher temperature
environment (cooling mode like an air conditioner) or it functions as a heater
transferring heat into a warm place from a cooler one (heating mode like a window unit
that heats a house by extracting thermal energy from the cold outdoors) The coefficient
of performance of a refrigerator or of a heat pump in cooling mode is defined to be
COP cooling modecQ
W
10
For a heat pump in heating mode the coefficient of performance is
COP heating modehQ
W
Note that we donrsquot have to use the engineering work here because in these systems
work is done on the system and W is naturally positive
A perfect refrigerator would take in heat Qc and exhaust the same amount of heat without
doing any work giving an infinite coefficient of performance Energy conservation alone
allows a perfect refrigerator but the second law requires COP lt infin A perfect heat pump
in heating mode would transfer Qh into the house without doing any work and so would
also have an infinite coefficient of performance The second law forbids this too A good
coefficient of performance for a real device would be around 5 or 6
Carnot Cycle
The most efficient of all possible engines is one that uses the Carnot cycle This cycle
employs an ideal gas has no friction and operates very slowly so that the gas can be in
thermal equilibrium at all parts of the cycle This means of course that it canrsquot
possibly be built and even if it could be built it would not run fast enough to be useful
Nevertheless this cycle is very important because it gives an upper bound on the
efficiency of real engines There cannot possibly be an engine that is more efficient
than one based on the Carnot cycle This cycle consists of the following four steps
1 The ideal gas absorbs heat Qh at constant temperature Th while the gas increases its
volume The reason that heat is absorbed is that expansion tends to cool the gas but
thermal contact with the environment at Th keeps the temperature high by heat
conduction into the ideal gas
2 The ideal gas further increases its volume by an adiabatic expansion This expansion
causes the gas to cool so at the end of this part of the cycle the gas is at temperature Tc
3 The gas exhausts heat Qc at constant temperature Tc while the gas decreases in
volume The reason that heat is exhausted is that compression tends to heat the gas
but thermal contact with the environment at Tc keeps the temperature low by heat
conduction out of the ideal gas
4 The gas is adiabatically compressed back to its original volume (the volume it started
with in step 1) This compression heats the gas from Tc up to Th
11
The efficiency of a Carnot engine is given by the very simple formula
1 cC
h
Te
T
where the temperatures must in be Kelvin No real engine can be more efficient than
this
The coefficient of performance of a Carnot refrigerator or heat pump in cooling mode
is given by
COP cooling mode cC
h c
T
T T
and no real refrigerator can have a coefficient of performance greater than this
The coefficient of performance of a Carnot heat pump in heating mode is
COP heating mode hC
h c
T
T T
So why donrsquot we just use these wonderful Carnot engines and have perfect efficiency
To make the reversible steps in the cycle really reversible they would have to occur
infinitely slowly So the price you pay for making a perfect engine is that it takes
forever to get it to do any work
Entropy
The entropy of a system is defined in terms of its molecular makeup and measures
roughly the disorder of the system If the system is packed into a very small volume
then it is quite ordered and the entropy will be low If it occupies a large volume the
entropy is high (To see what this has to do with disorder note that socks in a drawer
occupy a small volume while socks on the bed in the corner by the door and
hanging from the chandelier occupy a large volume) If the system is very cold then
the molecules hardly move and may even reach out to each other and form a crystal
This is a highly ordered state and therefore has low entropy If the system is very hot
with rapidly speeding molecules crashing into the container walls and bouncing off
each other things are disordered and the entropy is high
It is possible to calculate the entropy of a system in terms of its macroscopic thermody-
namic properties ie pressure volume temperature number of moles etc The key to this
calculation is the concept of a reversible process A reversible process is one that is
12
carried out without friction and so slowly that the process can be reversed at any stage by
making an infinitesimal change in the environment of the system The slow expansion of the
gas at Th in the Carnot cycle with tiny amounts of heat being transferred to the environment
is reversible If we turned around and began slowly to compress the gas would just slowly
exhaust heat to the environment in the exact reverse way that it absorbed it during expansion
Most processes however are irreversible For example if a gas-filled box were suddenly
increased in size so that the particles were free to wander into the void created by the sudden
expansion then the gas would eventually fill the new volume uniformly at the same
temperature as before the expansion (The temperature is unchanged in this imaginary
process because the kinetic energy of the molecules would be unaffected by such an
instantaneous expansion of the container walls) This imaginary but highly thought-
stimulating process is called a free expansion and it is impossible to reverse it During the
expansion we didnrsquot push on any of the molecules so reversing this process would mean
making them go back into their original volume without pushing on them they simply will
not cooperate to this extent Another way to see that just pushing them back where they came
from does not reverse the free expansion is to think about what would happen if we just
compressed either adiabatically or isothermally An adiabatic compression back to the
original volume would heat the gas above its original temperature and an isothermal
compression would require that heat be exhausted to the environment But the free expansion
involved neither temperature changes nor heat exchanges so neither of these two processes
is the reverse of the free expansion It is simply impossible to reverse this rapid expansion in
a way that takes us back to the initial state
It is possible to calculate the change of entropy for both reversible and irreversible processes
Letrsquos consider a reversible process first In a reversible process the entropy change is given
by the formula
dQS
T
where dQ is the amount of heat added to the system during a small step of the process
The total energy change during the process may then simply be calculated by integration
f f
f i i i
dQS S S dS
T
13
(Just as in the case of energy we are mainly interested in differences rather than in
absolute magnitudes)
This integration method does not work for irreversible processes For instance in the
rapid free expansion discussed above no heat is added to the system but its disorder
obviously goes up We need to find some other way to calculate the entropy The key
is the fact that the entropy of a system depends only on its current state and not at all
on how it arrived there So to calculate the entropy change in an irreversible process
first find out what the initial and final conditions of the process are Then invent a
reversible process that takes the system from the initial state to the final state Since
the entropy depends only on the state of system and not on the process the entropy
change for the reversible process is the same as that for the irreversible process
Entropy is thus like pressure or temperature if the pressure changes from Pl to to P2
then the pressure difference is P2 - Pl regardless of how the pressure change was
made This seems obvious for pressure but not for entropy only because you arent
comfortable with entropy yet
Entropy of an Ideal Gas
For n moles of an ideal gas the difference in entropy between a state with temperature
T and volume V and some standard state with temperature To and volume Vo is given
by
ln lno o Vo o
T VS T V S T V nC nR
T V
Entropy in a Phase Change
Calculating entropy change in a phase change like melting or boiling is easy It is
QT Where Q is in the latent heat for example the flows in to cause the melting and
T is the temperature of the phase change There is NO integral to do
14
Serway Chapter 35
Angle of Reflection
If plane waves are incident on a reflecting surface with the propagation direction of
the waves making angle θ1 with the normal direction to the surface then the reflected
angle θ1prime relative to the surface normal is simply
1 1
ie the incident angles and reflected angles are the same
Refraction
If plane waves traveling through medium 1 are incident on a plane interface between
medium 1 and medium 2 then the angle of incidence of the incoming wave θ1 and
the angle of refraction of the transmitted wave θ2 are related by Snellrsquos law
2 2
1 1
sin
sin
v
v
where v1 and v2 are the wave speeds in medium 1 and medium 2 The angle of
incidence and the angle of refraction are both measured between the wave propagation
direction and the normal to the interface In terms of indices of refraction in the case
of light waves Snellrsquos law takes the more familiar form
1 1 2 2sin sinn n
where ni = cvi
Total Internal Reflection
If a wave is incident from a medium of low wave speed into a medium of high wave
speed the law of refraction requires that the angle of refraction be greater than the
angle of incidence If the angle of refraction is required to be greater than 90deg then no
refracted wave can exist and total internal reflection occurs The critical incident
angle θc beyond which total internal reflection occurs is given by
2
1
sin c
n
n
15
Dispersion of Light
In addition to the speed of light varying from material to material it also varies with
wavelength within each material This means that the index of refraction is generally a
function of wavelength
c
nv
Since the wave speed is not constant such a medium is dispersive meaning in this
context that refraction actually disperses white light into its various colors because
Snellrsquos law gives a different angle for each wavelength In most materials the
variation with wavelength is quite small but this small effect is responsible for some
of the most spectacular color effects we ever see including rainbows a flashing
crystal chandelier and the colored fire of a diamond solitaire by candlelight
16
Serway Chapter 36
Real and Virtual Images
When light rays are focused at a certain plane producing an image if a sheet of white
paper is placed there we call the image a real image The images produced by film
projectors and overhead projectors are examples of real images
When light rays appear to come from a certain location but no image is produced
when a screen is placed there we say that there is a virtual image at that location For
instance when you look in a mirror it appears that someone is behind the mirror but
a screen placed back there in the dark would show nothing Your image in the mirror
is a virtual image
Ray Tracing
There are lots of rules about how to find the images in optical systems but the best
way to keep things straight is to learn how to draw the principal rays for curved
mirrors and lenses The rays for convex and concave mirrors are shown in Fig 3615
The rays for converging and diverging thin lenses are shown in Fig 3627 You should
memorize the principal rays and know how to use them to locate images
Curved Mirrors
The focal length of a curved mirror with radius of curvature R is given by
2
Rf
The relation between object distance p image distance q and focal length f is
1 1 1
p q f
If the mirror is a diverging mirror f should be negative and if q should turn out to be
negative the image is virtual
Lateral Magnification
The lateral magnification in an optical system is defined by the ratio of the image size
to the object size
17
Image height
Object height
qM
p
for curved mirrors
As usual there are sign conventions here too but this formula just gives the
magnitude It is better to keep keep track of upright images versus inverted images by
means of ray diagrams rather than by memorizing sign conventions
Thin Lenses
The focal length of a thin lens is related to the radii of curvature of the two faces R1
and R2 of the lens by
1 2
1 1 11n
f R R
Note that this formula differs from Eq (3611) in the text by not having a minus sign
between the two R-terms We like this form better because for a simple converging
lens like a magnifying glass we just use positive values of R for both surfaces If one
of the faces is concave producing divergence use a negative value for R And if a
surface is flat use R = infin If the face is flat the radius is infinite
The relation between the image object and focal distances for a thin lens is the same
as that for a curved mirror
1 1 1
q p f
Use a negative focal length if the lens is diverging
The lateral magnification for a thin lens is the same as for a curved mirror
Image height
Object height
qM
p
Camera
The lens system in a camera projects a real image of an object onto the film (or CCD
array in a digital camera) The position of the image is adjusted to be on the film by
moving the lens into or out of the camera
18
Eye
The eye is like a camera in that a real image is formed on the retina Unlike a camera
the image position is adjusted by changing the focal length of the lens This is done by
the ciliary muscle which squeezes the lens changing its shape
Near Point The near point is the closest distance from the eye for which the lens can
focus an image on the retina It is usually 18-25 cm for young persons
Far Point The far point is the greatest distance from the eye for which the lens can
focus an image on the retina For a person with normal vision the far point is at
infinity
Nearsightedness A person is nearsighted if their far point is at some finite distance
less than infinity This condition can be corrected with a lens that takes an object at
infinity and produces a virtual image at the personrsquos far point
Farsightedness and Presbyopia A person is farsighted if their near point is too far
away for comfortable near work like reading or knitting This can be corrected by a
lens which takes an object at a normal near point distance of 18-25 cm and produces a
virtual image at the personrsquos natural near point Presbyopia involves a similar
problem which nearly all people experience as they age The ciliary muscle becomes
too weak and the lens becomes too stiff to allow the eye to provide for both near and
far vision The solution for this problem is either reading glasses or bifocal lenses
Reading classes are just weak magnifying glasses mounted on eyeglass frames
Bifocal lenses are split into upper and lower halves The lower half is a lens which
gives the proper correction for near work and the upper half is a different lens for
proper focusing at infinity
Angular Size
When an object is brought closer to the eye it appears to be larger because the image
on the retina is larger The size of this image is directly proportional to the objectrsquos
angular size which is the angle subtended by the object measured from the center of
the lens of the eye In optical instruments which are to be used with the eye the
angular size of the final image is whatrsquos important because it determines how large
the image will appear to the viewer
19
Simple Magnifier
A simple magnifier is a single converging lens or magnifying glass It takes an object
closer to the eye than a normal near point and produces a virtual image at or beyond
this near point The angular magnification is defined to be the ratio of the angular size
when viewed through the lens to the angular size of the object when viewed at the
normal near point (without aid of the lens)
Microscope
This instrument has two lenses (1) The objective is near the object being viewed and
produces a greatly magnified real image (2) The eyepiece is a simple magnifier which
the viewer uses to closely examine the image from (1)
Telescope
This instrument also has two lenses (1) The objective at the front of the telescope
takes light from a distant object and produces a real inverted image (which is rather
small) near its focal point (2) This small real image is then examined by the eyepiece
functioning as a simple magnifier to produce a virtual image with a larger angular
size
20
Serway Chapter 37
Two-Slit Interference
If light is incident on two closely spaced narrow slits a pattern of light and dark
stripes is produced beyond the slits The bright stripes or fringes are caused by
constructive interference of the two waves coming from the slits Constructive
interference occurs whenever two waves arrive at a location in phase with each other
This occurs when the distance x1 from slit 1 to a point P on the screen and the
distance x2 from slit 2 to point P differ by in integral number of wavelengths
1 2 where 0 1 2x x m m
where λ is the wavelength of the light When x1 and x2 are much larger than the slit
spacing d this condition reduces to
sind m
where θ is the angle between the direction of the incident light and the direction of the
light arriving at the screen
Thin Films
When light is partially reflected and partially transmitted by a thin film of transparent
material it is possible to have interference between the wave reflected from the front
of the film and light reflected from the back of the film (The colored reflections from
the thin film of oil on the water in a rain-soaked parking lot are an example of this
effect) It is difficult to write down formulas that will work in all cases so we will
just review the important principles here
1 If the two reflected waves are in phase with each other the film has enhanced
reflection (constructive interference) but if the two reflected waves are out of phase
with each other reflection is diminished (destructive interference) Phase shifts occur
due to reflection and due to the extra path length through the film of the wave
reflected from the back of the film
2 The phase change due to reflection is determined by the difference in index of
refraction between the two media involved in the reflection If the wave is incident
21
from a medium with a low index of refraction into a medium with a high index of
refraction a phase change of 180deg occurs and the reflected wave is inverted If
incident from high to low no phase shift occurs and the reflected wave is non-
inverted
3 The extra path length through the film of the wave reflected from the back of the
film is equal to 2t where t is the thickness of the film (The incident light is assumed
to be normal to the surface of the film) The number of wavelengths contained in the
extra path length is equal to 2tλn where λn = λn is the wavelength of the light in the
film
4 Rules for reflection from thin films
If one ray is inverted and the other is not then we have
12 constructive
2 nt m
2 destructivent m
If either both rays are inverted or both are non-inverted we have
2 constructivent m
12 destructive
2 nt m
22
Serway Chapter 38
Diffraction Grating
A diffraction grating is simply a fancy version of two-slit interference with the two
slits replaced by thousands of slits Just as in the two-slit case the bright fringes
occur at angles given by
sin md
but in this case d the distance between neighboring slits is made to be very small
The effect of having many slits instead of two is to make each bright fringe highly
localized with wide dark regions between neighboring maxima
Single Slit Diffraction
When light passes through an opening in an opaque screen an interference pattern is
produced beyond the opening To understand why we may replace the single opening
by many small coherent sources of light These many sources interfere with each
other producing a pattern known as a diffraction pattern (Note that many authors do
not distinguish between interference and diffraction treating them as interchangeable
terms) If the opening is a slit of width a then the diffraction pattern far from the slit
will have a bright central maximum with a succession of minima and weaker maxima
on either side The angle between the incident direction and the minima is given by
sin ma
where m = plusmn1 plusmn2
If the opening is circular with diameter D the angle between the incident direction
and the first minimum is given by
sin 122D
Optical Resolution and Rayleighrsquos Criterion
Two point sources can just be resolved (distinguished from each other) if the peak of
the diffraction image of the first source overlies the first minimum of the diffraction
image of the second source For circular holes of the kind usually encountered in
23
optical devices this condition is approximately satisfied when the angular separation
between the two sources as viewed from the optical instrument is greater than or equal
to the critical angle
min 122D
where λ is the wavelength of the light and where D is the diameter of the aperture in
the instrument
Polarization
We say that an electromagnetic wave is polarized if its electric field vector doesnrsquot
change direction in a random fashion The simplest kind of polarization is linear
polarization in which the electric field vector oscillates back and forth along the
same axis in space Polarized light can be produced from normal unpolarized light by
selective absorption (as in Polaroid sunglasses) by passing light through crystals that
have different indices of refraction for different polarizations (double refraction) by
scattering (the blue sky is polarized) and by reflection (glare) Polarization by
reflection occurs when light reflects from a shiny insulating (non-metallic) surface
The amount of polarization is greatest for reflection at Brewsterrsquos angle
2
1
tan P
n
n
where θP is the incidence angle of light from medium 1 onto medium 2 and where n1
and n2 are the indices of refraction for the media (In our everyday experience n1 = 1
since the light comes in through the air and n2 is the index of refraction of the shiny
insulating material producing the glare eg water glass plastic paint etc)
Malusrsquos Law
The intensity of transmitted polarized light through a perfect polarizer is related to the
incident intensity of polarized light by Malusrsquos law
2cosoI I
where Io is the intensity of the incident light and where θ is the angle between the
electric field vector in the incident wave and the transmission axis the polarizer
24
Serway Chapter 39
Principles of Relativity
All of the weirdness of relativity flows from two simple principles
(1) The laws of physics must be the same in all inertial (non-accelerating) reference
frames
(2) The speed of light in vacuum has the same value in all inertial reference frames
Note that (1) seems reasonable but (2) is very odd It says that if two space ships are
approaching each other at nearly the speed of light and a laser pulse is shot from ship
1 toward ship 2 then when the people on ship 2 measure the speed of the pulse as it
goes by it is moving at 3 times 108 ms the same as if the ships were stationary
Simultaneity
If observer 1 sees two events in her own frame as simultaneous at two different
locations a moving observer 2 will see these two events happening at different times
Relativistic gamma
The factor γ (gamma) appears regularly in the formulas of relativity
2
2
1
1 vc
where v is the relative speed between two inertial frames
Time Dilation
If observer 1 sees two events at the same location in space separated by time Δtp in his
own frame then observer 2 moving at speed v relative to observer 1 will see these
two events separated by a longer time Δt
pt t
Moving clocks run slow
Length Contraction
If observer 1 measures the length of an object along the x-direction at some instant of
time in her frame to be Lp then observer 2 moving in the x-direction at speed v will
measure the length L of the object to be shorter
pLL
25
Moving meter sticks are short
Velocity Addition
If a particle is moving at velocity vac relative to frame c and if frame c is moving at
speed vcb with respect to frame b (with both velocities directed along the same line)
what would be the velocity of the particle in frame b In classical physics the answer is
ab ac cbv v v
but if any of these velocities are an appreciable fraction of the speed of light the
answer changes
21ac cb
abac cb
v Vv
v v c
In using this equation make sure that vac has the same sign in the numerator and in the
denominator and that vbc also has the same sign in both places Note that this equation
is different (and in our opinion easier to understand) than the equations in the text
Momentum and Energy
The momentum and total energy of a particle moving at velocity v (its speed is v v )
are given by
22
2 2
22
1 1 vvc c
m mcm E mc
vp v
Rest Energy
When a particle is at rest (v = 0 and γ = 1) then E = mc2 is called the rest energy It
includes the potential energy of the particle so that if we change the potential energy
of a particle we change its rest mass
Kinetic Energy The kinetic energy is the difference between the total energy and the
rest energy
2 2K mc mc
For v ltlt c this reduces to K = mv22
General Relativity
Einsteinrsquos principle of equivalence in the theory of general relativity is In the vicinity
of a point a gravitational field is equivalent to an accelerated frame of reference in
26
the absence of gravitational effects For example there is no difference whatever in
the physics experienced by an astronaut on earth and one who is in interstellar space
far away from any stars or planets but who has just turned on her spaceshiprsquos engines
and is accelerating at 98 ms2
27
Serway Chapter 40
Photons
We now know that even though light behaves like a wave it is better described as
consisting of small packets of energy called photons The energy of a photon is related
to its frequency by
E hf
where h is Planckrsquos constant
346626 10 J sh
Photons also carry momentum given by
hf hp
c
Since the amount of energy in each photon is fixed the difference between dim light
and intense light (of the same wavelength) is that dim light consists of fewer photons
than intense light
Photoelectric Effect
In the photoelectric effect light shining onto a metal surface gives the electrons in the
metal enough energy to escape and be detected It requires a certain amount of energy
(called the work function typically = 3 ndash 5 eV) just to kick the electron out of the
metal so the light must deliver at least this much energy to an electron to produce the
effect It is observed that red light no matter how intense never produces electrons
But ultraviolet light even if quite dim will eject electrons from the metal Since red
light consists of 2 eV photons and ultraviolet light has photons with energies around
4-6 eV the photon idea explains the behavior of the photoelectric effect The
maximum energy that an ejected electron can have is
maxK hf
where f is the photon frequency The minimum frequency that light can have and cause
electrons to be ejected is called the cutoff energy and is found by setting Kmax = 0
cf h
28
Compton Effect
When high-frequency light interacts with free electrons the scattered light does not
have the same wavelength as the incident light contrary to what classical
electromagnetism would predict But the photon picture in which photons have
momentum and energy allows us to treat scattering as a collision between the photon
and the electron In this collision the electron and the scattered photon both have a
different momentum and energy than they did before And since p = hλ for a photon
if the momentum is different the wavelength will also be different
after before 1 cose
h
m c
where the angle θ is the angle between the incoming photon direction and the
direction of the scattered photon
Particles are Waves
Since photons behave like particles sometimes it is not surprising that elementary
particles can behave like waves sometimes The wavelength of a particle with
momentum p is given by
h
p
Wave-particle Duality
Both photons and elementary particles have a dual nature sometimes they behave like
particles and sometimes they behave like waves It is hard for us to comprehend the
nature of such an object by making mental pictures but experiments definitively show
that this is the case Since we have no direct experience with photons and elementary
particles (because their energies are so much smaller than the energies of the everyday
objects we encounter) it is perhaps not surprising that we have a hard time forming a
mental picture of how they behave
Electron Interference
Since an electron is both a particle and a wave just like a photon it should be able to
produce an interference pattern This is observed If an electron beam is shot at two
closely-spaced slits and if the electrons that pass through the slits are detected
downstream it is found that there are some locations where electrons are never detected
29
and others where lots of electrons are detected The pattern is exactly the same as the
one observed for light waves provided that we use the electron wavelength λ = hp in
place of the wavelength of light This pattern is observed even though each electron is
detected as a single dot on the screen Only after many such dots are collected does the
pattern emerge And if we try to understand how this effect could possibly work by
looking closely at each slit to see which one the electron came through the pattern
disappears the act of measurement destroys the interference This means that each
single electron somehow comes through both slits (as a wave would)
Uncertainty Principle
In classical physics we always imagine that the positions and momenta of moving
particles have definite values It might be hard to measure them but surely at each
instant of time a particle should be precisely located at some point in space and have a
similarly precise momentum This turns out not to be true Instead both position and
momentum are required to be uncertain with their uncertainties Δx and Δp satisfying
the Heisenberg uncertainty relation
2x p
So if the particle were known to be precisely at some particular location (so that Δx =
0) we couldnrsquot know anything about its momentum (Δp = infin) and if its momentum
were exactly known we couldnrsquot know its position
There is a similar relation involving the particlersquos energy E and the time interval Δt
over which this energy is measured
2E t
This means that energy is not actually conserved if we are considering very small time
intervals and this brief non-conservation of energy has been observed
30
Serway Chapter 41
Wave Function ψ and Probability
Quantum mechanics does not predict exactly what an electron or a photon will do Instead
it specifies the wave function or probability amplitude ψ of an electron or a photon
This wave function is a complex-valued function of space and time whose squared
magnitude is the probability density P for finding a particle at a particular place in
space at a certain time
2P
where is the complex conjugate of ψ
For example the wave function of an electron with perfectly specified momentum p
would have a wavelength given by
h
p
and its probability amplitude would be proportional to
2i x ipxe e
The corresponding probability density would then be
21ipx ip ipxP e e e
which means that the electron is equally probable to be anywhere along the x axis
This is in accord with the uncertainty principle since we specified the momentum
precisely we canrsquot have any idea about the position of the electron
As another example you have probably seen ldquofuzzy ballrdquo drawings of electron
orbitals in a chemistry book These fuzzy balls are meant to indicate the distribution
of the probability density 2
P in the orbital
Particle in a Box
A simple example in which we can calculate the wave function is the case of a particle of
mass m constrained to be inside a 1-dimensional box between x = 0 and x = L In this
case the general wave function is a linear superposition of wave functions ψn of the form
sin niE tn
n xx t A e
L
31
where A is a positive constant where n = 1 2 3 and where the energy associated
with each of the quantum states ψn is given by
22
28n
hE n
mL
This wave function is zero at x = 0 and x = L which means that the particle will never
be found at the walls of the box The wave function has maximum values in the
interior and at these places the particle is most likely to be found and it also has
places where it is zero and at these places the particle will also never be found as
expressed by the formula
2sinn x
PL
The particle in the box is interfering with itself producing a probability interference
pattern across the box just like the interference patterns we studied with light and
sound
32
Serway Chapter 42
Atoms
Many people picture an atom as a miniature solar system where electrons orbit around
a massive nucleus at the center This picture is misleading because in atoms the wave
nature of electrons dominates The electrons form 3-dimensional standing waves
(called orbitals) centered at the nucleus And if you ask what is it thatrsquos waving the
answer is ψ the probability amplitude (see Chapter 41)
Standing waves on a string can occur only for certain frequencies (the fundamental
and higher harmonics) Similarly atomic orbitals occur only for certain energies For
the hydrogen atom the energies of the orbitals have the a particularly simple form
2
1136 eVnE
n
where n = 1 2 3 is called the principal quantum number For other atoms the
determination of the orbital energies requires numerical calculation by computers
Atomic Spectra
If an electron is somehow given extra extra energy (we say that it is excited) so that it
occupies a higher orbital it will eventually ldquofallrdquo back down to a lower orbital Each
time an electron falls to a lower orbital it loses the difference in energy between the
two orbitals in the form of a photon Since the orbital energies are discrete so are the
energy differences and so are the wavelengths of the emitted light The entire set of
these discrete wavelengths is called the atomic spectrum and it is unique to each type
of atom For hydrogen the spectrum can be simply written as
H 2 2
1 1 1
f i
Rn n
where RH = 10973732 times 107 m-1 For other atoms the spectrum cannot be expressed as
a formula
Orbital Angular Momentum Electrons in atoms also have quantized values of
angular momentum The orbital quantum number ℓ specifies the value of this
quantized angular momentum through the formula
1L
33
If we want to know the value of the angular momentum along some direction in space
say the z direction the answer is not L but rather
zL m
where mℓ is another quantum number which runs from
1 1m
This quantum number is important when an atom sits in a magnetic field
Spin Angular Momentum It has been found experimentally that electrons and other
charged particles also carry internal angular momentum which we call spin
Electrons have an intrinsic spin angular momentum s along a specified axis that is
extremely quantized it can only take on 2 possible values
1
2zs s
We thus say that electrons have spin s = 12 or that electrons are rdquospin one-half parti-
cles Since s is an angular momentum it obeys the same rule as that for orbital angular
momentum namely that if its value along some axis is s then its total magnitude is
given by
31
2S s s
I know this seems weird but quantum mechanics is weird The only excuse for this
bizarre way of looking at the world is that it predicts what happens in experiments
Exclusion Principle The answer to the question of how many electrons (or any other
spin one-half particle) can be in one particular quantum state was discovered by
Wolfgang Pauli and is called the exclusion principle
ldquoNo two electrons can ever be in the same quantum state therefore no two electrons
in the same atom can have the same set of quantum numbersrdquo
This is the reason that we have atoms with different properties instead of every atom
simple having all of its electrons in the ground state All of the variety we see around
us in the world is the result of chemical differences and these differences would not
exist unless electrons obeyed this important principle The entire structure of the
periodic table (see pages 1377-1379 in Serway) is an expression of this principle
34
Serway Chapter 44
Nuclear Properties
The nucleus is composed of protons with charge +e and mass mp = 1007226 u and of
neutrons with zero charge and mass mn = 1008665 u where u is one atomic mass unit
-271 u=1660540 10 kg
The atomic number Z counts the number of protons in a nucleus while the neutron
number N counts the number of neutrons The mass number A is the sum of the two
A N Z
Protons and nuetrons have about the same mass and some times called baryons (heavy
ones) or nucleons (partcles in the nucleus) A is sometimes called the baryon
number The chemical elements are distinguished by Z ie hydrogen has Z = 1 iron
has Z = 26 uranium has Z = 92 etc But for a given Z there might be several nuclei
with different numbers of neutrons These nuclear siblings are called isotopes
These numbers are used to label nuclei according to the pattern
5626 ie FeA
Z X
denotes the isotope of iron with N = 56 ndash 26 = 30 neutrons
The nucleus is roughly spherical with a radius given approximately by
1 3 150 0where 12 10 mr r A r
Nuclear Stability
Because the positively charged protons electrically repel each other with an enormous
force at distances as small as 10-15 m there must be some really strong force that that
overcomes electrical repulsion to hold protons and neutrons together This force is
called with some lack of imagination the strong force It is a very short-range force
(it only acts over a distance of about 2 x 10-15 m) and attracts protons to protons
neutrons to neutrons and neutrons to protons But in spite of this strong nuclear force
the coulomb repulsion of the protons is still present so anything that might keep the
protons from being right next to each other would help keep the nucleus from
35
exploding This role is played by the neutrons and for nuclei with Z le 20 the stable
nuclei roughly have N = Z
For Z gt 20 the coulomb repulsion force is more powerful and more neutrons are
needed to dilute it up to about N = 15Z around Z = 80 For Z gt 83 no amount of
neutrons can help and these nuclei are unstable (radioactive) There is an ldquoisland of
(relative) stabilityrdquo around Z=90-92 (thorium and uranium respectively) These
elements have one or two isotopes with half-lifes of billions of years so there are
substantial amounts of such elements on Earth This fact makes it possible to have
practical fission devices
Radioactive Decay
There are three types of radioactive decay
Alpha decay the nucleus kicks out a helium nucleus (N = 2 Z = 2 A = 4) So if the
original nucleus is called X and the new nucleus is called Y then the decay would look
like this
4 42 2X Y+ HeA A
Z Z
Beta decay the nucleus either kicks out an electron (endash) or its positively-charged
antimatter twin the positron (e+) plus either an electron neutrino v or an electron
anti-neutrino v
1X Y+e electron decayA AZ Z v
1X Y+e positron decayA AZ Z v
A neutrino is a particle with no charge hardly any mass (much less than the electron
mass) and interacts so weakly with matter that most neutrinos upon encountering the
planet earth just pass right through it as if it werenrsquot there
Gamma decay the nucleons in the nucleus X are in an excited energy state X
(perhaps as a result of having undergone alpha or beta decay) and they drop down to a
lower energy state shedding the energy as a high frequency photon
X XA AZ Z
36
This process is exactly analogous to the way that the electrons in atoms emit photons
Decay Rate and Half Life
There is no way to predict exactly when an unstable or excited nucleus will decay but
there is an average rate at which this decay occurs called the decay constant λ The
meaning of this constant is that if there are a large number N of nuclei in a sample
then the number of decays per second that will be observed (called the decay rate R)
is R = λN In mathematical language
dMR N
dt
This simple differential equation has for its solution
0tN t N e
where N0 is the number of nuclei in the sample at time t = 0
The half-life is the time it takes for half of the nuclei in the sample to decay and is
related to the decay constant by
1 2
ln 2 0693T
Disintegration Energy
When a nucleus decays it is making a transition to an overall state of lower energy
which means according to Einsteinrsquos famous formula E = mc2 that the sum of the
masses after the decay must be less than the mass before with the lost mass appearing
as kinetic energy among the decay products For example in alpha decay this kinetic
energy called the disintegration energy Q is given by
2X YQ M M M c
37
Serway Chapter 45
Nuclear energy
There are two ways to extract energy from the nucleus fission and fusion For nuclei
with Z greater than 26 breaking the nucleus apart into pieces leads to a lower
2mc energy than the original nucleus so energy can be extracted by fission For
nuclei with Z less than 26 a lower energy is achieved by combining nuclei so energy
can be extracted by fusion
Fission
Since neutrons have no charge they are not repelled from nuclei as protons are for
this reason their behavior is the key to understanding how fission works
Fast neutrons mostly bounce off other nuclei with each collision slowing the neutron
down This elastic energy loss is most effective if the other nuclei have low mass (like
hydrogen) and these materials are called moderators because of their ability to slow
down fast neutrons
The reason that slowing neutrons is important is that slow neutrons are much more
likely to be absorbed by a nucleus which then leads to nuclear reactions of various
kinds For a few very large nuclei like uranium-235 and some plutonium isotopes
absorption of a slow neutron causes the nucleus to split into two large fragments plus
2 or 3 fast neutrons (energy is released in the process too) Thus one slow neutron
can produce energy plus 2 or 3 more neutrons which if moderated from fast to slow
can split 2 or 3 more nuclei leading to a runaway chain reaction If the reaction is
allowed to proceed unchecked it produces a large explosion If a neutron absorbing
material is added to the mix (like the cadmium in reactor control rods) it is possible to
keep the reaction under control and to extract the released energy as heat to drive
steam turbines and produce electricity
Fusion
Fusion involves mashing two nuclei together and since they are both charged and repel
each other this reaction is much harder to make go The nuclei must have enough energy to
overcome the coulomb repulsion which is why this reaction requires a high temperature
(hundreds of millions of degrees K) like that in the sun (which is burning hydrogen to
helium via fusion) or like that in the center of the fission explosion that is used to detonate
38
a hydrogen bomb
This reaction is of interest for power production in spite of this difficult temperature
requirement because of the abundance of fusion fuel on the planet There are about
012 g of deuterium ( 21H ) in every gallon of water on earth and it only costs about 4
cents to extract it The fusion energy available from this minuscule amount of
deuterium would run a 1000-Megawatt power plant for 10 seconds Water is so
abundant on earth that if fusion were to work we would have an essentially
inexhaustible source of energy
So why donrsquot we have fusion power plants Well the fuel is cheap but the match is
incredibly expensive The only way we know to control this difficult high-temperature
reaction is with large and expensive pieces of equipment involving either large
magnetic fields and complex high-power electromagnetic antennas or with gigantic
(football-fieldsized) laser facilities involving more than a hundred of the highest-
energy lasers ever built Power plants based on these current methods for controlling
fusion are unattractive to the fiscally-minded people who run the electric power
industry Hopefully better designs will be discovered as experiments continue
39
c actinium 68 Er erbium 101 Md mendelevium 104 Rf rutherfordium l aluminum 63 Eu europium 80 Hg mercury 62 Sm samarium m americium 100 Fm fermium 42 Mo molybdenum 21 Sc scandium b antimony 9 F fluorine 60 Nd neodymium 106 Sg seaborgium r argon 87 Fr francium 10 Ne neon 34 Se selenium s arsenic 64 Gd gadolinium 93 Np neptunium 14 Si silicon t astatine 31 Ga gallium 28 Ni nickel 47 Ag silver a barium 32 Ge germanium 41 Nb niobium 11 Na sodium k berkelium 79 Au gold 7 N nitrogen 38 Sr strontium e beryllium 72 Hf hafnium 102 No nobelium 16 S sulfur i bismuth 108 Hs hassium 76 Os osmium 73 Ta tantalum h bohrium 2 He helium 8 O oxygen 43 Tc technetium
boron 67 Ho holmium 46 Pd palladium 52 Te tellurium r bromine 1 H hydrogen 15 P phosphorus 65 Tb terbium d cadmium 49 In indium 78 Pt platinum 81 Tl thallium a calcium 53 I iodine 94 Pu plutonium 90 Th thorium f californium 77 Ir iridium 84 Po polonium 69 Tm thulium
carbon 26 Fe iron 19 K potassium 50 Sn tin e cerium 36 Kr krypton 59 Pr praseodymium 22 Ti titanium s cesium 57 La lanthanum 61 Pm promethium 74 W tungsten l chlorine 103 Lr lawrencium 91 Pa protactinium 92 U uranium r chromium 82 Pb lead 88 Ra radium 23 V vanadium o cobalt 3 Li lithium 86 Rn radon 54 Xe xenon u copper 71 Lu lutetium 75 Re rhenium 70 Yb ytterbium m curium 12 Mg magnesium 45 Rh rhodium 39 Y yttrium b dubnium 25 Mn manganese 37 Rb rubidium 30 Zn zinc y dysprosium 109 Mt meitnerium 44 Ru ruthenium 40 Zr zirconium s einsteinium
40
UNITS The SI units are given in parentheses Other commonly used units are given in terms of the SI units Symbols conform with the recommendations of the American National Standards Institute (ANSI) and the American Institute of Physics (AIP) Commonly used multiples of SI units
prefix symbol factor tera T 1012 giga G 109
mega M 106 kilo k 103
centi c 10-2 milli m 10-3
micro μ 10-6 nano n 10-9 pico p 10-12
femto f 10-15
length meter (m) angstrom 1 Ǻ = 10-10 m inch 1 in = 254 x 10-2 m foot 1 ft = 03048 m mile 1 mi = 1609 m
mass kilogram (kg) atomic mass unit 1 u = 1661 x 10-27 kgslug 1 slug = 1459 kg
force newton (N = kg bull ms2) dyne 1 dyn = 10-5 N pound 1 lb = 4448 N
pressure Pascal (Pa = kgm bull s2) atmosphere 1 atm = 1013 x 105 Pa poundssquare inch 1 psi = 6895 Pa cm of mercury 1 cm Hg = 1333 Pa bar 1 bar = 1000 x 105 Pa torr 1 torr = 1333 Pa
time second (s) minute 1 min = 60 s hour 1 h =3600 s
frequency hertz (Hz = s-1) radianssecond 1 rads = 12π Hz
energy joule (J = kg bull m2s2) erg 1 erg = 10-7 J electron volt 1 eV = 1602 x 10-19 J calorie 1 cal = 4187 J kilowatt-hour 1 kWh = 36 x 106 J British thermal unit 1 Btu = 1055 J
power watt (W = kg bull m2s3) horsepower 1 hp = 7457 W
charge coulomb (C = A bull s)
electric potential volt (V = kg bull m2 s3 bull A)
current ampere (A)
resistance ohm (Ω = kg bull m2s3 bull A2)
capacitance farad (F = s4 bull A2kg bull m2)
magnetic field tesla (T = kgs2 bull A) gauss 1 G = 10-4 T
magnetic flux weber (Wb = kg m2s2 bull A) maxwell 1 Mx = 10-8 Wb
magnetic inductance henry (H = kg bull m2s2 bullA2)
temperature kelvin (K) degrees Celsius 0degC = 27315 K
angle radian (rad) degree 1 deg = π180 rad revolution 1 rev = 2π rad
41
Some Physical Constantsa Quantity Symbol Valueb Atomic mass unit μ 1660 538 73 (13) x 10-27 kg
931494 013 (37) MeVc2 Avogadros number NA 6022 141 99 (47) x 1023 particlesmol Avogadrorsquos number 6022 x 1023 mol Bohr magneton
2B e
em 9274 008 99 (37) x 10-24 JT
Bohr magneton μB 927 x 10-24 JT Bohr radius
0
2
2e em e k
a 5291 772 083 (19) x 10-11 m
Boltzmanns constant B A
RNk 1380 650 3 (24) x 10-23 JK
Boltzmannrsquos constant kB 1380 x 10-23 JK Compton wavelength
C eh
m c 2426 310 215 (18) x 10-12 m
Deuteron mass md 3343 583 09 (26) x 10-27 kg 2013 553 212 71 (35) u
electron charge e 1602 x 10-19 C Electron mass me 9109 381 88 (72) x 10-31 kg
5485 799 110 (12) x 10-4 u 0510 998 902 (21) MeVc2
electron mass 911 x 10-31 kg Electron volt eV 1602 176 462 (63) x 10-19 J Elementary charge e 1602 176 462 (63) x 10-19 C Gas constant R 8314 472 (15) JKmol Gravitational constant G 6673 (10) x 10-11 Nm2kg2 Neutron mass mn 1674 927 16 (13) x 10-27 kg
1008 664 915 78 (55) u 939565 330 (38) MeVc2
neutron mass 1675 x 10-27 kg Nuclear magneton
2n p
em 5050 783 17 (20) x 10-27 JT
permeability constant μ0 1257 x 10-6 Hm permittivity constant ε0 8854 x 10-12 Fm Plancks constant h
2h
6626 068 76 (52) x 10-34 Js 1054 571 596 (82) x 10-34 Js
Planckrsquos constant h ħ
6626 x 10-34 Js 1055 x 10-34 Js
Proton mass mp 1672621 58 (13) x 10-27 kg 1007 276 466 88 (13) u 938271 998 (38) MeVc2
proton mass 1673 x 10-27 kg Rydberg constant RH 1097 373 156 854 9 (83) x 107 m-1 Speed of light in vacuum c 2997 92458 x 108 ms (exact) speed of light c 300 x 108 ms a
These constants are the values recommended in 1998 by CODATA based on a least-squares adjustment of data from different measurements For a more complete list see P J Mohr and B N Taylor Rev Mod Phys 72351 2000 b The numbers in parentheses for the values above represent the uncertainties of the last two digits
NOTE The ones in red are the ones from Appendix 13 The ones in yellow are the ones that are from the Serway book table but that were already on the Appendix 13 table
42
INDEX
Absolute zero 9
Adiabatic 17
Adiabatic exponent 18
Adiabatic process 17
Alpha decay 46
Amplitude 4
Angle of reflection 25
Angular frequency 3
Angular magnification 30
Angular momentum atomic 43
Angular size 29
Archimedes Principle 1
Atomic spectra 43
Atoms 43
Avogadrorsquos number 10
Beats 8
Bernoullirsquos Equation 2
Beta decay 46
Bifocals 29
Brewsterrsquos angle 34
British Thermal Unit Btu 11
Buoyancy 2
calorie 11
Calorie 11
Camera 28
Carnot cycle 21
Carnot efficiency 22
Celsius scale 9
Ciliary muscle 29
Coefficient of performance 21
Compton effect 39
Constructive interference 7
Continuity equation of 2
Contraction length 35
Convection 14
Curved mirrors 27
Decay constant 46
Decay rate radioactive 46
Decibel scale 5
Degrees of freedom 16
Density 1
Destructive interference 7
Diffraction single slit 33
Diffraction grating 33
Dilation time 35
Disintegration energy 47
Dispersion of light 26
Doppler effect 6
Double slit interference 31
Efficiency 20
Electron interference 39
Emissivity 15
Energy nuclear decay 47
Energy relativistic 36
43
Engineering work 20
Entropy 22
Entropy ideal gas 24
Equipartition of energy 18
Equivalence principle 36
Exclusion principle 44
Expansion thermal 9
Expansion coefficient linear 9
Eye 29
Fahrenheit scale 9
Far point 29
Farsightedness 29
First Law of Thermodynamics 12
Fission 48
Flux volume 2
Free expansion 23
Fringes 31
Fusion 48
Fusion heat of 11
Gamma relativistic 35
Gamma decay 46
General relativity 36
Half-life 46
Heat 11
Heat capacity 11
Heat conduction 13
Heat engine 20
Heat of fusion 11
Heat of vaporization 11
Heat pump 20
Heats of transformation 11
Hydrostatics 1
Ideal Gas Law 10
Images real and virtual 27
Intensity sound 5
Interference 7
Interference two-slit 31
Internal energy 12
Internal energy degrees of freedom 16
Irreversible process 19 23
Isotopes 45
Joule 11
Kelvin scale 9
Kinetic energy relativity 36
Kinetic theory 16
Latent heat 11
Length contraction 35
Linear expansion coefficient 9
Linear polarization 34
Linear superposition 7
Longitudinal wave 3
Loudness 5
Magnification lateral 27
Magnifying glass 30
Malusrsquos law 34
Microscope 30
Momentum relativistic 36
Muscle ciliary 29
44
Musical instruments 8
Musical scale 8
Near point 29
Nearsightedness 29
Nonlinear 7
Nuclear energy 48
Nuclear properties 45
Nuclear stability 45
Octave 8
Optical resolution 33
Orbital quantum number ℓ 43
Orbitals 43
Particle in a Box 41
Particles are waves 39
Pascalrsquos Principle 1
Period 3
Photoelectric effect 38
Photons 38
Pitch 8
Polarization 34
Power sound 5
Presbyopia 29
Pressure 1
Principal quantum number n 43
Principle of equivalence 36
Principle of linear superposition 7
Probability amplitude ψ 41
Processes thermodynamic 13
R-value 14
Radiation thermal 14
Radioactive decay 46
Radioactive decay rate 46
Ray tracing 27
Rayleighrsquos criterion 33
Reading glasses 29
Real image 27
Refraction 25
Refrigerator 20
Relativistic gamma 35
Relativity principles 35
Resolved for light sources 33
Rest energy 36
Reversible process 19 22
Rope wave speed 4
Second Law of Thermodynamics 19
Shock waves 6
Simple magnifier 30
Simultaneity 35
Single slit diffraction 33
Snellrsquos law 25
Sound speed 5
Specific heat 11
Spin Angular momentum 44
Standing waves 7
Stefanrsquos law 14
Strong force 45
Telescope 30
Temperature 9
45
Temperature Scales 9
Thermal conductivity 14
Thermal energy 12
Thermal expansion 9
Thin film interference 31
Thin lenses 28
Time dilation 35
Tone musical 8
Total internal reflection 25
Transverse wave 3
Traveling Waves 3
Two-slit interference 31
Uncertainty principle 40
Vaporization heat of 11
Velocity addition relativity 36
Virtual image 27
Volume flux 2
Wave function 41
Wave Function ψ and Probability 41
Wave speed 3
Wave-particle duality 39
Wavelength 3
Wavenumber 3
Wien Displacement Law 16
Work 11
6
Heat Capacities of an Ideal Gas The heat capacity of a gas is described by means of
molar heat capacities CV and CP These are the heat capacities per mole and the
subscript V on CV means that the volume is being held constant while for CP the
pressure is held constant For example to raise the temperature of n moles of a gas
whose pressure is held constant by 10 K we would have to supply an amount of heat Q
= nCP (10) K
Molar Specific Heat of an Ideal Gas at Constant Volume
VQ nC T
3monatomic
2VC R
5diatomic
2VC R
5polyatomic
2VC
Real gases deviate from these formulas because in addition to the translational and ro-
tational degrees of freedom they also have vibrational and electronic degrees of
freedom These are unimportant at low temperatures due to quantum mechanical
effects but become increasingly important at higher temperatures The rough rule is
No of degrees of freedom
2VC R
Molar Specific Heat of an Ideal Gas at Constant Pressure
PQ nC T
P VC C R
The internal energy of an ideal gas depends only on the temperature
int VE nC T
Adiabatic Processes in an Ideal Gas
7
An adiabatic process is one in which no heat is exchanged between the system and the
environment When an ideal gas expands or contracts adiabatically not only does its
pressure change as expected from the ideal gas law but its temperature changes as
well Under these conditions the final pressure Pf can be computed from the initial
pressure Pi and from the final and initial volumes Vf and Vi by
or constantf f i iP V PV PV
where γ = CPCV The quantity γ is called the adiabatic exponent Note that this
doesnrsquot mean that the ideal gas law no longer holds it does and in fact it can be
combined with the adiabatic law for pressure given above to obtain the adiabatic law
for temperatures
1 constantTV
Compressions in sound waves are adiabatic because they happen too rapidly for any
appreciable amount of heat to flow This is why the adiabatic exponent appears in the
formula for the speed of sound in an ideal gas
RTv
M
Note that v depends only on T and not on P Because it depends only on the
temperature the speed of sound is the same in Provo as at sea level in spite of the
lower pressure here due to the difference in elevation
Equipartition of Energy
Every kind of molecule has a certain number of degrees of freedom which are
independent ways in which it can store energy Each such degree of freedom has
associated with it ndash on average ndash an energy of 12 Bk T per molecule (or 1
2 RT per mole)
(Note since a molecule has so many possible degrees of freedom it would seem that
there should be a lot of 12 sBk T to spread around But because energy is quantized
some of these degrees of freedom are not ldquoactiverdquo until the temperature becomes high
enough that 12 Bk T is as big as the lowest quantum of energy
8
Serway Chapter 22
Second Law of Thermodynamics
There are several equivalent forms of this important law
Kelvin It is not possible to change heat completely into work with no other change
taking place Or in other words there are no perfect heat engines
Clausius It is not possible for heat to flow from one body to another body at a higher
temperature with no other change taking place Or in other words there are no
perfect refrigerators
Entropy In any thermodynamic process that proceeds from one equilibrium state to
another the entropy of the system + environment either remains unchanged or
increases The total entropy never decreases This law is a bit of an oddity among the
laws of physics because it is not absolute Things are forbidden by the second law not
because it is impossible for them to happen but because it is extremely unlikely for
them to happen (See below for more information about entropy)
Reversible and Irreversible Processes
A reversible process is one which occurs so slowly that it is in thermal equilibrium (or
very nearly so) at all times A hallmark of such processes is that a motion picture of
them looks perfectly normal whether run forward or backward Imagine for instance
the slow expansion of a gas at constant temperature in a cylinder whose volume is being
increased by a slowly moving piston Run the movie backwards and what do you see
You see the slow compression of a gas at constant temperature which looks perfectly
normal
An irreversible process is one which occurs in such a way that thermal equilibrium is
not maintained throughout the process The mark of this kind of process is that a motion
picture of it looks very odd when run backward Imagine the sudden expansion of a gas
into a previously evacuated chamber because a hole was punched in the wall between a
pressurized chamber and the evacuated one Run the movie backward and what do you
see You see the gas in the soon-to-be-evacuated chamber gather itself together and
stream through a tiny hole into a chamber in which there is already plenty of gas If you
have ever seen this happen get in touch with the support group for those who have
witnessed the spontaneous combustion of large mammals by calling 1-800-PYROCOW
9
Heat Engines
Because of the vexing difference in sign between work done on and system and work done
by a system we will invent a new work variable Weng Heat engines do work and so the net
W for these engines is negative But in engineering applications hidden minus signs are
regarded as evil so for heat engines we donrsquot talk about W instead we talk about its
magnitude engW W So for heat engines the first law is
int engE Q W
But for heat pumps and refrigerators work is done on the system so we use the usual work
W when we talk about these systems
A heat engine is a machine that absorbs heat converts part of it to work and exhausts the
rest The heat must be absorbed at high temperature and exhausted at low temperature If the
absorbed heat is Qh the exhausted heat is Qc and the work done by the engine is Weng then
eng h cW Q Q
and the efficiency of the engine is defined to be
eng
h
We
Q
A perfect engine would convert the heat hQ completely into work Weng giving an effi-
ciency of e = 1 Energy conservation alone allows a perfect engine but the second law
requires e lt 1
Refrigerators and Heat Pumps
A refrigerator is a machine that absorbs heat at low temperature and exhausts it at high
temperature the ldquobackwardsrdquo heat transfer being driven by the work done on the machine
by some source of power A heat pump is a machine that either works like a refrigerator
keeping a place cold by transferring heat from this cold place to a higher temperature
environment (cooling mode like an air conditioner) or it functions as a heater
transferring heat into a warm place from a cooler one (heating mode like a window unit
that heats a house by extracting thermal energy from the cold outdoors) The coefficient
of performance of a refrigerator or of a heat pump in cooling mode is defined to be
COP cooling modecQ
W
10
For a heat pump in heating mode the coefficient of performance is
COP heating modehQ
W
Note that we donrsquot have to use the engineering work here because in these systems
work is done on the system and W is naturally positive
A perfect refrigerator would take in heat Qc and exhaust the same amount of heat without
doing any work giving an infinite coefficient of performance Energy conservation alone
allows a perfect refrigerator but the second law requires COP lt infin A perfect heat pump
in heating mode would transfer Qh into the house without doing any work and so would
also have an infinite coefficient of performance The second law forbids this too A good
coefficient of performance for a real device would be around 5 or 6
Carnot Cycle
The most efficient of all possible engines is one that uses the Carnot cycle This cycle
employs an ideal gas has no friction and operates very slowly so that the gas can be in
thermal equilibrium at all parts of the cycle This means of course that it canrsquot
possibly be built and even if it could be built it would not run fast enough to be useful
Nevertheless this cycle is very important because it gives an upper bound on the
efficiency of real engines There cannot possibly be an engine that is more efficient
than one based on the Carnot cycle This cycle consists of the following four steps
1 The ideal gas absorbs heat Qh at constant temperature Th while the gas increases its
volume The reason that heat is absorbed is that expansion tends to cool the gas but
thermal contact with the environment at Th keeps the temperature high by heat
conduction into the ideal gas
2 The ideal gas further increases its volume by an adiabatic expansion This expansion
causes the gas to cool so at the end of this part of the cycle the gas is at temperature Tc
3 The gas exhausts heat Qc at constant temperature Tc while the gas decreases in
volume The reason that heat is exhausted is that compression tends to heat the gas
but thermal contact with the environment at Tc keeps the temperature low by heat
conduction out of the ideal gas
4 The gas is adiabatically compressed back to its original volume (the volume it started
with in step 1) This compression heats the gas from Tc up to Th
11
The efficiency of a Carnot engine is given by the very simple formula
1 cC
h
Te
T
where the temperatures must in be Kelvin No real engine can be more efficient than
this
The coefficient of performance of a Carnot refrigerator or heat pump in cooling mode
is given by
COP cooling mode cC
h c
T
T T
and no real refrigerator can have a coefficient of performance greater than this
The coefficient of performance of a Carnot heat pump in heating mode is
COP heating mode hC
h c
T
T T
So why donrsquot we just use these wonderful Carnot engines and have perfect efficiency
To make the reversible steps in the cycle really reversible they would have to occur
infinitely slowly So the price you pay for making a perfect engine is that it takes
forever to get it to do any work
Entropy
The entropy of a system is defined in terms of its molecular makeup and measures
roughly the disorder of the system If the system is packed into a very small volume
then it is quite ordered and the entropy will be low If it occupies a large volume the
entropy is high (To see what this has to do with disorder note that socks in a drawer
occupy a small volume while socks on the bed in the corner by the door and
hanging from the chandelier occupy a large volume) If the system is very cold then
the molecules hardly move and may even reach out to each other and form a crystal
This is a highly ordered state and therefore has low entropy If the system is very hot
with rapidly speeding molecules crashing into the container walls and bouncing off
each other things are disordered and the entropy is high
It is possible to calculate the entropy of a system in terms of its macroscopic thermody-
namic properties ie pressure volume temperature number of moles etc The key to this
calculation is the concept of a reversible process A reversible process is one that is
12
carried out without friction and so slowly that the process can be reversed at any stage by
making an infinitesimal change in the environment of the system The slow expansion of the
gas at Th in the Carnot cycle with tiny amounts of heat being transferred to the environment
is reversible If we turned around and began slowly to compress the gas would just slowly
exhaust heat to the environment in the exact reverse way that it absorbed it during expansion
Most processes however are irreversible For example if a gas-filled box were suddenly
increased in size so that the particles were free to wander into the void created by the sudden
expansion then the gas would eventually fill the new volume uniformly at the same
temperature as before the expansion (The temperature is unchanged in this imaginary
process because the kinetic energy of the molecules would be unaffected by such an
instantaneous expansion of the container walls) This imaginary but highly thought-
stimulating process is called a free expansion and it is impossible to reverse it During the
expansion we didnrsquot push on any of the molecules so reversing this process would mean
making them go back into their original volume without pushing on them they simply will
not cooperate to this extent Another way to see that just pushing them back where they came
from does not reverse the free expansion is to think about what would happen if we just
compressed either adiabatically or isothermally An adiabatic compression back to the
original volume would heat the gas above its original temperature and an isothermal
compression would require that heat be exhausted to the environment But the free expansion
involved neither temperature changes nor heat exchanges so neither of these two processes
is the reverse of the free expansion It is simply impossible to reverse this rapid expansion in
a way that takes us back to the initial state
It is possible to calculate the change of entropy for both reversible and irreversible processes
Letrsquos consider a reversible process first In a reversible process the entropy change is given
by the formula
dQS
T
where dQ is the amount of heat added to the system during a small step of the process
The total energy change during the process may then simply be calculated by integration
f f
f i i i
dQS S S dS
T
13
(Just as in the case of energy we are mainly interested in differences rather than in
absolute magnitudes)
This integration method does not work for irreversible processes For instance in the
rapid free expansion discussed above no heat is added to the system but its disorder
obviously goes up We need to find some other way to calculate the entropy The key
is the fact that the entropy of a system depends only on its current state and not at all
on how it arrived there So to calculate the entropy change in an irreversible process
first find out what the initial and final conditions of the process are Then invent a
reversible process that takes the system from the initial state to the final state Since
the entropy depends only on the state of system and not on the process the entropy
change for the reversible process is the same as that for the irreversible process
Entropy is thus like pressure or temperature if the pressure changes from Pl to to P2
then the pressure difference is P2 - Pl regardless of how the pressure change was
made This seems obvious for pressure but not for entropy only because you arent
comfortable with entropy yet
Entropy of an Ideal Gas
For n moles of an ideal gas the difference in entropy between a state with temperature
T and volume V and some standard state with temperature To and volume Vo is given
by
ln lno o Vo o
T VS T V S T V nC nR
T V
Entropy in a Phase Change
Calculating entropy change in a phase change like melting or boiling is easy It is
QT Where Q is in the latent heat for example the flows in to cause the melting and
T is the temperature of the phase change There is NO integral to do
14
Serway Chapter 35
Angle of Reflection
If plane waves are incident on a reflecting surface with the propagation direction of
the waves making angle θ1 with the normal direction to the surface then the reflected
angle θ1prime relative to the surface normal is simply
1 1
ie the incident angles and reflected angles are the same
Refraction
If plane waves traveling through medium 1 are incident on a plane interface between
medium 1 and medium 2 then the angle of incidence of the incoming wave θ1 and
the angle of refraction of the transmitted wave θ2 are related by Snellrsquos law
2 2
1 1
sin
sin
v
v
where v1 and v2 are the wave speeds in medium 1 and medium 2 The angle of
incidence and the angle of refraction are both measured between the wave propagation
direction and the normal to the interface In terms of indices of refraction in the case
of light waves Snellrsquos law takes the more familiar form
1 1 2 2sin sinn n
where ni = cvi
Total Internal Reflection
If a wave is incident from a medium of low wave speed into a medium of high wave
speed the law of refraction requires that the angle of refraction be greater than the
angle of incidence If the angle of refraction is required to be greater than 90deg then no
refracted wave can exist and total internal reflection occurs The critical incident
angle θc beyond which total internal reflection occurs is given by
2
1
sin c
n
n
15
Dispersion of Light
In addition to the speed of light varying from material to material it also varies with
wavelength within each material This means that the index of refraction is generally a
function of wavelength
c
nv
Since the wave speed is not constant such a medium is dispersive meaning in this
context that refraction actually disperses white light into its various colors because
Snellrsquos law gives a different angle for each wavelength In most materials the
variation with wavelength is quite small but this small effect is responsible for some
of the most spectacular color effects we ever see including rainbows a flashing
crystal chandelier and the colored fire of a diamond solitaire by candlelight
16
Serway Chapter 36
Real and Virtual Images
When light rays are focused at a certain plane producing an image if a sheet of white
paper is placed there we call the image a real image The images produced by film
projectors and overhead projectors are examples of real images
When light rays appear to come from a certain location but no image is produced
when a screen is placed there we say that there is a virtual image at that location For
instance when you look in a mirror it appears that someone is behind the mirror but
a screen placed back there in the dark would show nothing Your image in the mirror
is a virtual image
Ray Tracing
There are lots of rules about how to find the images in optical systems but the best
way to keep things straight is to learn how to draw the principal rays for curved
mirrors and lenses The rays for convex and concave mirrors are shown in Fig 3615
The rays for converging and diverging thin lenses are shown in Fig 3627 You should
memorize the principal rays and know how to use them to locate images
Curved Mirrors
The focal length of a curved mirror with radius of curvature R is given by
2
Rf
The relation between object distance p image distance q and focal length f is
1 1 1
p q f
If the mirror is a diverging mirror f should be negative and if q should turn out to be
negative the image is virtual
Lateral Magnification
The lateral magnification in an optical system is defined by the ratio of the image size
to the object size
17
Image height
Object height
qM
p
for curved mirrors
As usual there are sign conventions here too but this formula just gives the
magnitude It is better to keep keep track of upright images versus inverted images by
means of ray diagrams rather than by memorizing sign conventions
Thin Lenses
The focal length of a thin lens is related to the radii of curvature of the two faces R1
and R2 of the lens by
1 2
1 1 11n
f R R
Note that this formula differs from Eq (3611) in the text by not having a minus sign
between the two R-terms We like this form better because for a simple converging
lens like a magnifying glass we just use positive values of R for both surfaces If one
of the faces is concave producing divergence use a negative value for R And if a
surface is flat use R = infin If the face is flat the radius is infinite
The relation between the image object and focal distances for a thin lens is the same
as that for a curved mirror
1 1 1
q p f
Use a negative focal length if the lens is diverging
The lateral magnification for a thin lens is the same as for a curved mirror
Image height
Object height
qM
p
Camera
The lens system in a camera projects a real image of an object onto the film (or CCD
array in a digital camera) The position of the image is adjusted to be on the film by
moving the lens into or out of the camera
18
Eye
The eye is like a camera in that a real image is formed on the retina Unlike a camera
the image position is adjusted by changing the focal length of the lens This is done by
the ciliary muscle which squeezes the lens changing its shape
Near Point The near point is the closest distance from the eye for which the lens can
focus an image on the retina It is usually 18-25 cm for young persons
Far Point The far point is the greatest distance from the eye for which the lens can
focus an image on the retina For a person with normal vision the far point is at
infinity
Nearsightedness A person is nearsighted if their far point is at some finite distance
less than infinity This condition can be corrected with a lens that takes an object at
infinity and produces a virtual image at the personrsquos far point
Farsightedness and Presbyopia A person is farsighted if their near point is too far
away for comfortable near work like reading or knitting This can be corrected by a
lens which takes an object at a normal near point distance of 18-25 cm and produces a
virtual image at the personrsquos natural near point Presbyopia involves a similar
problem which nearly all people experience as they age The ciliary muscle becomes
too weak and the lens becomes too stiff to allow the eye to provide for both near and
far vision The solution for this problem is either reading glasses or bifocal lenses
Reading classes are just weak magnifying glasses mounted on eyeglass frames
Bifocal lenses are split into upper and lower halves The lower half is a lens which
gives the proper correction for near work and the upper half is a different lens for
proper focusing at infinity
Angular Size
When an object is brought closer to the eye it appears to be larger because the image
on the retina is larger The size of this image is directly proportional to the objectrsquos
angular size which is the angle subtended by the object measured from the center of
the lens of the eye In optical instruments which are to be used with the eye the
angular size of the final image is whatrsquos important because it determines how large
the image will appear to the viewer
19
Simple Magnifier
A simple magnifier is a single converging lens or magnifying glass It takes an object
closer to the eye than a normal near point and produces a virtual image at or beyond
this near point The angular magnification is defined to be the ratio of the angular size
when viewed through the lens to the angular size of the object when viewed at the
normal near point (without aid of the lens)
Microscope
This instrument has two lenses (1) The objective is near the object being viewed and
produces a greatly magnified real image (2) The eyepiece is a simple magnifier which
the viewer uses to closely examine the image from (1)
Telescope
This instrument also has two lenses (1) The objective at the front of the telescope
takes light from a distant object and produces a real inverted image (which is rather
small) near its focal point (2) This small real image is then examined by the eyepiece
functioning as a simple magnifier to produce a virtual image with a larger angular
size
20
Serway Chapter 37
Two-Slit Interference
If light is incident on two closely spaced narrow slits a pattern of light and dark
stripes is produced beyond the slits The bright stripes or fringes are caused by
constructive interference of the two waves coming from the slits Constructive
interference occurs whenever two waves arrive at a location in phase with each other
This occurs when the distance x1 from slit 1 to a point P on the screen and the
distance x2 from slit 2 to point P differ by in integral number of wavelengths
1 2 where 0 1 2x x m m
where λ is the wavelength of the light When x1 and x2 are much larger than the slit
spacing d this condition reduces to
sind m
where θ is the angle between the direction of the incident light and the direction of the
light arriving at the screen
Thin Films
When light is partially reflected and partially transmitted by a thin film of transparent
material it is possible to have interference between the wave reflected from the front
of the film and light reflected from the back of the film (The colored reflections from
the thin film of oil on the water in a rain-soaked parking lot are an example of this
effect) It is difficult to write down formulas that will work in all cases so we will
just review the important principles here
1 If the two reflected waves are in phase with each other the film has enhanced
reflection (constructive interference) but if the two reflected waves are out of phase
with each other reflection is diminished (destructive interference) Phase shifts occur
due to reflection and due to the extra path length through the film of the wave
reflected from the back of the film
2 The phase change due to reflection is determined by the difference in index of
refraction between the two media involved in the reflection If the wave is incident
21
from a medium with a low index of refraction into a medium with a high index of
refraction a phase change of 180deg occurs and the reflected wave is inverted If
incident from high to low no phase shift occurs and the reflected wave is non-
inverted
3 The extra path length through the film of the wave reflected from the back of the
film is equal to 2t where t is the thickness of the film (The incident light is assumed
to be normal to the surface of the film) The number of wavelengths contained in the
extra path length is equal to 2tλn where λn = λn is the wavelength of the light in the
film
4 Rules for reflection from thin films
If one ray is inverted and the other is not then we have
12 constructive
2 nt m
2 destructivent m
If either both rays are inverted or both are non-inverted we have
2 constructivent m
12 destructive
2 nt m
22
Serway Chapter 38
Diffraction Grating
A diffraction grating is simply a fancy version of two-slit interference with the two
slits replaced by thousands of slits Just as in the two-slit case the bright fringes
occur at angles given by
sin md
but in this case d the distance between neighboring slits is made to be very small
The effect of having many slits instead of two is to make each bright fringe highly
localized with wide dark regions between neighboring maxima
Single Slit Diffraction
When light passes through an opening in an opaque screen an interference pattern is
produced beyond the opening To understand why we may replace the single opening
by many small coherent sources of light These many sources interfere with each
other producing a pattern known as a diffraction pattern (Note that many authors do
not distinguish between interference and diffraction treating them as interchangeable
terms) If the opening is a slit of width a then the diffraction pattern far from the slit
will have a bright central maximum with a succession of minima and weaker maxima
on either side The angle between the incident direction and the minima is given by
sin ma
where m = plusmn1 plusmn2
If the opening is circular with diameter D the angle between the incident direction
and the first minimum is given by
sin 122D
Optical Resolution and Rayleighrsquos Criterion
Two point sources can just be resolved (distinguished from each other) if the peak of
the diffraction image of the first source overlies the first minimum of the diffraction
image of the second source For circular holes of the kind usually encountered in
23
optical devices this condition is approximately satisfied when the angular separation
between the two sources as viewed from the optical instrument is greater than or equal
to the critical angle
min 122D
where λ is the wavelength of the light and where D is the diameter of the aperture in
the instrument
Polarization
We say that an electromagnetic wave is polarized if its electric field vector doesnrsquot
change direction in a random fashion The simplest kind of polarization is linear
polarization in which the electric field vector oscillates back and forth along the
same axis in space Polarized light can be produced from normal unpolarized light by
selective absorption (as in Polaroid sunglasses) by passing light through crystals that
have different indices of refraction for different polarizations (double refraction) by
scattering (the blue sky is polarized) and by reflection (glare) Polarization by
reflection occurs when light reflects from a shiny insulating (non-metallic) surface
The amount of polarization is greatest for reflection at Brewsterrsquos angle
2
1
tan P
n
n
where θP is the incidence angle of light from medium 1 onto medium 2 and where n1
and n2 are the indices of refraction for the media (In our everyday experience n1 = 1
since the light comes in through the air and n2 is the index of refraction of the shiny
insulating material producing the glare eg water glass plastic paint etc)
Malusrsquos Law
The intensity of transmitted polarized light through a perfect polarizer is related to the
incident intensity of polarized light by Malusrsquos law
2cosoI I
where Io is the intensity of the incident light and where θ is the angle between the
electric field vector in the incident wave and the transmission axis the polarizer
24
Serway Chapter 39
Principles of Relativity
All of the weirdness of relativity flows from two simple principles
(1) The laws of physics must be the same in all inertial (non-accelerating) reference
frames
(2) The speed of light in vacuum has the same value in all inertial reference frames
Note that (1) seems reasonable but (2) is very odd It says that if two space ships are
approaching each other at nearly the speed of light and a laser pulse is shot from ship
1 toward ship 2 then when the people on ship 2 measure the speed of the pulse as it
goes by it is moving at 3 times 108 ms the same as if the ships were stationary
Simultaneity
If observer 1 sees two events in her own frame as simultaneous at two different
locations a moving observer 2 will see these two events happening at different times
Relativistic gamma
The factor γ (gamma) appears regularly in the formulas of relativity
2
2
1
1 vc
where v is the relative speed between two inertial frames
Time Dilation
If observer 1 sees two events at the same location in space separated by time Δtp in his
own frame then observer 2 moving at speed v relative to observer 1 will see these
two events separated by a longer time Δt
pt t
Moving clocks run slow
Length Contraction
If observer 1 measures the length of an object along the x-direction at some instant of
time in her frame to be Lp then observer 2 moving in the x-direction at speed v will
measure the length L of the object to be shorter
pLL
25
Moving meter sticks are short
Velocity Addition
If a particle is moving at velocity vac relative to frame c and if frame c is moving at
speed vcb with respect to frame b (with both velocities directed along the same line)
what would be the velocity of the particle in frame b In classical physics the answer is
ab ac cbv v v
but if any of these velocities are an appreciable fraction of the speed of light the
answer changes
21ac cb
abac cb
v Vv
v v c
In using this equation make sure that vac has the same sign in the numerator and in the
denominator and that vbc also has the same sign in both places Note that this equation
is different (and in our opinion easier to understand) than the equations in the text
Momentum and Energy
The momentum and total energy of a particle moving at velocity v (its speed is v v )
are given by
22
2 2
22
1 1 vvc c
m mcm E mc
vp v
Rest Energy
When a particle is at rest (v = 0 and γ = 1) then E = mc2 is called the rest energy It
includes the potential energy of the particle so that if we change the potential energy
of a particle we change its rest mass
Kinetic Energy The kinetic energy is the difference between the total energy and the
rest energy
2 2K mc mc
For v ltlt c this reduces to K = mv22
General Relativity
Einsteinrsquos principle of equivalence in the theory of general relativity is In the vicinity
of a point a gravitational field is equivalent to an accelerated frame of reference in
26
the absence of gravitational effects For example there is no difference whatever in
the physics experienced by an astronaut on earth and one who is in interstellar space
far away from any stars or planets but who has just turned on her spaceshiprsquos engines
and is accelerating at 98 ms2
27
Serway Chapter 40
Photons
We now know that even though light behaves like a wave it is better described as
consisting of small packets of energy called photons The energy of a photon is related
to its frequency by
E hf
where h is Planckrsquos constant
346626 10 J sh
Photons also carry momentum given by
hf hp
c
Since the amount of energy in each photon is fixed the difference between dim light
and intense light (of the same wavelength) is that dim light consists of fewer photons
than intense light
Photoelectric Effect
In the photoelectric effect light shining onto a metal surface gives the electrons in the
metal enough energy to escape and be detected It requires a certain amount of energy
(called the work function typically = 3 ndash 5 eV) just to kick the electron out of the
metal so the light must deliver at least this much energy to an electron to produce the
effect It is observed that red light no matter how intense never produces electrons
But ultraviolet light even if quite dim will eject electrons from the metal Since red
light consists of 2 eV photons and ultraviolet light has photons with energies around
4-6 eV the photon idea explains the behavior of the photoelectric effect The
maximum energy that an ejected electron can have is
maxK hf
where f is the photon frequency The minimum frequency that light can have and cause
electrons to be ejected is called the cutoff energy and is found by setting Kmax = 0
cf h
28
Compton Effect
When high-frequency light interacts with free electrons the scattered light does not
have the same wavelength as the incident light contrary to what classical
electromagnetism would predict But the photon picture in which photons have
momentum and energy allows us to treat scattering as a collision between the photon
and the electron In this collision the electron and the scattered photon both have a
different momentum and energy than they did before And since p = hλ for a photon
if the momentum is different the wavelength will also be different
after before 1 cose
h
m c
where the angle θ is the angle between the incoming photon direction and the
direction of the scattered photon
Particles are Waves
Since photons behave like particles sometimes it is not surprising that elementary
particles can behave like waves sometimes The wavelength of a particle with
momentum p is given by
h
p
Wave-particle Duality
Both photons and elementary particles have a dual nature sometimes they behave like
particles and sometimes they behave like waves It is hard for us to comprehend the
nature of such an object by making mental pictures but experiments definitively show
that this is the case Since we have no direct experience with photons and elementary
particles (because their energies are so much smaller than the energies of the everyday
objects we encounter) it is perhaps not surprising that we have a hard time forming a
mental picture of how they behave
Electron Interference
Since an electron is both a particle and a wave just like a photon it should be able to
produce an interference pattern This is observed If an electron beam is shot at two
closely-spaced slits and if the electrons that pass through the slits are detected
downstream it is found that there are some locations where electrons are never detected
29
and others where lots of electrons are detected The pattern is exactly the same as the
one observed for light waves provided that we use the electron wavelength λ = hp in
place of the wavelength of light This pattern is observed even though each electron is
detected as a single dot on the screen Only after many such dots are collected does the
pattern emerge And if we try to understand how this effect could possibly work by
looking closely at each slit to see which one the electron came through the pattern
disappears the act of measurement destroys the interference This means that each
single electron somehow comes through both slits (as a wave would)
Uncertainty Principle
In classical physics we always imagine that the positions and momenta of moving
particles have definite values It might be hard to measure them but surely at each
instant of time a particle should be precisely located at some point in space and have a
similarly precise momentum This turns out not to be true Instead both position and
momentum are required to be uncertain with their uncertainties Δx and Δp satisfying
the Heisenberg uncertainty relation
2x p
So if the particle were known to be precisely at some particular location (so that Δx =
0) we couldnrsquot know anything about its momentum (Δp = infin) and if its momentum
were exactly known we couldnrsquot know its position
There is a similar relation involving the particlersquos energy E and the time interval Δt
over which this energy is measured
2E t
This means that energy is not actually conserved if we are considering very small time
intervals and this brief non-conservation of energy has been observed
30
Serway Chapter 41
Wave Function ψ and Probability
Quantum mechanics does not predict exactly what an electron or a photon will do Instead
it specifies the wave function or probability amplitude ψ of an electron or a photon
This wave function is a complex-valued function of space and time whose squared
magnitude is the probability density P for finding a particle at a particular place in
space at a certain time
2P
where is the complex conjugate of ψ
For example the wave function of an electron with perfectly specified momentum p
would have a wavelength given by
h
p
and its probability amplitude would be proportional to
2i x ipxe e
The corresponding probability density would then be
21ipx ip ipxP e e e
which means that the electron is equally probable to be anywhere along the x axis
This is in accord with the uncertainty principle since we specified the momentum
precisely we canrsquot have any idea about the position of the electron
As another example you have probably seen ldquofuzzy ballrdquo drawings of electron
orbitals in a chemistry book These fuzzy balls are meant to indicate the distribution
of the probability density 2
P in the orbital
Particle in a Box
A simple example in which we can calculate the wave function is the case of a particle of
mass m constrained to be inside a 1-dimensional box between x = 0 and x = L In this
case the general wave function is a linear superposition of wave functions ψn of the form
sin niE tn
n xx t A e
L
31
where A is a positive constant where n = 1 2 3 and where the energy associated
with each of the quantum states ψn is given by
22
28n
hE n
mL
This wave function is zero at x = 0 and x = L which means that the particle will never
be found at the walls of the box The wave function has maximum values in the
interior and at these places the particle is most likely to be found and it also has
places where it is zero and at these places the particle will also never be found as
expressed by the formula
2sinn x
PL
The particle in the box is interfering with itself producing a probability interference
pattern across the box just like the interference patterns we studied with light and
sound
32
Serway Chapter 42
Atoms
Many people picture an atom as a miniature solar system where electrons orbit around
a massive nucleus at the center This picture is misleading because in atoms the wave
nature of electrons dominates The electrons form 3-dimensional standing waves
(called orbitals) centered at the nucleus And if you ask what is it thatrsquos waving the
answer is ψ the probability amplitude (see Chapter 41)
Standing waves on a string can occur only for certain frequencies (the fundamental
and higher harmonics) Similarly atomic orbitals occur only for certain energies For
the hydrogen atom the energies of the orbitals have the a particularly simple form
2
1136 eVnE
n
where n = 1 2 3 is called the principal quantum number For other atoms the
determination of the orbital energies requires numerical calculation by computers
Atomic Spectra
If an electron is somehow given extra extra energy (we say that it is excited) so that it
occupies a higher orbital it will eventually ldquofallrdquo back down to a lower orbital Each
time an electron falls to a lower orbital it loses the difference in energy between the
two orbitals in the form of a photon Since the orbital energies are discrete so are the
energy differences and so are the wavelengths of the emitted light The entire set of
these discrete wavelengths is called the atomic spectrum and it is unique to each type
of atom For hydrogen the spectrum can be simply written as
H 2 2
1 1 1
f i
Rn n
where RH = 10973732 times 107 m-1 For other atoms the spectrum cannot be expressed as
a formula
Orbital Angular Momentum Electrons in atoms also have quantized values of
angular momentum The orbital quantum number ℓ specifies the value of this
quantized angular momentum through the formula
1L
33
If we want to know the value of the angular momentum along some direction in space
say the z direction the answer is not L but rather
zL m
where mℓ is another quantum number which runs from
1 1m
This quantum number is important when an atom sits in a magnetic field
Spin Angular Momentum It has been found experimentally that electrons and other
charged particles also carry internal angular momentum which we call spin
Electrons have an intrinsic spin angular momentum s along a specified axis that is
extremely quantized it can only take on 2 possible values
1
2zs s
We thus say that electrons have spin s = 12 or that electrons are rdquospin one-half parti-
cles Since s is an angular momentum it obeys the same rule as that for orbital angular
momentum namely that if its value along some axis is s then its total magnitude is
given by
31
2S s s
I know this seems weird but quantum mechanics is weird The only excuse for this
bizarre way of looking at the world is that it predicts what happens in experiments
Exclusion Principle The answer to the question of how many electrons (or any other
spin one-half particle) can be in one particular quantum state was discovered by
Wolfgang Pauli and is called the exclusion principle
ldquoNo two electrons can ever be in the same quantum state therefore no two electrons
in the same atom can have the same set of quantum numbersrdquo
This is the reason that we have atoms with different properties instead of every atom
simple having all of its electrons in the ground state All of the variety we see around
us in the world is the result of chemical differences and these differences would not
exist unless electrons obeyed this important principle The entire structure of the
periodic table (see pages 1377-1379 in Serway) is an expression of this principle
34
Serway Chapter 44
Nuclear Properties
The nucleus is composed of protons with charge +e and mass mp = 1007226 u and of
neutrons with zero charge and mass mn = 1008665 u where u is one atomic mass unit
-271 u=1660540 10 kg
The atomic number Z counts the number of protons in a nucleus while the neutron
number N counts the number of neutrons The mass number A is the sum of the two
A N Z
Protons and nuetrons have about the same mass and some times called baryons (heavy
ones) or nucleons (partcles in the nucleus) A is sometimes called the baryon
number The chemical elements are distinguished by Z ie hydrogen has Z = 1 iron
has Z = 26 uranium has Z = 92 etc But for a given Z there might be several nuclei
with different numbers of neutrons These nuclear siblings are called isotopes
These numbers are used to label nuclei according to the pattern
5626 ie FeA
Z X
denotes the isotope of iron with N = 56 ndash 26 = 30 neutrons
The nucleus is roughly spherical with a radius given approximately by
1 3 150 0where 12 10 mr r A r
Nuclear Stability
Because the positively charged protons electrically repel each other with an enormous
force at distances as small as 10-15 m there must be some really strong force that that
overcomes electrical repulsion to hold protons and neutrons together This force is
called with some lack of imagination the strong force It is a very short-range force
(it only acts over a distance of about 2 x 10-15 m) and attracts protons to protons
neutrons to neutrons and neutrons to protons But in spite of this strong nuclear force
the coulomb repulsion of the protons is still present so anything that might keep the
protons from being right next to each other would help keep the nucleus from
35
exploding This role is played by the neutrons and for nuclei with Z le 20 the stable
nuclei roughly have N = Z
For Z gt 20 the coulomb repulsion force is more powerful and more neutrons are
needed to dilute it up to about N = 15Z around Z = 80 For Z gt 83 no amount of
neutrons can help and these nuclei are unstable (radioactive) There is an ldquoisland of
(relative) stabilityrdquo around Z=90-92 (thorium and uranium respectively) These
elements have one or two isotopes with half-lifes of billions of years so there are
substantial amounts of such elements on Earth This fact makes it possible to have
practical fission devices
Radioactive Decay
There are three types of radioactive decay
Alpha decay the nucleus kicks out a helium nucleus (N = 2 Z = 2 A = 4) So if the
original nucleus is called X and the new nucleus is called Y then the decay would look
like this
4 42 2X Y+ HeA A
Z Z
Beta decay the nucleus either kicks out an electron (endash) or its positively-charged
antimatter twin the positron (e+) plus either an electron neutrino v or an electron
anti-neutrino v
1X Y+e electron decayA AZ Z v
1X Y+e positron decayA AZ Z v
A neutrino is a particle with no charge hardly any mass (much less than the electron
mass) and interacts so weakly with matter that most neutrinos upon encountering the
planet earth just pass right through it as if it werenrsquot there
Gamma decay the nucleons in the nucleus X are in an excited energy state X
(perhaps as a result of having undergone alpha or beta decay) and they drop down to a
lower energy state shedding the energy as a high frequency photon
X XA AZ Z
36
This process is exactly analogous to the way that the electrons in atoms emit photons
Decay Rate and Half Life
There is no way to predict exactly when an unstable or excited nucleus will decay but
there is an average rate at which this decay occurs called the decay constant λ The
meaning of this constant is that if there are a large number N of nuclei in a sample
then the number of decays per second that will be observed (called the decay rate R)
is R = λN In mathematical language
dMR N
dt
This simple differential equation has for its solution
0tN t N e
where N0 is the number of nuclei in the sample at time t = 0
The half-life is the time it takes for half of the nuclei in the sample to decay and is
related to the decay constant by
1 2
ln 2 0693T
Disintegration Energy
When a nucleus decays it is making a transition to an overall state of lower energy
which means according to Einsteinrsquos famous formula E = mc2 that the sum of the
masses after the decay must be less than the mass before with the lost mass appearing
as kinetic energy among the decay products For example in alpha decay this kinetic
energy called the disintegration energy Q is given by
2X YQ M M M c
37
Serway Chapter 45
Nuclear energy
There are two ways to extract energy from the nucleus fission and fusion For nuclei
with Z greater than 26 breaking the nucleus apart into pieces leads to a lower
2mc energy than the original nucleus so energy can be extracted by fission For
nuclei with Z less than 26 a lower energy is achieved by combining nuclei so energy
can be extracted by fusion
Fission
Since neutrons have no charge they are not repelled from nuclei as protons are for
this reason their behavior is the key to understanding how fission works
Fast neutrons mostly bounce off other nuclei with each collision slowing the neutron
down This elastic energy loss is most effective if the other nuclei have low mass (like
hydrogen) and these materials are called moderators because of their ability to slow
down fast neutrons
The reason that slowing neutrons is important is that slow neutrons are much more
likely to be absorbed by a nucleus which then leads to nuclear reactions of various
kinds For a few very large nuclei like uranium-235 and some plutonium isotopes
absorption of a slow neutron causes the nucleus to split into two large fragments plus
2 or 3 fast neutrons (energy is released in the process too) Thus one slow neutron
can produce energy plus 2 or 3 more neutrons which if moderated from fast to slow
can split 2 or 3 more nuclei leading to a runaway chain reaction If the reaction is
allowed to proceed unchecked it produces a large explosion If a neutron absorbing
material is added to the mix (like the cadmium in reactor control rods) it is possible to
keep the reaction under control and to extract the released energy as heat to drive
steam turbines and produce electricity
Fusion
Fusion involves mashing two nuclei together and since they are both charged and repel
each other this reaction is much harder to make go The nuclei must have enough energy to
overcome the coulomb repulsion which is why this reaction requires a high temperature
(hundreds of millions of degrees K) like that in the sun (which is burning hydrogen to
helium via fusion) or like that in the center of the fission explosion that is used to detonate
38
a hydrogen bomb
This reaction is of interest for power production in spite of this difficult temperature
requirement because of the abundance of fusion fuel on the planet There are about
012 g of deuterium ( 21H ) in every gallon of water on earth and it only costs about 4
cents to extract it The fusion energy available from this minuscule amount of
deuterium would run a 1000-Megawatt power plant for 10 seconds Water is so
abundant on earth that if fusion were to work we would have an essentially
inexhaustible source of energy
So why donrsquot we have fusion power plants Well the fuel is cheap but the match is
incredibly expensive The only way we know to control this difficult high-temperature
reaction is with large and expensive pieces of equipment involving either large
magnetic fields and complex high-power electromagnetic antennas or with gigantic
(football-fieldsized) laser facilities involving more than a hundred of the highest-
energy lasers ever built Power plants based on these current methods for controlling
fusion are unattractive to the fiscally-minded people who run the electric power
industry Hopefully better designs will be discovered as experiments continue
39
c actinium 68 Er erbium 101 Md mendelevium 104 Rf rutherfordium l aluminum 63 Eu europium 80 Hg mercury 62 Sm samarium m americium 100 Fm fermium 42 Mo molybdenum 21 Sc scandium b antimony 9 F fluorine 60 Nd neodymium 106 Sg seaborgium r argon 87 Fr francium 10 Ne neon 34 Se selenium s arsenic 64 Gd gadolinium 93 Np neptunium 14 Si silicon t astatine 31 Ga gallium 28 Ni nickel 47 Ag silver a barium 32 Ge germanium 41 Nb niobium 11 Na sodium k berkelium 79 Au gold 7 N nitrogen 38 Sr strontium e beryllium 72 Hf hafnium 102 No nobelium 16 S sulfur i bismuth 108 Hs hassium 76 Os osmium 73 Ta tantalum h bohrium 2 He helium 8 O oxygen 43 Tc technetium
boron 67 Ho holmium 46 Pd palladium 52 Te tellurium r bromine 1 H hydrogen 15 P phosphorus 65 Tb terbium d cadmium 49 In indium 78 Pt platinum 81 Tl thallium a calcium 53 I iodine 94 Pu plutonium 90 Th thorium f californium 77 Ir iridium 84 Po polonium 69 Tm thulium
carbon 26 Fe iron 19 K potassium 50 Sn tin e cerium 36 Kr krypton 59 Pr praseodymium 22 Ti titanium s cesium 57 La lanthanum 61 Pm promethium 74 W tungsten l chlorine 103 Lr lawrencium 91 Pa protactinium 92 U uranium r chromium 82 Pb lead 88 Ra radium 23 V vanadium o cobalt 3 Li lithium 86 Rn radon 54 Xe xenon u copper 71 Lu lutetium 75 Re rhenium 70 Yb ytterbium m curium 12 Mg magnesium 45 Rh rhodium 39 Y yttrium b dubnium 25 Mn manganese 37 Rb rubidium 30 Zn zinc y dysprosium 109 Mt meitnerium 44 Ru ruthenium 40 Zr zirconium s einsteinium
40
UNITS The SI units are given in parentheses Other commonly used units are given in terms of the SI units Symbols conform with the recommendations of the American National Standards Institute (ANSI) and the American Institute of Physics (AIP) Commonly used multiples of SI units
prefix symbol factor tera T 1012 giga G 109
mega M 106 kilo k 103
centi c 10-2 milli m 10-3
micro μ 10-6 nano n 10-9 pico p 10-12
femto f 10-15
length meter (m) angstrom 1 Ǻ = 10-10 m inch 1 in = 254 x 10-2 m foot 1 ft = 03048 m mile 1 mi = 1609 m
mass kilogram (kg) atomic mass unit 1 u = 1661 x 10-27 kgslug 1 slug = 1459 kg
force newton (N = kg bull ms2) dyne 1 dyn = 10-5 N pound 1 lb = 4448 N
pressure Pascal (Pa = kgm bull s2) atmosphere 1 atm = 1013 x 105 Pa poundssquare inch 1 psi = 6895 Pa cm of mercury 1 cm Hg = 1333 Pa bar 1 bar = 1000 x 105 Pa torr 1 torr = 1333 Pa
time second (s) minute 1 min = 60 s hour 1 h =3600 s
frequency hertz (Hz = s-1) radianssecond 1 rads = 12π Hz
energy joule (J = kg bull m2s2) erg 1 erg = 10-7 J electron volt 1 eV = 1602 x 10-19 J calorie 1 cal = 4187 J kilowatt-hour 1 kWh = 36 x 106 J British thermal unit 1 Btu = 1055 J
power watt (W = kg bull m2s3) horsepower 1 hp = 7457 W
charge coulomb (C = A bull s)
electric potential volt (V = kg bull m2 s3 bull A)
current ampere (A)
resistance ohm (Ω = kg bull m2s3 bull A2)
capacitance farad (F = s4 bull A2kg bull m2)
magnetic field tesla (T = kgs2 bull A) gauss 1 G = 10-4 T
magnetic flux weber (Wb = kg m2s2 bull A) maxwell 1 Mx = 10-8 Wb
magnetic inductance henry (H = kg bull m2s2 bullA2)
temperature kelvin (K) degrees Celsius 0degC = 27315 K
angle radian (rad) degree 1 deg = π180 rad revolution 1 rev = 2π rad
41
Some Physical Constantsa Quantity Symbol Valueb Atomic mass unit μ 1660 538 73 (13) x 10-27 kg
931494 013 (37) MeVc2 Avogadros number NA 6022 141 99 (47) x 1023 particlesmol Avogadrorsquos number 6022 x 1023 mol Bohr magneton
2B e
em 9274 008 99 (37) x 10-24 JT
Bohr magneton μB 927 x 10-24 JT Bohr radius
0
2
2e em e k
a 5291 772 083 (19) x 10-11 m
Boltzmanns constant B A
RNk 1380 650 3 (24) x 10-23 JK
Boltzmannrsquos constant kB 1380 x 10-23 JK Compton wavelength
C eh
m c 2426 310 215 (18) x 10-12 m
Deuteron mass md 3343 583 09 (26) x 10-27 kg 2013 553 212 71 (35) u
electron charge e 1602 x 10-19 C Electron mass me 9109 381 88 (72) x 10-31 kg
5485 799 110 (12) x 10-4 u 0510 998 902 (21) MeVc2
electron mass 911 x 10-31 kg Electron volt eV 1602 176 462 (63) x 10-19 J Elementary charge e 1602 176 462 (63) x 10-19 C Gas constant R 8314 472 (15) JKmol Gravitational constant G 6673 (10) x 10-11 Nm2kg2 Neutron mass mn 1674 927 16 (13) x 10-27 kg
1008 664 915 78 (55) u 939565 330 (38) MeVc2
neutron mass 1675 x 10-27 kg Nuclear magneton
2n p
em 5050 783 17 (20) x 10-27 JT
permeability constant μ0 1257 x 10-6 Hm permittivity constant ε0 8854 x 10-12 Fm Plancks constant h
2h
6626 068 76 (52) x 10-34 Js 1054 571 596 (82) x 10-34 Js
Planckrsquos constant h ħ
6626 x 10-34 Js 1055 x 10-34 Js
Proton mass mp 1672621 58 (13) x 10-27 kg 1007 276 466 88 (13) u 938271 998 (38) MeVc2
proton mass 1673 x 10-27 kg Rydberg constant RH 1097 373 156 854 9 (83) x 107 m-1 Speed of light in vacuum c 2997 92458 x 108 ms (exact) speed of light c 300 x 108 ms a
These constants are the values recommended in 1998 by CODATA based on a least-squares adjustment of data from different measurements For a more complete list see P J Mohr and B N Taylor Rev Mod Phys 72351 2000 b The numbers in parentheses for the values above represent the uncertainties of the last two digits
NOTE The ones in red are the ones from Appendix 13 The ones in yellow are the ones that are from the Serway book table but that were already on the Appendix 13 table
42
INDEX
Absolute zero 9
Adiabatic 17
Adiabatic exponent 18
Adiabatic process 17
Alpha decay 46
Amplitude 4
Angle of reflection 25
Angular frequency 3
Angular magnification 30
Angular momentum atomic 43
Angular size 29
Archimedes Principle 1
Atomic spectra 43
Atoms 43
Avogadrorsquos number 10
Beats 8
Bernoullirsquos Equation 2
Beta decay 46
Bifocals 29
Brewsterrsquos angle 34
British Thermal Unit Btu 11
Buoyancy 2
calorie 11
Calorie 11
Camera 28
Carnot cycle 21
Carnot efficiency 22
Celsius scale 9
Ciliary muscle 29
Coefficient of performance 21
Compton effect 39
Constructive interference 7
Continuity equation of 2
Contraction length 35
Convection 14
Curved mirrors 27
Decay constant 46
Decay rate radioactive 46
Decibel scale 5
Degrees of freedom 16
Density 1
Destructive interference 7
Diffraction single slit 33
Diffraction grating 33
Dilation time 35
Disintegration energy 47
Dispersion of light 26
Doppler effect 6
Double slit interference 31
Efficiency 20
Electron interference 39
Emissivity 15
Energy nuclear decay 47
Energy relativistic 36
43
Engineering work 20
Entropy 22
Entropy ideal gas 24
Equipartition of energy 18
Equivalence principle 36
Exclusion principle 44
Expansion thermal 9
Expansion coefficient linear 9
Eye 29
Fahrenheit scale 9
Far point 29
Farsightedness 29
First Law of Thermodynamics 12
Fission 48
Flux volume 2
Free expansion 23
Fringes 31
Fusion 48
Fusion heat of 11
Gamma relativistic 35
Gamma decay 46
General relativity 36
Half-life 46
Heat 11
Heat capacity 11
Heat conduction 13
Heat engine 20
Heat of fusion 11
Heat of vaporization 11
Heat pump 20
Heats of transformation 11
Hydrostatics 1
Ideal Gas Law 10
Images real and virtual 27
Intensity sound 5
Interference 7
Interference two-slit 31
Internal energy 12
Internal energy degrees of freedom 16
Irreversible process 19 23
Isotopes 45
Joule 11
Kelvin scale 9
Kinetic energy relativity 36
Kinetic theory 16
Latent heat 11
Length contraction 35
Linear expansion coefficient 9
Linear polarization 34
Linear superposition 7
Longitudinal wave 3
Loudness 5
Magnification lateral 27
Magnifying glass 30
Malusrsquos law 34
Microscope 30
Momentum relativistic 36
Muscle ciliary 29
44
Musical instruments 8
Musical scale 8
Near point 29
Nearsightedness 29
Nonlinear 7
Nuclear energy 48
Nuclear properties 45
Nuclear stability 45
Octave 8
Optical resolution 33
Orbital quantum number ℓ 43
Orbitals 43
Particle in a Box 41
Particles are waves 39
Pascalrsquos Principle 1
Period 3
Photoelectric effect 38
Photons 38
Pitch 8
Polarization 34
Power sound 5
Presbyopia 29
Pressure 1
Principal quantum number n 43
Principle of equivalence 36
Principle of linear superposition 7
Probability amplitude ψ 41
Processes thermodynamic 13
R-value 14
Radiation thermal 14
Radioactive decay 46
Radioactive decay rate 46
Ray tracing 27
Rayleighrsquos criterion 33
Reading glasses 29
Real image 27
Refraction 25
Refrigerator 20
Relativistic gamma 35
Relativity principles 35
Resolved for light sources 33
Rest energy 36
Reversible process 19 22
Rope wave speed 4
Second Law of Thermodynamics 19
Shock waves 6
Simple magnifier 30
Simultaneity 35
Single slit diffraction 33
Snellrsquos law 25
Sound speed 5
Specific heat 11
Spin Angular momentum 44
Standing waves 7
Stefanrsquos law 14
Strong force 45
Telescope 30
Temperature 9
45
Temperature Scales 9
Thermal conductivity 14
Thermal energy 12
Thermal expansion 9
Thin film interference 31
Thin lenses 28
Time dilation 35
Tone musical 8
Total internal reflection 25
Transverse wave 3
Traveling Waves 3
Two-slit interference 31
Uncertainty principle 40
Vaporization heat of 11
Velocity addition relativity 36
Virtual image 27
Volume flux 2
Wave function 41
Wave Function ψ and Probability 41
Wave speed 3
Wave-particle duality 39
Wavelength 3
Wavenumber 3
Wien Displacement Law 16
Work 11
7
An adiabatic process is one in which no heat is exchanged between the system and the
environment When an ideal gas expands or contracts adiabatically not only does its
pressure change as expected from the ideal gas law but its temperature changes as
well Under these conditions the final pressure Pf can be computed from the initial
pressure Pi and from the final and initial volumes Vf and Vi by
or constantf f i iP V PV PV
where γ = CPCV The quantity γ is called the adiabatic exponent Note that this
doesnrsquot mean that the ideal gas law no longer holds it does and in fact it can be
combined with the adiabatic law for pressure given above to obtain the adiabatic law
for temperatures
1 constantTV
Compressions in sound waves are adiabatic because they happen too rapidly for any
appreciable amount of heat to flow This is why the adiabatic exponent appears in the
formula for the speed of sound in an ideal gas
RTv
M
Note that v depends only on T and not on P Because it depends only on the
temperature the speed of sound is the same in Provo as at sea level in spite of the
lower pressure here due to the difference in elevation
Equipartition of Energy
Every kind of molecule has a certain number of degrees of freedom which are
independent ways in which it can store energy Each such degree of freedom has
associated with it ndash on average ndash an energy of 12 Bk T per molecule (or 1
2 RT per mole)
(Note since a molecule has so many possible degrees of freedom it would seem that
there should be a lot of 12 sBk T to spread around But because energy is quantized
some of these degrees of freedom are not ldquoactiverdquo until the temperature becomes high
enough that 12 Bk T is as big as the lowest quantum of energy
8
Serway Chapter 22
Second Law of Thermodynamics
There are several equivalent forms of this important law
Kelvin It is not possible to change heat completely into work with no other change
taking place Or in other words there are no perfect heat engines
Clausius It is not possible for heat to flow from one body to another body at a higher
temperature with no other change taking place Or in other words there are no
perfect refrigerators
Entropy In any thermodynamic process that proceeds from one equilibrium state to
another the entropy of the system + environment either remains unchanged or
increases The total entropy never decreases This law is a bit of an oddity among the
laws of physics because it is not absolute Things are forbidden by the second law not
because it is impossible for them to happen but because it is extremely unlikely for
them to happen (See below for more information about entropy)
Reversible and Irreversible Processes
A reversible process is one which occurs so slowly that it is in thermal equilibrium (or
very nearly so) at all times A hallmark of such processes is that a motion picture of
them looks perfectly normal whether run forward or backward Imagine for instance
the slow expansion of a gas at constant temperature in a cylinder whose volume is being
increased by a slowly moving piston Run the movie backwards and what do you see
You see the slow compression of a gas at constant temperature which looks perfectly
normal
An irreversible process is one which occurs in such a way that thermal equilibrium is
not maintained throughout the process The mark of this kind of process is that a motion
picture of it looks very odd when run backward Imagine the sudden expansion of a gas
into a previously evacuated chamber because a hole was punched in the wall between a
pressurized chamber and the evacuated one Run the movie backward and what do you
see You see the gas in the soon-to-be-evacuated chamber gather itself together and
stream through a tiny hole into a chamber in which there is already plenty of gas If you
have ever seen this happen get in touch with the support group for those who have
witnessed the spontaneous combustion of large mammals by calling 1-800-PYROCOW
9
Heat Engines
Because of the vexing difference in sign between work done on and system and work done
by a system we will invent a new work variable Weng Heat engines do work and so the net
W for these engines is negative But in engineering applications hidden minus signs are
regarded as evil so for heat engines we donrsquot talk about W instead we talk about its
magnitude engW W So for heat engines the first law is
int engE Q W
But for heat pumps and refrigerators work is done on the system so we use the usual work
W when we talk about these systems
A heat engine is a machine that absorbs heat converts part of it to work and exhausts the
rest The heat must be absorbed at high temperature and exhausted at low temperature If the
absorbed heat is Qh the exhausted heat is Qc and the work done by the engine is Weng then
eng h cW Q Q
and the efficiency of the engine is defined to be
eng
h
We
Q
A perfect engine would convert the heat hQ completely into work Weng giving an effi-
ciency of e = 1 Energy conservation alone allows a perfect engine but the second law
requires e lt 1
Refrigerators and Heat Pumps
A refrigerator is a machine that absorbs heat at low temperature and exhausts it at high
temperature the ldquobackwardsrdquo heat transfer being driven by the work done on the machine
by some source of power A heat pump is a machine that either works like a refrigerator
keeping a place cold by transferring heat from this cold place to a higher temperature
environment (cooling mode like an air conditioner) or it functions as a heater
transferring heat into a warm place from a cooler one (heating mode like a window unit
that heats a house by extracting thermal energy from the cold outdoors) The coefficient
of performance of a refrigerator or of a heat pump in cooling mode is defined to be
COP cooling modecQ
W
10
For a heat pump in heating mode the coefficient of performance is
COP heating modehQ
W
Note that we donrsquot have to use the engineering work here because in these systems
work is done on the system and W is naturally positive
A perfect refrigerator would take in heat Qc and exhaust the same amount of heat without
doing any work giving an infinite coefficient of performance Energy conservation alone
allows a perfect refrigerator but the second law requires COP lt infin A perfect heat pump
in heating mode would transfer Qh into the house without doing any work and so would
also have an infinite coefficient of performance The second law forbids this too A good
coefficient of performance for a real device would be around 5 or 6
Carnot Cycle
The most efficient of all possible engines is one that uses the Carnot cycle This cycle
employs an ideal gas has no friction and operates very slowly so that the gas can be in
thermal equilibrium at all parts of the cycle This means of course that it canrsquot
possibly be built and even if it could be built it would not run fast enough to be useful
Nevertheless this cycle is very important because it gives an upper bound on the
efficiency of real engines There cannot possibly be an engine that is more efficient
than one based on the Carnot cycle This cycle consists of the following four steps
1 The ideal gas absorbs heat Qh at constant temperature Th while the gas increases its
volume The reason that heat is absorbed is that expansion tends to cool the gas but
thermal contact with the environment at Th keeps the temperature high by heat
conduction into the ideal gas
2 The ideal gas further increases its volume by an adiabatic expansion This expansion
causes the gas to cool so at the end of this part of the cycle the gas is at temperature Tc
3 The gas exhausts heat Qc at constant temperature Tc while the gas decreases in
volume The reason that heat is exhausted is that compression tends to heat the gas
but thermal contact with the environment at Tc keeps the temperature low by heat
conduction out of the ideal gas
4 The gas is adiabatically compressed back to its original volume (the volume it started
with in step 1) This compression heats the gas from Tc up to Th
11
The efficiency of a Carnot engine is given by the very simple formula
1 cC
h
Te
T
where the temperatures must in be Kelvin No real engine can be more efficient than
this
The coefficient of performance of a Carnot refrigerator or heat pump in cooling mode
is given by
COP cooling mode cC
h c
T
T T
and no real refrigerator can have a coefficient of performance greater than this
The coefficient of performance of a Carnot heat pump in heating mode is
COP heating mode hC
h c
T
T T
So why donrsquot we just use these wonderful Carnot engines and have perfect efficiency
To make the reversible steps in the cycle really reversible they would have to occur
infinitely slowly So the price you pay for making a perfect engine is that it takes
forever to get it to do any work
Entropy
The entropy of a system is defined in terms of its molecular makeup and measures
roughly the disorder of the system If the system is packed into a very small volume
then it is quite ordered and the entropy will be low If it occupies a large volume the
entropy is high (To see what this has to do with disorder note that socks in a drawer
occupy a small volume while socks on the bed in the corner by the door and
hanging from the chandelier occupy a large volume) If the system is very cold then
the molecules hardly move and may even reach out to each other and form a crystal
This is a highly ordered state and therefore has low entropy If the system is very hot
with rapidly speeding molecules crashing into the container walls and bouncing off
each other things are disordered and the entropy is high
It is possible to calculate the entropy of a system in terms of its macroscopic thermody-
namic properties ie pressure volume temperature number of moles etc The key to this
calculation is the concept of a reversible process A reversible process is one that is
12
carried out without friction and so slowly that the process can be reversed at any stage by
making an infinitesimal change in the environment of the system The slow expansion of the
gas at Th in the Carnot cycle with tiny amounts of heat being transferred to the environment
is reversible If we turned around and began slowly to compress the gas would just slowly
exhaust heat to the environment in the exact reverse way that it absorbed it during expansion
Most processes however are irreversible For example if a gas-filled box were suddenly
increased in size so that the particles were free to wander into the void created by the sudden
expansion then the gas would eventually fill the new volume uniformly at the same
temperature as before the expansion (The temperature is unchanged in this imaginary
process because the kinetic energy of the molecules would be unaffected by such an
instantaneous expansion of the container walls) This imaginary but highly thought-
stimulating process is called a free expansion and it is impossible to reverse it During the
expansion we didnrsquot push on any of the molecules so reversing this process would mean
making them go back into their original volume without pushing on them they simply will
not cooperate to this extent Another way to see that just pushing them back where they came
from does not reverse the free expansion is to think about what would happen if we just
compressed either adiabatically or isothermally An adiabatic compression back to the
original volume would heat the gas above its original temperature and an isothermal
compression would require that heat be exhausted to the environment But the free expansion
involved neither temperature changes nor heat exchanges so neither of these two processes
is the reverse of the free expansion It is simply impossible to reverse this rapid expansion in
a way that takes us back to the initial state
It is possible to calculate the change of entropy for both reversible and irreversible processes
Letrsquos consider a reversible process first In a reversible process the entropy change is given
by the formula
dQS
T
where dQ is the amount of heat added to the system during a small step of the process
The total energy change during the process may then simply be calculated by integration
f f
f i i i
dQS S S dS
T
13
(Just as in the case of energy we are mainly interested in differences rather than in
absolute magnitudes)
This integration method does not work for irreversible processes For instance in the
rapid free expansion discussed above no heat is added to the system but its disorder
obviously goes up We need to find some other way to calculate the entropy The key
is the fact that the entropy of a system depends only on its current state and not at all
on how it arrived there So to calculate the entropy change in an irreversible process
first find out what the initial and final conditions of the process are Then invent a
reversible process that takes the system from the initial state to the final state Since
the entropy depends only on the state of system and not on the process the entropy
change for the reversible process is the same as that for the irreversible process
Entropy is thus like pressure or temperature if the pressure changes from Pl to to P2
then the pressure difference is P2 - Pl regardless of how the pressure change was
made This seems obvious for pressure but not for entropy only because you arent
comfortable with entropy yet
Entropy of an Ideal Gas
For n moles of an ideal gas the difference in entropy between a state with temperature
T and volume V and some standard state with temperature To and volume Vo is given
by
ln lno o Vo o
T VS T V S T V nC nR
T V
Entropy in a Phase Change
Calculating entropy change in a phase change like melting or boiling is easy It is
QT Where Q is in the latent heat for example the flows in to cause the melting and
T is the temperature of the phase change There is NO integral to do
14
Serway Chapter 35
Angle of Reflection
If plane waves are incident on a reflecting surface with the propagation direction of
the waves making angle θ1 with the normal direction to the surface then the reflected
angle θ1prime relative to the surface normal is simply
1 1
ie the incident angles and reflected angles are the same
Refraction
If plane waves traveling through medium 1 are incident on a plane interface between
medium 1 and medium 2 then the angle of incidence of the incoming wave θ1 and
the angle of refraction of the transmitted wave θ2 are related by Snellrsquos law
2 2
1 1
sin
sin
v
v
where v1 and v2 are the wave speeds in medium 1 and medium 2 The angle of
incidence and the angle of refraction are both measured between the wave propagation
direction and the normal to the interface In terms of indices of refraction in the case
of light waves Snellrsquos law takes the more familiar form
1 1 2 2sin sinn n
where ni = cvi
Total Internal Reflection
If a wave is incident from a medium of low wave speed into a medium of high wave
speed the law of refraction requires that the angle of refraction be greater than the
angle of incidence If the angle of refraction is required to be greater than 90deg then no
refracted wave can exist and total internal reflection occurs The critical incident
angle θc beyond which total internal reflection occurs is given by
2
1
sin c
n
n
15
Dispersion of Light
In addition to the speed of light varying from material to material it also varies with
wavelength within each material This means that the index of refraction is generally a
function of wavelength
c
nv
Since the wave speed is not constant such a medium is dispersive meaning in this
context that refraction actually disperses white light into its various colors because
Snellrsquos law gives a different angle for each wavelength In most materials the
variation with wavelength is quite small but this small effect is responsible for some
of the most spectacular color effects we ever see including rainbows a flashing
crystal chandelier and the colored fire of a diamond solitaire by candlelight
16
Serway Chapter 36
Real and Virtual Images
When light rays are focused at a certain plane producing an image if a sheet of white
paper is placed there we call the image a real image The images produced by film
projectors and overhead projectors are examples of real images
When light rays appear to come from a certain location but no image is produced
when a screen is placed there we say that there is a virtual image at that location For
instance when you look in a mirror it appears that someone is behind the mirror but
a screen placed back there in the dark would show nothing Your image in the mirror
is a virtual image
Ray Tracing
There are lots of rules about how to find the images in optical systems but the best
way to keep things straight is to learn how to draw the principal rays for curved
mirrors and lenses The rays for convex and concave mirrors are shown in Fig 3615
The rays for converging and diverging thin lenses are shown in Fig 3627 You should
memorize the principal rays and know how to use them to locate images
Curved Mirrors
The focal length of a curved mirror with radius of curvature R is given by
2
Rf
The relation between object distance p image distance q and focal length f is
1 1 1
p q f
If the mirror is a diverging mirror f should be negative and if q should turn out to be
negative the image is virtual
Lateral Magnification
The lateral magnification in an optical system is defined by the ratio of the image size
to the object size
17
Image height
Object height
qM
p
for curved mirrors
As usual there are sign conventions here too but this formula just gives the
magnitude It is better to keep keep track of upright images versus inverted images by
means of ray diagrams rather than by memorizing sign conventions
Thin Lenses
The focal length of a thin lens is related to the radii of curvature of the two faces R1
and R2 of the lens by
1 2
1 1 11n
f R R
Note that this formula differs from Eq (3611) in the text by not having a minus sign
between the two R-terms We like this form better because for a simple converging
lens like a magnifying glass we just use positive values of R for both surfaces If one
of the faces is concave producing divergence use a negative value for R And if a
surface is flat use R = infin If the face is flat the radius is infinite
The relation between the image object and focal distances for a thin lens is the same
as that for a curved mirror
1 1 1
q p f
Use a negative focal length if the lens is diverging
The lateral magnification for a thin lens is the same as for a curved mirror
Image height
Object height
qM
p
Camera
The lens system in a camera projects a real image of an object onto the film (or CCD
array in a digital camera) The position of the image is adjusted to be on the film by
moving the lens into or out of the camera
18
Eye
The eye is like a camera in that a real image is formed on the retina Unlike a camera
the image position is adjusted by changing the focal length of the lens This is done by
the ciliary muscle which squeezes the lens changing its shape
Near Point The near point is the closest distance from the eye for which the lens can
focus an image on the retina It is usually 18-25 cm for young persons
Far Point The far point is the greatest distance from the eye for which the lens can
focus an image on the retina For a person with normal vision the far point is at
infinity
Nearsightedness A person is nearsighted if their far point is at some finite distance
less than infinity This condition can be corrected with a lens that takes an object at
infinity and produces a virtual image at the personrsquos far point
Farsightedness and Presbyopia A person is farsighted if their near point is too far
away for comfortable near work like reading or knitting This can be corrected by a
lens which takes an object at a normal near point distance of 18-25 cm and produces a
virtual image at the personrsquos natural near point Presbyopia involves a similar
problem which nearly all people experience as they age The ciliary muscle becomes
too weak and the lens becomes too stiff to allow the eye to provide for both near and
far vision The solution for this problem is either reading glasses or bifocal lenses
Reading classes are just weak magnifying glasses mounted on eyeglass frames
Bifocal lenses are split into upper and lower halves The lower half is a lens which
gives the proper correction for near work and the upper half is a different lens for
proper focusing at infinity
Angular Size
When an object is brought closer to the eye it appears to be larger because the image
on the retina is larger The size of this image is directly proportional to the objectrsquos
angular size which is the angle subtended by the object measured from the center of
the lens of the eye In optical instruments which are to be used with the eye the
angular size of the final image is whatrsquos important because it determines how large
the image will appear to the viewer
19
Simple Magnifier
A simple magnifier is a single converging lens or magnifying glass It takes an object
closer to the eye than a normal near point and produces a virtual image at or beyond
this near point The angular magnification is defined to be the ratio of the angular size
when viewed through the lens to the angular size of the object when viewed at the
normal near point (without aid of the lens)
Microscope
This instrument has two lenses (1) The objective is near the object being viewed and
produces a greatly magnified real image (2) The eyepiece is a simple magnifier which
the viewer uses to closely examine the image from (1)
Telescope
This instrument also has two lenses (1) The objective at the front of the telescope
takes light from a distant object and produces a real inverted image (which is rather
small) near its focal point (2) This small real image is then examined by the eyepiece
functioning as a simple magnifier to produce a virtual image with a larger angular
size
20
Serway Chapter 37
Two-Slit Interference
If light is incident on two closely spaced narrow slits a pattern of light and dark
stripes is produced beyond the slits The bright stripes or fringes are caused by
constructive interference of the two waves coming from the slits Constructive
interference occurs whenever two waves arrive at a location in phase with each other
This occurs when the distance x1 from slit 1 to a point P on the screen and the
distance x2 from slit 2 to point P differ by in integral number of wavelengths
1 2 where 0 1 2x x m m
where λ is the wavelength of the light When x1 and x2 are much larger than the slit
spacing d this condition reduces to
sind m
where θ is the angle between the direction of the incident light and the direction of the
light arriving at the screen
Thin Films
When light is partially reflected and partially transmitted by a thin film of transparent
material it is possible to have interference between the wave reflected from the front
of the film and light reflected from the back of the film (The colored reflections from
the thin film of oil on the water in a rain-soaked parking lot are an example of this
effect) It is difficult to write down formulas that will work in all cases so we will
just review the important principles here
1 If the two reflected waves are in phase with each other the film has enhanced
reflection (constructive interference) but if the two reflected waves are out of phase
with each other reflection is diminished (destructive interference) Phase shifts occur
due to reflection and due to the extra path length through the film of the wave
reflected from the back of the film
2 The phase change due to reflection is determined by the difference in index of
refraction between the two media involved in the reflection If the wave is incident
21
from a medium with a low index of refraction into a medium with a high index of
refraction a phase change of 180deg occurs and the reflected wave is inverted If
incident from high to low no phase shift occurs and the reflected wave is non-
inverted
3 The extra path length through the film of the wave reflected from the back of the
film is equal to 2t where t is the thickness of the film (The incident light is assumed
to be normal to the surface of the film) The number of wavelengths contained in the
extra path length is equal to 2tλn where λn = λn is the wavelength of the light in the
film
4 Rules for reflection from thin films
If one ray is inverted and the other is not then we have
12 constructive
2 nt m
2 destructivent m
If either both rays are inverted or both are non-inverted we have
2 constructivent m
12 destructive
2 nt m
22
Serway Chapter 38
Diffraction Grating
A diffraction grating is simply a fancy version of two-slit interference with the two
slits replaced by thousands of slits Just as in the two-slit case the bright fringes
occur at angles given by
sin md
but in this case d the distance between neighboring slits is made to be very small
The effect of having many slits instead of two is to make each bright fringe highly
localized with wide dark regions between neighboring maxima
Single Slit Diffraction
When light passes through an opening in an opaque screen an interference pattern is
produced beyond the opening To understand why we may replace the single opening
by many small coherent sources of light These many sources interfere with each
other producing a pattern known as a diffraction pattern (Note that many authors do
not distinguish between interference and diffraction treating them as interchangeable
terms) If the opening is a slit of width a then the diffraction pattern far from the slit
will have a bright central maximum with a succession of minima and weaker maxima
on either side The angle between the incident direction and the minima is given by
sin ma
where m = plusmn1 plusmn2
If the opening is circular with diameter D the angle between the incident direction
and the first minimum is given by
sin 122D
Optical Resolution and Rayleighrsquos Criterion
Two point sources can just be resolved (distinguished from each other) if the peak of
the diffraction image of the first source overlies the first minimum of the diffraction
image of the second source For circular holes of the kind usually encountered in
23
optical devices this condition is approximately satisfied when the angular separation
between the two sources as viewed from the optical instrument is greater than or equal
to the critical angle
min 122D
where λ is the wavelength of the light and where D is the diameter of the aperture in
the instrument
Polarization
We say that an electromagnetic wave is polarized if its electric field vector doesnrsquot
change direction in a random fashion The simplest kind of polarization is linear
polarization in which the electric field vector oscillates back and forth along the
same axis in space Polarized light can be produced from normal unpolarized light by
selective absorption (as in Polaroid sunglasses) by passing light through crystals that
have different indices of refraction for different polarizations (double refraction) by
scattering (the blue sky is polarized) and by reflection (glare) Polarization by
reflection occurs when light reflects from a shiny insulating (non-metallic) surface
The amount of polarization is greatest for reflection at Brewsterrsquos angle
2
1
tan P
n
n
where θP is the incidence angle of light from medium 1 onto medium 2 and where n1
and n2 are the indices of refraction for the media (In our everyday experience n1 = 1
since the light comes in through the air and n2 is the index of refraction of the shiny
insulating material producing the glare eg water glass plastic paint etc)
Malusrsquos Law
The intensity of transmitted polarized light through a perfect polarizer is related to the
incident intensity of polarized light by Malusrsquos law
2cosoI I
where Io is the intensity of the incident light and where θ is the angle between the
electric field vector in the incident wave and the transmission axis the polarizer
24
Serway Chapter 39
Principles of Relativity
All of the weirdness of relativity flows from two simple principles
(1) The laws of physics must be the same in all inertial (non-accelerating) reference
frames
(2) The speed of light in vacuum has the same value in all inertial reference frames
Note that (1) seems reasonable but (2) is very odd It says that if two space ships are
approaching each other at nearly the speed of light and a laser pulse is shot from ship
1 toward ship 2 then when the people on ship 2 measure the speed of the pulse as it
goes by it is moving at 3 times 108 ms the same as if the ships were stationary
Simultaneity
If observer 1 sees two events in her own frame as simultaneous at two different
locations a moving observer 2 will see these two events happening at different times
Relativistic gamma
The factor γ (gamma) appears regularly in the formulas of relativity
2
2
1
1 vc
where v is the relative speed between two inertial frames
Time Dilation
If observer 1 sees two events at the same location in space separated by time Δtp in his
own frame then observer 2 moving at speed v relative to observer 1 will see these
two events separated by a longer time Δt
pt t
Moving clocks run slow
Length Contraction
If observer 1 measures the length of an object along the x-direction at some instant of
time in her frame to be Lp then observer 2 moving in the x-direction at speed v will
measure the length L of the object to be shorter
pLL
25
Moving meter sticks are short
Velocity Addition
If a particle is moving at velocity vac relative to frame c and if frame c is moving at
speed vcb with respect to frame b (with both velocities directed along the same line)
what would be the velocity of the particle in frame b In classical physics the answer is
ab ac cbv v v
but if any of these velocities are an appreciable fraction of the speed of light the
answer changes
21ac cb
abac cb
v Vv
v v c
In using this equation make sure that vac has the same sign in the numerator and in the
denominator and that vbc also has the same sign in both places Note that this equation
is different (and in our opinion easier to understand) than the equations in the text
Momentum and Energy
The momentum and total energy of a particle moving at velocity v (its speed is v v )
are given by
22
2 2
22
1 1 vvc c
m mcm E mc
vp v
Rest Energy
When a particle is at rest (v = 0 and γ = 1) then E = mc2 is called the rest energy It
includes the potential energy of the particle so that if we change the potential energy
of a particle we change its rest mass
Kinetic Energy The kinetic energy is the difference between the total energy and the
rest energy
2 2K mc mc
For v ltlt c this reduces to K = mv22
General Relativity
Einsteinrsquos principle of equivalence in the theory of general relativity is In the vicinity
of a point a gravitational field is equivalent to an accelerated frame of reference in
26
the absence of gravitational effects For example there is no difference whatever in
the physics experienced by an astronaut on earth and one who is in interstellar space
far away from any stars or planets but who has just turned on her spaceshiprsquos engines
and is accelerating at 98 ms2
27
Serway Chapter 40
Photons
We now know that even though light behaves like a wave it is better described as
consisting of small packets of energy called photons The energy of a photon is related
to its frequency by
E hf
where h is Planckrsquos constant
346626 10 J sh
Photons also carry momentum given by
hf hp
c
Since the amount of energy in each photon is fixed the difference between dim light
and intense light (of the same wavelength) is that dim light consists of fewer photons
than intense light
Photoelectric Effect
In the photoelectric effect light shining onto a metal surface gives the electrons in the
metal enough energy to escape and be detected It requires a certain amount of energy
(called the work function typically = 3 ndash 5 eV) just to kick the electron out of the
metal so the light must deliver at least this much energy to an electron to produce the
effect It is observed that red light no matter how intense never produces electrons
But ultraviolet light even if quite dim will eject electrons from the metal Since red
light consists of 2 eV photons and ultraviolet light has photons with energies around
4-6 eV the photon idea explains the behavior of the photoelectric effect The
maximum energy that an ejected electron can have is
maxK hf
where f is the photon frequency The minimum frequency that light can have and cause
electrons to be ejected is called the cutoff energy and is found by setting Kmax = 0
cf h
28
Compton Effect
When high-frequency light interacts with free electrons the scattered light does not
have the same wavelength as the incident light contrary to what classical
electromagnetism would predict But the photon picture in which photons have
momentum and energy allows us to treat scattering as a collision between the photon
and the electron In this collision the electron and the scattered photon both have a
different momentum and energy than they did before And since p = hλ for a photon
if the momentum is different the wavelength will also be different
after before 1 cose
h
m c
where the angle θ is the angle between the incoming photon direction and the
direction of the scattered photon
Particles are Waves
Since photons behave like particles sometimes it is not surprising that elementary
particles can behave like waves sometimes The wavelength of a particle with
momentum p is given by
h
p
Wave-particle Duality
Both photons and elementary particles have a dual nature sometimes they behave like
particles and sometimes they behave like waves It is hard for us to comprehend the
nature of such an object by making mental pictures but experiments definitively show
that this is the case Since we have no direct experience with photons and elementary
particles (because their energies are so much smaller than the energies of the everyday
objects we encounter) it is perhaps not surprising that we have a hard time forming a
mental picture of how they behave
Electron Interference
Since an electron is both a particle and a wave just like a photon it should be able to
produce an interference pattern This is observed If an electron beam is shot at two
closely-spaced slits and if the electrons that pass through the slits are detected
downstream it is found that there are some locations where electrons are never detected
29
and others where lots of electrons are detected The pattern is exactly the same as the
one observed for light waves provided that we use the electron wavelength λ = hp in
place of the wavelength of light This pattern is observed even though each electron is
detected as a single dot on the screen Only after many such dots are collected does the
pattern emerge And if we try to understand how this effect could possibly work by
looking closely at each slit to see which one the electron came through the pattern
disappears the act of measurement destroys the interference This means that each
single electron somehow comes through both slits (as a wave would)
Uncertainty Principle
In classical physics we always imagine that the positions and momenta of moving
particles have definite values It might be hard to measure them but surely at each
instant of time a particle should be precisely located at some point in space and have a
similarly precise momentum This turns out not to be true Instead both position and
momentum are required to be uncertain with their uncertainties Δx and Δp satisfying
the Heisenberg uncertainty relation
2x p
So if the particle were known to be precisely at some particular location (so that Δx =
0) we couldnrsquot know anything about its momentum (Δp = infin) and if its momentum
were exactly known we couldnrsquot know its position
There is a similar relation involving the particlersquos energy E and the time interval Δt
over which this energy is measured
2E t
This means that energy is not actually conserved if we are considering very small time
intervals and this brief non-conservation of energy has been observed
30
Serway Chapter 41
Wave Function ψ and Probability
Quantum mechanics does not predict exactly what an electron or a photon will do Instead
it specifies the wave function or probability amplitude ψ of an electron or a photon
This wave function is a complex-valued function of space and time whose squared
magnitude is the probability density P for finding a particle at a particular place in
space at a certain time
2P
where is the complex conjugate of ψ
For example the wave function of an electron with perfectly specified momentum p
would have a wavelength given by
h
p
and its probability amplitude would be proportional to
2i x ipxe e
The corresponding probability density would then be
21ipx ip ipxP e e e
which means that the electron is equally probable to be anywhere along the x axis
This is in accord with the uncertainty principle since we specified the momentum
precisely we canrsquot have any idea about the position of the electron
As another example you have probably seen ldquofuzzy ballrdquo drawings of electron
orbitals in a chemistry book These fuzzy balls are meant to indicate the distribution
of the probability density 2
P in the orbital
Particle in a Box
A simple example in which we can calculate the wave function is the case of a particle of
mass m constrained to be inside a 1-dimensional box between x = 0 and x = L In this
case the general wave function is a linear superposition of wave functions ψn of the form
sin niE tn
n xx t A e
L
31
where A is a positive constant where n = 1 2 3 and where the energy associated
with each of the quantum states ψn is given by
22
28n
hE n
mL
This wave function is zero at x = 0 and x = L which means that the particle will never
be found at the walls of the box The wave function has maximum values in the
interior and at these places the particle is most likely to be found and it also has
places where it is zero and at these places the particle will also never be found as
expressed by the formula
2sinn x
PL
The particle in the box is interfering with itself producing a probability interference
pattern across the box just like the interference patterns we studied with light and
sound
32
Serway Chapter 42
Atoms
Many people picture an atom as a miniature solar system where electrons orbit around
a massive nucleus at the center This picture is misleading because in atoms the wave
nature of electrons dominates The electrons form 3-dimensional standing waves
(called orbitals) centered at the nucleus And if you ask what is it thatrsquos waving the
answer is ψ the probability amplitude (see Chapter 41)
Standing waves on a string can occur only for certain frequencies (the fundamental
and higher harmonics) Similarly atomic orbitals occur only for certain energies For
the hydrogen atom the energies of the orbitals have the a particularly simple form
2
1136 eVnE
n
where n = 1 2 3 is called the principal quantum number For other atoms the
determination of the orbital energies requires numerical calculation by computers
Atomic Spectra
If an electron is somehow given extra extra energy (we say that it is excited) so that it
occupies a higher orbital it will eventually ldquofallrdquo back down to a lower orbital Each
time an electron falls to a lower orbital it loses the difference in energy between the
two orbitals in the form of a photon Since the orbital energies are discrete so are the
energy differences and so are the wavelengths of the emitted light The entire set of
these discrete wavelengths is called the atomic spectrum and it is unique to each type
of atom For hydrogen the spectrum can be simply written as
H 2 2
1 1 1
f i
Rn n
where RH = 10973732 times 107 m-1 For other atoms the spectrum cannot be expressed as
a formula
Orbital Angular Momentum Electrons in atoms also have quantized values of
angular momentum The orbital quantum number ℓ specifies the value of this
quantized angular momentum through the formula
1L
33
If we want to know the value of the angular momentum along some direction in space
say the z direction the answer is not L but rather
zL m
where mℓ is another quantum number which runs from
1 1m
This quantum number is important when an atom sits in a magnetic field
Spin Angular Momentum It has been found experimentally that electrons and other
charged particles also carry internal angular momentum which we call spin
Electrons have an intrinsic spin angular momentum s along a specified axis that is
extremely quantized it can only take on 2 possible values
1
2zs s
We thus say that electrons have spin s = 12 or that electrons are rdquospin one-half parti-
cles Since s is an angular momentum it obeys the same rule as that for orbital angular
momentum namely that if its value along some axis is s then its total magnitude is
given by
31
2S s s
I know this seems weird but quantum mechanics is weird The only excuse for this
bizarre way of looking at the world is that it predicts what happens in experiments
Exclusion Principle The answer to the question of how many electrons (or any other
spin one-half particle) can be in one particular quantum state was discovered by
Wolfgang Pauli and is called the exclusion principle
ldquoNo two electrons can ever be in the same quantum state therefore no two electrons
in the same atom can have the same set of quantum numbersrdquo
This is the reason that we have atoms with different properties instead of every atom
simple having all of its electrons in the ground state All of the variety we see around
us in the world is the result of chemical differences and these differences would not
exist unless electrons obeyed this important principle The entire structure of the
periodic table (see pages 1377-1379 in Serway) is an expression of this principle
34
Serway Chapter 44
Nuclear Properties
The nucleus is composed of protons with charge +e and mass mp = 1007226 u and of
neutrons with zero charge and mass mn = 1008665 u where u is one atomic mass unit
-271 u=1660540 10 kg
The atomic number Z counts the number of protons in a nucleus while the neutron
number N counts the number of neutrons The mass number A is the sum of the two
A N Z
Protons and nuetrons have about the same mass and some times called baryons (heavy
ones) or nucleons (partcles in the nucleus) A is sometimes called the baryon
number The chemical elements are distinguished by Z ie hydrogen has Z = 1 iron
has Z = 26 uranium has Z = 92 etc But for a given Z there might be several nuclei
with different numbers of neutrons These nuclear siblings are called isotopes
These numbers are used to label nuclei according to the pattern
5626 ie FeA
Z X
denotes the isotope of iron with N = 56 ndash 26 = 30 neutrons
The nucleus is roughly spherical with a radius given approximately by
1 3 150 0where 12 10 mr r A r
Nuclear Stability
Because the positively charged protons electrically repel each other with an enormous
force at distances as small as 10-15 m there must be some really strong force that that
overcomes electrical repulsion to hold protons and neutrons together This force is
called with some lack of imagination the strong force It is a very short-range force
(it only acts over a distance of about 2 x 10-15 m) and attracts protons to protons
neutrons to neutrons and neutrons to protons But in spite of this strong nuclear force
the coulomb repulsion of the protons is still present so anything that might keep the
protons from being right next to each other would help keep the nucleus from
35
exploding This role is played by the neutrons and for nuclei with Z le 20 the stable
nuclei roughly have N = Z
For Z gt 20 the coulomb repulsion force is more powerful and more neutrons are
needed to dilute it up to about N = 15Z around Z = 80 For Z gt 83 no amount of
neutrons can help and these nuclei are unstable (radioactive) There is an ldquoisland of
(relative) stabilityrdquo around Z=90-92 (thorium and uranium respectively) These
elements have one or two isotopes with half-lifes of billions of years so there are
substantial amounts of such elements on Earth This fact makes it possible to have
practical fission devices
Radioactive Decay
There are three types of radioactive decay
Alpha decay the nucleus kicks out a helium nucleus (N = 2 Z = 2 A = 4) So if the
original nucleus is called X and the new nucleus is called Y then the decay would look
like this
4 42 2X Y+ HeA A
Z Z
Beta decay the nucleus either kicks out an electron (endash) or its positively-charged
antimatter twin the positron (e+) plus either an electron neutrino v or an electron
anti-neutrino v
1X Y+e electron decayA AZ Z v
1X Y+e positron decayA AZ Z v
A neutrino is a particle with no charge hardly any mass (much less than the electron
mass) and interacts so weakly with matter that most neutrinos upon encountering the
planet earth just pass right through it as if it werenrsquot there
Gamma decay the nucleons in the nucleus X are in an excited energy state X
(perhaps as a result of having undergone alpha or beta decay) and they drop down to a
lower energy state shedding the energy as a high frequency photon
X XA AZ Z
36
This process is exactly analogous to the way that the electrons in atoms emit photons
Decay Rate and Half Life
There is no way to predict exactly when an unstable or excited nucleus will decay but
there is an average rate at which this decay occurs called the decay constant λ The
meaning of this constant is that if there are a large number N of nuclei in a sample
then the number of decays per second that will be observed (called the decay rate R)
is R = λN In mathematical language
dMR N
dt
This simple differential equation has for its solution
0tN t N e
where N0 is the number of nuclei in the sample at time t = 0
The half-life is the time it takes for half of the nuclei in the sample to decay and is
related to the decay constant by
1 2
ln 2 0693T
Disintegration Energy
When a nucleus decays it is making a transition to an overall state of lower energy
which means according to Einsteinrsquos famous formula E = mc2 that the sum of the
masses after the decay must be less than the mass before with the lost mass appearing
as kinetic energy among the decay products For example in alpha decay this kinetic
energy called the disintegration energy Q is given by
2X YQ M M M c
37
Serway Chapter 45
Nuclear energy
There are two ways to extract energy from the nucleus fission and fusion For nuclei
with Z greater than 26 breaking the nucleus apart into pieces leads to a lower
2mc energy than the original nucleus so energy can be extracted by fission For
nuclei with Z less than 26 a lower energy is achieved by combining nuclei so energy
can be extracted by fusion
Fission
Since neutrons have no charge they are not repelled from nuclei as protons are for
this reason their behavior is the key to understanding how fission works
Fast neutrons mostly bounce off other nuclei with each collision slowing the neutron
down This elastic energy loss is most effective if the other nuclei have low mass (like
hydrogen) and these materials are called moderators because of their ability to slow
down fast neutrons
The reason that slowing neutrons is important is that slow neutrons are much more
likely to be absorbed by a nucleus which then leads to nuclear reactions of various
kinds For a few very large nuclei like uranium-235 and some plutonium isotopes
absorption of a slow neutron causes the nucleus to split into two large fragments plus
2 or 3 fast neutrons (energy is released in the process too) Thus one slow neutron
can produce energy plus 2 or 3 more neutrons which if moderated from fast to slow
can split 2 or 3 more nuclei leading to a runaway chain reaction If the reaction is
allowed to proceed unchecked it produces a large explosion If a neutron absorbing
material is added to the mix (like the cadmium in reactor control rods) it is possible to
keep the reaction under control and to extract the released energy as heat to drive
steam turbines and produce electricity
Fusion
Fusion involves mashing two nuclei together and since they are both charged and repel
each other this reaction is much harder to make go The nuclei must have enough energy to
overcome the coulomb repulsion which is why this reaction requires a high temperature
(hundreds of millions of degrees K) like that in the sun (which is burning hydrogen to
helium via fusion) or like that in the center of the fission explosion that is used to detonate
38
a hydrogen bomb
This reaction is of interest for power production in spite of this difficult temperature
requirement because of the abundance of fusion fuel on the planet There are about
012 g of deuterium ( 21H ) in every gallon of water on earth and it only costs about 4
cents to extract it The fusion energy available from this minuscule amount of
deuterium would run a 1000-Megawatt power plant for 10 seconds Water is so
abundant on earth that if fusion were to work we would have an essentially
inexhaustible source of energy
So why donrsquot we have fusion power plants Well the fuel is cheap but the match is
incredibly expensive The only way we know to control this difficult high-temperature
reaction is with large and expensive pieces of equipment involving either large
magnetic fields and complex high-power electromagnetic antennas or with gigantic
(football-fieldsized) laser facilities involving more than a hundred of the highest-
energy lasers ever built Power plants based on these current methods for controlling
fusion are unattractive to the fiscally-minded people who run the electric power
industry Hopefully better designs will be discovered as experiments continue
39
c actinium 68 Er erbium 101 Md mendelevium 104 Rf rutherfordium l aluminum 63 Eu europium 80 Hg mercury 62 Sm samarium m americium 100 Fm fermium 42 Mo molybdenum 21 Sc scandium b antimony 9 F fluorine 60 Nd neodymium 106 Sg seaborgium r argon 87 Fr francium 10 Ne neon 34 Se selenium s arsenic 64 Gd gadolinium 93 Np neptunium 14 Si silicon t astatine 31 Ga gallium 28 Ni nickel 47 Ag silver a barium 32 Ge germanium 41 Nb niobium 11 Na sodium k berkelium 79 Au gold 7 N nitrogen 38 Sr strontium e beryllium 72 Hf hafnium 102 No nobelium 16 S sulfur i bismuth 108 Hs hassium 76 Os osmium 73 Ta tantalum h bohrium 2 He helium 8 O oxygen 43 Tc technetium
boron 67 Ho holmium 46 Pd palladium 52 Te tellurium r bromine 1 H hydrogen 15 P phosphorus 65 Tb terbium d cadmium 49 In indium 78 Pt platinum 81 Tl thallium a calcium 53 I iodine 94 Pu plutonium 90 Th thorium f californium 77 Ir iridium 84 Po polonium 69 Tm thulium
carbon 26 Fe iron 19 K potassium 50 Sn tin e cerium 36 Kr krypton 59 Pr praseodymium 22 Ti titanium s cesium 57 La lanthanum 61 Pm promethium 74 W tungsten l chlorine 103 Lr lawrencium 91 Pa protactinium 92 U uranium r chromium 82 Pb lead 88 Ra radium 23 V vanadium o cobalt 3 Li lithium 86 Rn radon 54 Xe xenon u copper 71 Lu lutetium 75 Re rhenium 70 Yb ytterbium m curium 12 Mg magnesium 45 Rh rhodium 39 Y yttrium b dubnium 25 Mn manganese 37 Rb rubidium 30 Zn zinc y dysprosium 109 Mt meitnerium 44 Ru ruthenium 40 Zr zirconium s einsteinium
40
UNITS The SI units are given in parentheses Other commonly used units are given in terms of the SI units Symbols conform with the recommendations of the American National Standards Institute (ANSI) and the American Institute of Physics (AIP) Commonly used multiples of SI units
prefix symbol factor tera T 1012 giga G 109
mega M 106 kilo k 103
centi c 10-2 milli m 10-3
micro μ 10-6 nano n 10-9 pico p 10-12
femto f 10-15
length meter (m) angstrom 1 Ǻ = 10-10 m inch 1 in = 254 x 10-2 m foot 1 ft = 03048 m mile 1 mi = 1609 m
mass kilogram (kg) atomic mass unit 1 u = 1661 x 10-27 kgslug 1 slug = 1459 kg
force newton (N = kg bull ms2) dyne 1 dyn = 10-5 N pound 1 lb = 4448 N
pressure Pascal (Pa = kgm bull s2) atmosphere 1 atm = 1013 x 105 Pa poundssquare inch 1 psi = 6895 Pa cm of mercury 1 cm Hg = 1333 Pa bar 1 bar = 1000 x 105 Pa torr 1 torr = 1333 Pa
time second (s) minute 1 min = 60 s hour 1 h =3600 s
frequency hertz (Hz = s-1) radianssecond 1 rads = 12π Hz
energy joule (J = kg bull m2s2) erg 1 erg = 10-7 J electron volt 1 eV = 1602 x 10-19 J calorie 1 cal = 4187 J kilowatt-hour 1 kWh = 36 x 106 J British thermal unit 1 Btu = 1055 J
power watt (W = kg bull m2s3) horsepower 1 hp = 7457 W
charge coulomb (C = A bull s)
electric potential volt (V = kg bull m2 s3 bull A)
current ampere (A)
resistance ohm (Ω = kg bull m2s3 bull A2)
capacitance farad (F = s4 bull A2kg bull m2)
magnetic field tesla (T = kgs2 bull A) gauss 1 G = 10-4 T
magnetic flux weber (Wb = kg m2s2 bull A) maxwell 1 Mx = 10-8 Wb
magnetic inductance henry (H = kg bull m2s2 bullA2)
temperature kelvin (K) degrees Celsius 0degC = 27315 K
angle radian (rad) degree 1 deg = π180 rad revolution 1 rev = 2π rad
41
Some Physical Constantsa Quantity Symbol Valueb Atomic mass unit μ 1660 538 73 (13) x 10-27 kg
931494 013 (37) MeVc2 Avogadros number NA 6022 141 99 (47) x 1023 particlesmol Avogadrorsquos number 6022 x 1023 mol Bohr magneton
2B e
em 9274 008 99 (37) x 10-24 JT
Bohr magneton μB 927 x 10-24 JT Bohr radius
0
2
2e em e k
a 5291 772 083 (19) x 10-11 m
Boltzmanns constant B A
RNk 1380 650 3 (24) x 10-23 JK
Boltzmannrsquos constant kB 1380 x 10-23 JK Compton wavelength
C eh
m c 2426 310 215 (18) x 10-12 m
Deuteron mass md 3343 583 09 (26) x 10-27 kg 2013 553 212 71 (35) u
electron charge e 1602 x 10-19 C Electron mass me 9109 381 88 (72) x 10-31 kg
5485 799 110 (12) x 10-4 u 0510 998 902 (21) MeVc2
electron mass 911 x 10-31 kg Electron volt eV 1602 176 462 (63) x 10-19 J Elementary charge e 1602 176 462 (63) x 10-19 C Gas constant R 8314 472 (15) JKmol Gravitational constant G 6673 (10) x 10-11 Nm2kg2 Neutron mass mn 1674 927 16 (13) x 10-27 kg
1008 664 915 78 (55) u 939565 330 (38) MeVc2
neutron mass 1675 x 10-27 kg Nuclear magneton
2n p
em 5050 783 17 (20) x 10-27 JT
permeability constant μ0 1257 x 10-6 Hm permittivity constant ε0 8854 x 10-12 Fm Plancks constant h
2h
6626 068 76 (52) x 10-34 Js 1054 571 596 (82) x 10-34 Js
Planckrsquos constant h ħ
6626 x 10-34 Js 1055 x 10-34 Js
Proton mass mp 1672621 58 (13) x 10-27 kg 1007 276 466 88 (13) u 938271 998 (38) MeVc2
proton mass 1673 x 10-27 kg Rydberg constant RH 1097 373 156 854 9 (83) x 107 m-1 Speed of light in vacuum c 2997 92458 x 108 ms (exact) speed of light c 300 x 108 ms a
These constants are the values recommended in 1998 by CODATA based on a least-squares adjustment of data from different measurements For a more complete list see P J Mohr and B N Taylor Rev Mod Phys 72351 2000 b The numbers in parentheses for the values above represent the uncertainties of the last two digits
NOTE The ones in red are the ones from Appendix 13 The ones in yellow are the ones that are from the Serway book table but that were already on the Appendix 13 table
42
INDEX
Absolute zero 9
Adiabatic 17
Adiabatic exponent 18
Adiabatic process 17
Alpha decay 46
Amplitude 4
Angle of reflection 25
Angular frequency 3
Angular magnification 30
Angular momentum atomic 43
Angular size 29
Archimedes Principle 1
Atomic spectra 43
Atoms 43
Avogadrorsquos number 10
Beats 8
Bernoullirsquos Equation 2
Beta decay 46
Bifocals 29
Brewsterrsquos angle 34
British Thermal Unit Btu 11
Buoyancy 2
calorie 11
Calorie 11
Camera 28
Carnot cycle 21
Carnot efficiency 22
Celsius scale 9
Ciliary muscle 29
Coefficient of performance 21
Compton effect 39
Constructive interference 7
Continuity equation of 2
Contraction length 35
Convection 14
Curved mirrors 27
Decay constant 46
Decay rate radioactive 46
Decibel scale 5
Degrees of freedom 16
Density 1
Destructive interference 7
Diffraction single slit 33
Diffraction grating 33
Dilation time 35
Disintegration energy 47
Dispersion of light 26
Doppler effect 6
Double slit interference 31
Efficiency 20
Electron interference 39
Emissivity 15
Energy nuclear decay 47
Energy relativistic 36
43
Engineering work 20
Entropy 22
Entropy ideal gas 24
Equipartition of energy 18
Equivalence principle 36
Exclusion principle 44
Expansion thermal 9
Expansion coefficient linear 9
Eye 29
Fahrenheit scale 9
Far point 29
Farsightedness 29
First Law of Thermodynamics 12
Fission 48
Flux volume 2
Free expansion 23
Fringes 31
Fusion 48
Fusion heat of 11
Gamma relativistic 35
Gamma decay 46
General relativity 36
Half-life 46
Heat 11
Heat capacity 11
Heat conduction 13
Heat engine 20
Heat of fusion 11
Heat of vaporization 11
Heat pump 20
Heats of transformation 11
Hydrostatics 1
Ideal Gas Law 10
Images real and virtual 27
Intensity sound 5
Interference 7
Interference two-slit 31
Internal energy 12
Internal energy degrees of freedom 16
Irreversible process 19 23
Isotopes 45
Joule 11
Kelvin scale 9
Kinetic energy relativity 36
Kinetic theory 16
Latent heat 11
Length contraction 35
Linear expansion coefficient 9
Linear polarization 34
Linear superposition 7
Longitudinal wave 3
Loudness 5
Magnification lateral 27
Magnifying glass 30
Malusrsquos law 34
Microscope 30
Momentum relativistic 36
Muscle ciliary 29
44
Musical instruments 8
Musical scale 8
Near point 29
Nearsightedness 29
Nonlinear 7
Nuclear energy 48
Nuclear properties 45
Nuclear stability 45
Octave 8
Optical resolution 33
Orbital quantum number ℓ 43
Orbitals 43
Particle in a Box 41
Particles are waves 39
Pascalrsquos Principle 1
Period 3
Photoelectric effect 38
Photons 38
Pitch 8
Polarization 34
Power sound 5
Presbyopia 29
Pressure 1
Principal quantum number n 43
Principle of equivalence 36
Principle of linear superposition 7
Probability amplitude ψ 41
Processes thermodynamic 13
R-value 14
Radiation thermal 14
Radioactive decay 46
Radioactive decay rate 46
Ray tracing 27
Rayleighrsquos criterion 33
Reading glasses 29
Real image 27
Refraction 25
Refrigerator 20
Relativistic gamma 35
Relativity principles 35
Resolved for light sources 33
Rest energy 36
Reversible process 19 22
Rope wave speed 4
Second Law of Thermodynamics 19
Shock waves 6
Simple magnifier 30
Simultaneity 35
Single slit diffraction 33
Snellrsquos law 25
Sound speed 5
Specific heat 11
Spin Angular momentum 44
Standing waves 7
Stefanrsquos law 14
Strong force 45
Telescope 30
Temperature 9
45
Temperature Scales 9
Thermal conductivity 14
Thermal energy 12
Thermal expansion 9
Thin film interference 31
Thin lenses 28
Time dilation 35
Tone musical 8
Total internal reflection 25
Transverse wave 3
Traveling Waves 3
Two-slit interference 31
Uncertainty principle 40
Vaporization heat of 11
Velocity addition relativity 36
Virtual image 27
Volume flux 2
Wave function 41
Wave Function ψ and Probability 41
Wave speed 3
Wave-particle duality 39
Wavelength 3
Wavenumber 3
Wien Displacement Law 16
Work 11
8
Serway Chapter 22
Second Law of Thermodynamics
There are several equivalent forms of this important law
Kelvin It is not possible to change heat completely into work with no other change
taking place Or in other words there are no perfect heat engines
Clausius It is not possible for heat to flow from one body to another body at a higher
temperature with no other change taking place Or in other words there are no
perfect refrigerators
Entropy In any thermodynamic process that proceeds from one equilibrium state to
another the entropy of the system + environment either remains unchanged or
increases The total entropy never decreases This law is a bit of an oddity among the
laws of physics because it is not absolute Things are forbidden by the second law not
because it is impossible for them to happen but because it is extremely unlikely for
them to happen (See below for more information about entropy)
Reversible and Irreversible Processes
A reversible process is one which occurs so slowly that it is in thermal equilibrium (or
very nearly so) at all times A hallmark of such processes is that a motion picture of
them looks perfectly normal whether run forward or backward Imagine for instance
the slow expansion of a gas at constant temperature in a cylinder whose volume is being
increased by a slowly moving piston Run the movie backwards and what do you see
You see the slow compression of a gas at constant temperature which looks perfectly
normal
An irreversible process is one which occurs in such a way that thermal equilibrium is
not maintained throughout the process The mark of this kind of process is that a motion
picture of it looks very odd when run backward Imagine the sudden expansion of a gas
into a previously evacuated chamber because a hole was punched in the wall between a
pressurized chamber and the evacuated one Run the movie backward and what do you
see You see the gas in the soon-to-be-evacuated chamber gather itself together and
stream through a tiny hole into a chamber in which there is already plenty of gas If you
have ever seen this happen get in touch with the support group for those who have
witnessed the spontaneous combustion of large mammals by calling 1-800-PYROCOW
9
Heat Engines
Because of the vexing difference in sign between work done on and system and work done
by a system we will invent a new work variable Weng Heat engines do work and so the net
W for these engines is negative But in engineering applications hidden minus signs are
regarded as evil so for heat engines we donrsquot talk about W instead we talk about its
magnitude engW W So for heat engines the first law is
int engE Q W
But for heat pumps and refrigerators work is done on the system so we use the usual work
W when we talk about these systems
A heat engine is a machine that absorbs heat converts part of it to work and exhausts the
rest The heat must be absorbed at high temperature and exhausted at low temperature If the
absorbed heat is Qh the exhausted heat is Qc and the work done by the engine is Weng then
eng h cW Q Q
and the efficiency of the engine is defined to be
eng
h
We
Q
A perfect engine would convert the heat hQ completely into work Weng giving an effi-
ciency of e = 1 Energy conservation alone allows a perfect engine but the second law
requires e lt 1
Refrigerators and Heat Pumps
A refrigerator is a machine that absorbs heat at low temperature and exhausts it at high
temperature the ldquobackwardsrdquo heat transfer being driven by the work done on the machine
by some source of power A heat pump is a machine that either works like a refrigerator
keeping a place cold by transferring heat from this cold place to a higher temperature
environment (cooling mode like an air conditioner) or it functions as a heater
transferring heat into a warm place from a cooler one (heating mode like a window unit
that heats a house by extracting thermal energy from the cold outdoors) The coefficient
of performance of a refrigerator or of a heat pump in cooling mode is defined to be
COP cooling modecQ
W
10
For a heat pump in heating mode the coefficient of performance is
COP heating modehQ
W
Note that we donrsquot have to use the engineering work here because in these systems
work is done on the system and W is naturally positive
A perfect refrigerator would take in heat Qc and exhaust the same amount of heat without
doing any work giving an infinite coefficient of performance Energy conservation alone
allows a perfect refrigerator but the second law requires COP lt infin A perfect heat pump
in heating mode would transfer Qh into the house without doing any work and so would
also have an infinite coefficient of performance The second law forbids this too A good
coefficient of performance for a real device would be around 5 or 6
Carnot Cycle
The most efficient of all possible engines is one that uses the Carnot cycle This cycle
employs an ideal gas has no friction and operates very slowly so that the gas can be in
thermal equilibrium at all parts of the cycle This means of course that it canrsquot
possibly be built and even if it could be built it would not run fast enough to be useful
Nevertheless this cycle is very important because it gives an upper bound on the
efficiency of real engines There cannot possibly be an engine that is more efficient
than one based on the Carnot cycle This cycle consists of the following four steps
1 The ideal gas absorbs heat Qh at constant temperature Th while the gas increases its
volume The reason that heat is absorbed is that expansion tends to cool the gas but
thermal contact with the environment at Th keeps the temperature high by heat
conduction into the ideal gas
2 The ideal gas further increases its volume by an adiabatic expansion This expansion
causes the gas to cool so at the end of this part of the cycle the gas is at temperature Tc
3 The gas exhausts heat Qc at constant temperature Tc while the gas decreases in
volume The reason that heat is exhausted is that compression tends to heat the gas
but thermal contact with the environment at Tc keeps the temperature low by heat
conduction out of the ideal gas
4 The gas is adiabatically compressed back to its original volume (the volume it started
with in step 1) This compression heats the gas from Tc up to Th
11
The efficiency of a Carnot engine is given by the very simple formula
1 cC
h
Te
T
where the temperatures must in be Kelvin No real engine can be more efficient than
this
The coefficient of performance of a Carnot refrigerator or heat pump in cooling mode
is given by
COP cooling mode cC
h c
T
T T
and no real refrigerator can have a coefficient of performance greater than this
The coefficient of performance of a Carnot heat pump in heating mode is
COP heating mode hC
h c
T
T T
So why donrsquot we just use these wonderful Carnot engines and have perfect efficiency
To make the reversible steps in the cycle really reversible they would have to occur
infinitely slowly So the price you pay for making a perfect engine is that it takes
forever to get it to do any work
Entropy
The entropy of a system is defined in terms of its molecular makeup and measures
roughly the disorder of the system If the system is packed into a very small volume
then it is quite ordered and the entropy will be low If it occupies a large volume the
entropy is high (To see what this has to do with disorder note that socks in a drawer
occupy a small volume while socks on the bed in the corner by the door and
hanging from the chandelier occupy a large volume) If the system is very cold then
the molecules hardly move and may even reach out to each other and form a crystal
This is a highly ordered state and therefore has low entropy If the system is very hot
with rapidly speeding molecules crashing into the container walls and bouncing off
each other things are disordered and the entropy is high
It is possible to calculate the entropy of a system in terms of its macroscopic thermody-
namic properties ie pressure volume temperature number of moles etc The key to this
calculation is the concept of a reversible process A reversible process is one that is
12
carried out without friction and so slowly that the process can be reversed at any stage by
making an infinitesimal change in the environment of the system The slow expansion of the
gas at Th in the Carnot cycle with tiny amounts of heat being transferred to the environment
is reversible If we turned around and began slowly to compress the gas would just slowly
exhaust heat to the environment in the exact reverse way that it absorbed it during expansion
Most processes however are irreversible For example if a gas-filled box were suddenly
increased in size so that the particles were free to wander into the void created by the sudden
expansion then the gas would eventually fill the new volume uniformly at the same
temperature as before the expansion (The temperature is unchanged in this imaginary
process because the kinetic energy of the molecules would be unaffected by such an
instantaneous expansion of the container walls) This imaginary but highly thought-
stimulating process is called a free expansion and it is impossible to reverse it During the
expansion we didnrsquot push on any of the molecules so reversing this process would mean
making them go back into their original volume without pushing on them they simply will
not cooperate to this extent Another way to see that just pushing them back where they came
from does not reverse the free expansion is to think about what would happen if we just
compressed either adiabatically or isothermally An adiabatic compression back to the
original volume would heat the gas above its original temperature and an isothermal
compression would require that heat be exhausted to the environment But the free expansion
involved neither temperature changes nor heat exchanges so neither of these two processes
is the reverse of the free expansion It is simply impossible to reverse this rapid expansion in
a way that takes us back to the initial state
It is possible to calculate the change of entropy for both reversible and irreversible processes
Letrsquos consider a reversible process first In a reversible process the entropy change is given
by the formula
dQS
T
where dQ is the amount of heat added to the system during a small step of the process
The total energy change during the process may then simply be calculated by integration
f f
f i i i
dQS S S dS
T
13
(Just as in the case of energy we are mainly interested in differences rather than in
absolute magnitudes)
This integration method does not work for irreversible processes For instance in the
rapid free expansion discussed above no heat is added to the system but its disorder
obviously goes up We need to find some other way to calculate the entropy The key
is the fact that the entropy of a system depends only on its current state and not at all
on how it arrived there So to calculate the entropy change in an irreversible process
first find out what the initial and final conditions of the process are Then invent a
reversible process that takes the system from the initial state to the final state Since
the entropy depends only on the state of system and not on the process the entropy
change for the reversible process is the same as that for the irreversible process
Entropy is thus like pressure or temperature if the pressure changes from Pl to to P2
then the pressure difference is P2 - Pl regardless of how the pressure change was
made This seems obvious for pressure but not for entropy only because you arent
comfortable with entropy yet
Entropy of an Ideal Gas
For n moles of an ideal gas the difference in entropy between a state with temperature
T and volume V and some standard state with temperature To and volume Vo is given
by
ln lno o Vo o
T VS T V S T V nC nR
T V
Entropy in a Phase Change
Calculating entropy change in a phase change like melting or boiling is easy It is
QT Where Q is in the latent heat for example the flows in to cause the melting and
T is the temperature of the phase change There is NO integral to do
14
Serway Chapter 35
Angle of Reflection
If plane waves are incident on a reflecting surface with the propagation direction of
the waves making angle θ1 with the normal direction to the surface then the reflected
angle θ1prime relative to the surface normal is simply
1 1
ie the incident angles and reflected angles are the same
Refraction
If plane waves traveling through medium 1 are incident on a plane interface between
medium 1 and medium 2 then the angle of incidence of the incoming wave θ1 and
the angle of refraction of the transmitted wave θ2 are related by Snellrsquos law
2 2
1 1
sin
sin
v
v
where v1 and v2 are the wave speeds in medium 1 and medium 2 The angle of
incidence and the angle of refraction are both measured between the wave propagation
direction and the normal to the interface In terms of indices of refraction in the case
of light waves Snellrsquos law takes the more familiar form
1 1 2 2sin sinn n
where ni = cvi
Total Internal Reflection
If a wave is incident from a medium of low wave speed into a medium of high wave
speed the law of refraction requires that the angle of refraction be greater than the
angle of incidence If the angle of refraction is required to be greater than 90deg then no
refracted wave can exist and total internal reflection occurs The critical incident
angle θc beyond which total internal reflection occurs is given by
2
1
sin c
n
n
15
Dispersion of Light
In addition to the speed of light varying from material to material it also varies with
wavelength within each material This means that the index of refraction is generally a
function of wavelength
c
nv
Since the wave speed is not constant such a medium is dispersive meaning in this
context that refraction actually disperses white light into its various colors because
Snellrsquos law gives a different angle for each wavelength In most materials the
variation with wavelength is quite small but this small effect is responsible for some
of the most spectacular color effects we ever see including rainbows a flashing
crystal chandelier and the colored fire of a diamond solitaire by candlelight
16
Serway Chapter 36
Real and Virtual Images
When light rays are focused at a certain plane producing an image if a sheet of white
paper is placed there we call the image a real image The images produced by film
projectors and overhead projectors are examples of real images
When light rays appear to come from a certain location but no image is produced
when a screen is placed there we say that there is a virtual image at that location For
instance when you look in a mirror it appears that someone is behind the mirror but
a screen placed back there in the dark would show nothing Your image in the mirror
is a virtual image
Ray Tracing
There are lots of rules about how to find the images in optical systems but the best
way to keep things straight is to learn how to draw the principal rays for curved
mirrors and lenses The rays for convex and concave mirrors are shown in Fig 3615
The rays for converging and diverging thin lenses are shown in Fig 3627 You should
memorize the principal rays and know how to use them to locate images
Curved Mirrors
The focal length of a curved mirror with radius of curvature R is given by
2
Rf
The relation between object distance p image distance q and focal length f is
1 1 1
p q f
If the mirror is a diverging mirror f should be negative and if q should turn out to be
negative the image is virtual
Lateral Magnification
The lateral magnification in an optical system is defined by the ratio of the image size
to the object size
17
Image height
Object height
qM
p
for curved mirrors
As usual there are sign conventions here too but this formula just gives the
magnitude It is better to keep keep track of upright images versus inverted images by
means of ray diagrams rather than by memorizing sign conventions
Thin Lenses
The focal length of a thin lens is related to the radii of curvature of the two faces R1
and R2 of the lens by
1 2
1 1 11n
f R R
Note that this formula differs from Eq (3611) in the text by not having a minus sign
between the two R-terms We like this form better because for a simple converging
lens like a magnifying glass we just use positive values of R for both surfaces If one
of the faces is concave producing divergence use a negative value for R And if a
surface is flat use R = infin If the face is flat the radius is infinite
The relation between the image object and focal distances for a thin lens is the same
as that for a curved mirror
1 1 1
q p f
Use a negative focal length if the lens is diverging
The lateral magnification for a thin lens is the same as for a curved mirror
Image height
Object height
qM
p
Camera
The lens system in a camera projects a real image of an object onto the film (or CCD
array in a digital camera) The position of the image is adjusted to be on the film by
moving the lens into or out of the camera
18
Eye
The eye is like a camera in that a real image is formed on the retina Unlike a camera
the image position is adjusted by changing the focal length of the lens This is done by
the ciliary muscle which squeezes the lens changing its shape
Near Point The near point is the closest distance from the eye for which the lens can
focus an image on the retina It is usually 18-25 cm for young persons
Far Point The far point is the greatest distance from the eye for which the lens can
focus an image on the retina For a person with normal vision the far point is at
infinity
Nearsightedness A person is nearsighted if their far point is at some finite distance
less than infinity This condition can be corrected with a lens that takes an object at
infinity and produces a virtual image at the personrsquos far point
Farsightedness and Presbyopia A person is farsighted if their near point is too far
away for comfortable near work like reading or knitting This can be corrected by a
lens which takes an object at a normal near point distance of 18-25 cm and produces a
virtual image at the personrsquos natural near point Presbyopia involves a similar
problem which nearly all people experience as they age The ciliary muscle becomes
too weak and the lens becomes too stiff to allow the eye to provide for both near and
far vision The solution for this problem is either reading glasses or bifocal lenses
Reading classes are just weak magnifying glasses mounted on eyeglass frames
Bifocal lenses are split into upper and lower halves The lower half is a lens which
gives the proper correction for near work and the upper half is a different lens for
proper focusing at infinity
Angular Size
When an object is brought closer to the eye it appears to be larger because the image
on the retina is larger The size of this image is directly proportional to the objectrsquos
angular size which is the angle subtended by the object measured from the center of
the lens of the eye In optical instruments which are to be used with the eye the
angular size of the final image is whatrsquos important because it determines how large
the image will appear to the viewer
19
Simple Magnifier
A simple magnifier is a single converging lens or magnifying glass It takes an object
closer to the eye than a normal near point and produces a virtual image at or beyond
this near point The angular magnification is defined to be the ratio of the angular size
when viewed through the lens to the angular size of the object when viewed at the
normal near point (without aid of the lens)
Microscope
This instrument has two lenses (1) The objective is near the object being viewed and
produces a greatly magnified real image (2) The eyepiece is a simple magnifier which
the viewer uses to closely examine the image from (1)
Telescope
This instrument also has two lenses (1) The objective at the front of the telescope
takes light from a distant object and produces a real inverted image (which is rather
small) near its focal point (2) This small real image is then examined by the eyepiece
functioning as a simple magnifier to produce a virtual image with a larger angular
size
20
Serway Chapter 37
Two-Slit Interference
If light is incident on two closely spaced narrow slits a pattern of light and dark
stripes is produced beyond the slits The bright stripes or fringes are caused by
constructive interference of the two waves coming from the slits Constructive
interference occurs whenever two waves arrive at a location in phase with each other
This occurs when the distance x1 from slit 1 to a point P on the screen and the
distance x2 from slit 2 to point P differ by in integral number of wavelengths
1 2 where 0 1 2x x m m
where λ is the wavelength of the light When x1 and x2 are much larger than the slit
spacing d this condition reduces to
sind m
where θ is the angle between the direction of the incident light and the direction of the
light arriving at the screen
Thin Films
When light is partially reflected and partially transmitted by a thin film of transparent
material it is possible to have interference between the wave reflected from the front
of the film and light reflected from the back of the film (The colored reflections from
the thin film of oil on the water in a rain-soaked parking lot are an example of this
effect) It is difficult to write down formulas that will work in all cases so we will
just review the important principles here
1 If the two reflected waves are in phase with each other the film has enhanced
reflection (constructive interference) but if the two reflected waves are out of phase
with each other reflection is diminished (destructive interference) Phase shifts occur
due to reflection and due to the extra path length through the film of the wave
reflected from the back of the film
2 The phase change due to reflection is determined by the difference in index of
refraction between the two media involved in the reflection If the wave is incident
21
from a medium with a low index of refraction into a medium with a high index of
refraction a phase change of 180deg occurs and the reflected wave is inverted If
incident from high to low no phase shift occurs and the reflected wave is non-
inverted
3 The extra path length through the film of the wave reflected from the back of the
film is equal to 2t where t is the thickness of the film (The incident light is assumed
to be normal to the surface of the film) The number of wavelengths contained in the
extra path length is equal to 2tλn where λn = λn is the wavelength of the light in the
film
4 Rules for reflection from thin films
If one ray is inverted and the other is not then we have
12 constructive
2 nt m
2 destructivent m
If either both rays are inverted or both are non-inverted we have
2 constructivent m
12 destructive
2 nt m
22
Serway Chapter 38
Diffraction Grating
A diffraction grating is simply a fancy version of two-slit interference with the two
slits replaced by thousands of slits Just as in the two-slit case the bright fringes
occur at angles given by
sin md
but in this case d the distance between neighboring slits is made to be very small
The effect of having many slits instead of two is to make each bright fringe highly
localized with wide dark regions between neighboring maxima
Single Slit Diffraction
When light passes through an opening in an opaque screen an interference pattern is
produced beyond the opening To understand why we may replace the single opening
by many small coherent sources of light These many sources interfere with each
other producing a pattern known as a diffraction pattern (Note that many authors do
not distinguish between interference and diffraction treating them as interchangeable
terms) If the opening is a slit of width a then the diffraction pattern far from the slit
will have a bright central maximum with a succession of minima and weaker maxima
on either side The angle between the incident direction and the minima is given by
sin ma
where m = plusmn1 plusmn2
If the opening is circular with diameter D the angle between the incident direction
and the first minimum is given by
sin 122D
Optical Resolution and Rayleighrsquos Criterion
Two point sources can just be resolved (distinguished from each other) if the peak of
the diffraction image of the first source overlies the first minimum of the diffraction
image of the second source For circular holes of the kind usually encountered in
23
optical devices this condition is approximately satisfied when the angular separation
between the two sources as viewed from the optical instrument is greater than or equal
to the critical angle
min 122D
where λ is the wavelength of the light and where D is the diameter of the aperture in
the instrument
Polarization
We say that an electromagnetic wave is polarized if its electric field vector doesnrsquot
change direction in a random fashion The simplest kind of polarization is linear
polarization in which the electric field vector oscillates back and forth along the
same axis in space Polarized light can be produced from normal unpolarized light by
selective absorption (as in Polaroid sunglasses) by passing light through crystals that
have different indices of refraction for different polarizations (double refraction) by
scattering (the blue sky is polarized) and by reflection (glare) Polarization by
reflection occurs when light reflects from a shiny insulating (non-metallic) surface
The amount of polarization is greatest for reflection at Brewsterrsquos angle
2
1
tan P
n
n
where θP is the incidence angle of light from medium 1 onto medium 2 and where n1
and n2 are the indices of refraction for the media (In our everyday experience n1 = 1
since the light comes in through the air and n2 is the index of refraction of the shiny
insulating material producing the glare eg water glass plastic paint etc)
Malusrsquos Law
The intensity of transmitted polarized light through a perfect polarizer is related to the
incident intensity of polarized light by Malusrsquos law
2cosoI I
where Io is the intensity of the incident light and where θ is the angle between the
electric field vector in the incident wave and the transmission axis the polarizer
24
Serway Chapter 39
Principles of Relativity
All of the weirdness of relativity flows from two simple principles
(1) The laws of physics must be the same in all inertial (non-accelerating) reference
frames
(2) The speed of light in vacuum has the same value in all inertial reference frames
Note that (1) seems reasonable but (2) is very odd It says that if two space ships are
approaching each other at nearly the speed of light and a laser pulse is shot from ship
1 toward ship 2 then when the people on ship 2 measure the speed of the pulse as it
goes by it is moving at 3 times 108 ms the same as if the ships were stationary
Simultaneity
If observer 1 sees two events in her own frame as simultaneous at two different
locations a moving observer 2 will see these two events happening at different times
Relativistic gamma
The factor γ (gamma) appears regularly in the formulas of relativity
2
2
1
1 vc
where v is the relative speed between two inertial frames
Time Dilation
If observer 1 sees two events at the same location in space separated by time Δtp in his
own frame then observer 2 moving at speed v relative to observer 1 will see these
two events separated by a longer time Δt
pt t
Moving clocks run slow
Length Contraction
If observer 1 measures the length of an object along the x-direction at some instant of
time in her frame to be Lp then observer 2 moving in the x-direction at speed v will
measure the length L of the object to be shorter
pLL
25
Moving meter sticks are short
Velocity Addition
If a particle is moving at velocity vac relative to frame c and if frame c is moving at
speed vcb with respect to frame b (with both velocities directed along the same line)
what would be the velocity of the particle in frame b In classical physics the answer is
ab ac cbv v v
but if any of these velocities are an appreciable fraction of the speed of light the
answer changes
21ac cb
abac cb
v Vv
v v c
In using this equation make sure that vac has the same sign in the numerator and in the
denominator and that vbc also has the same sign in both places Note that this equation
is different (and in our opinion easier to understand) than the equations in the text
Momentum and Energy
The momentum and total energy of a particle moving at velocity v (its speed is v v )
are given by
22
2 2
22
1 1 vvc c
m mcm E mc
vp v
Rest Energy
When a particle is at rest (v = 0 and γ = 1) then E = mc2 is called the rest energy It
includes the potential energy of the particle so that if we change the potential energy
of a particle we change its rest mass
Kinetic Energy The kinetic energy is the difference between the total energy and the
rest energy
2 2K mc mc
For v ltlt c this reduces to K = mv22
General Relativity
Einsteinrsquos principle of equivalence in the theory of general relativity is In the vicinity
of a point a gravitational field is equivalent to an accelerated frame of reference in
26
the absence of gravitational effects For example there is no difference whatever in
the physics experienced by an astronaut on earth and one who is in interstellar space
far away from any stars or planets but who has just turned on her spaceshiprsquos engines
and is accelerating at 98 ms2
27
Serway Chapter 40
Photons
We now know that even though light behaves like a wave it is better described as
consisting of small packets of energy called photons The energy of a photon is related
to its frequency by
E hf
where h is Planckrsquos constant
346626 10 J sh
Photons also carry momentum given by
hf hp
c
Since the amount of energy in each photon is fixed the difference between dim light
and intense light (of the same wavelength) is that dim light consists of fewer photons
than intense light
Photoelectric Effect
In the photoelectric effect light shining onto a metal surface gives the electrons in the
metal enough energy to escape and be detected It requires a certain amount of energy
(called the work function typically = 3 ndash 5 eV) just to kick the electron out of the
metal so the light must deliver at least this much energy to an electron to produce the
effect It is observed that red light no matter how intense never produces electrons
But ultraviolet light even if quite dim will eject electrons from the metal Since red
light consists of 2 eV photons and ultraviolet light has photons with energies around
4-6 eV the photon idea explains the behavior of the photoelectric effect The
maximum energy that an ejected electron can have is
maxK hf
where f is the photon frequency The minimum frequency that light can have and cause
electrons to be ejected is called the cutoff energy and is found by setting Kmax = 0
cf h
28
Compton Effect
When high-frequency light interacts with free electrons the scattered light does not
have the same wavelength as the incident light contrary to what classical
electromagnetism would predict But the photon picture in which photons have
momentum and energy allows us to treat scattering as a collision between the photon
and the electron In this collision the electron and the scattered photon both have a
different momentum and energy than they did before And since p = hλ for a photon
if the momentum is different the wavelength will also be different
after before 1 cose
h
m c
where the angle θ is the angle between the incoming photon direction and the
direction of the scattered photon
Particles are Waves
Since photons behave like particles sometimes it is not surprising that elementary
particles can behave like waves sometimes The wavelength of a particle with
momentum p is given by
h
p
Wave-particle Duality
Both photons and elementary particles have a dual nature sometimes they behave like
particles and sometimes they behave like waves It is hard for us to comprehend the
nature of such an object by making mental pictures but experiments definitively show
that this is the case Since we have no direct experience with photons and elementary
particles (because their energies are so much smaller than the energies of the everyday
objects we encounter) it is perhaps not surprising that we have a hard time forming a
mental picture of how they behave
Electron Interference
Since an electron is both a particle and a wave just like a photon it should be able to
produce an interference pattern This is observed If an electron beam is shot at two
closely-spaced slits and if the electrons that pass through the slits are detected
downstream it is found that there are some locations where electrons are never detected
29
and others where lots of electrons are detected The pattern is exactly the same as the
one observed for light waves provided that we use the electron wavelength λ = hp in
place of the wavelength of light This pattern is observed even though each electron is
detected as a single dot on the screen Only after many such dots are collected does the
pattern emerge And if we try to understand how this effect could possibly work by
looking closely at each slit to see which one the electron came through the pattern
disappears the act of measurement destroys the interference This means that each
single electron somehow comes through both slits (as a wave would)
Uncertainty Principle
In classical physics we always imagine that the positions and momenta of moving
particles have definite values It might be hard to measure them but surely at each
instant of time a particle should be precisely located at some point in space and have a
similarly precise momentum This turns out not to be true Instead both position and
momentum are required to be uncertain with their uncertainties Δx and Δp satisfying
the Heisenberg uncertainty relation
2x p
So if the particle were known to be precisely at some particular location (so that Δx =
0) we couldnrsquot know anything about its momentum (Δp = infin) and if its momentum
were exactly known we couldnrsquot know its position
There is a similar relation involving the particlersquos energy E and the time interval Δt
over which this energy is measured
2E t
This means that energy is not actually conserved if we are considering very small time
intervals and this brief non-conservation of energy has been observed
30
Serway Chapter 41
Wave Function ψ and Probability
Quantum mechanics does not predict exactly what an electron or a photon will do Instead
it specifies the wave function or probability amplitude ψ of an electron or a photon
This wave function is a complex-valued function of space and time whose squared
magnitude is the probability density P for finding a particle at a particular place in
space at a certain time
2P
where is the complex conjugate of ψ
For example the wave function of an electron with perfectly specified momentum p
would have a wavelength given by
h
p
and its probability amplitude would be proportional to
2i x ipxe e
The corresponding probability density would then be
21ipx ip ipxP e e e
which means that the electron is equally probable to be anywhere along the x axis
This is in accord with the uncertainty principle since we specified the momentum
precisely we canrsquot have any idea about the position of the electron
As another example you have probably seen ldquofuzzy ballrdquo drawings of electron
orbitals in a chemistry book These fuzzy balls are meant to indicate the distribution
of the probability density 2
P in the orbital
Particle in a Box
A simple example in which we can calculate the wave function is the case of a particle of
mass m constrained to be inside a 1-dimensional box between x = 0 and x = L In this
case the general wave function is a linear superposition of wave functions ψn of the form
sin niE tn
n xx t A e
L
31
where A is a positive constant where n = 1 2 3 and where the energy associated
with each of the quantum states ψn is given by
22
28n
hE n
mL
This wave function is zero at x = 0 and x = L which means that the particle will never
be found at the walls of the box The wave function has maximum values in the
interior and at these places the particle is most likely to be found and it also has
places where it is zero and at these places the particle will also never be found as
expressed by the formula
2sinn x
PL
The particle in the box is interfering with itself producing a probability interference
pattern across the box just like the interference patterns we studied with light and
sound
32
Serway Chapter 42
Atoms
Many people picture an atom as a miniature solar system where electrons orbit around
a massive nucleus at the center This picture is misleading because in atoms the wave
nature of electrons dominates The electrons form 3-dimensional standing waves
(called orbitals) centered at the nucleus And if you ask what is it thatrsquos waving the
answer is ψ the probability amplitude (see Chapter 41)
Standing waves on a string can occur only for certain frequencies (the fundamental
and higher harmonics) Similarly atomic orbitals occur only for certain energies For
the hydrogen atom the energies of the orbitals have the a particularly simple form
2
1136 eVnE
n
where n = 1 2 3 is called the principal quantum number For other atoms the
determination of the orbital energies requires numerical calculation by computers
Atomic Spectra
If an electron is somehow given extra extra energy (we say that it is excited) so that it
occupies a higher orbital it will eventually ldquofallrdquo back down to a lower orbital Each
time an electron falls to a lower orbital it loses the difference in energy between the
two orbitals in the form of a photon Since the orbital energies are discrete so are the
energy differences and so are the wavelengths of the emitted light The entire set of
these discrete wavelengths is called the atomic spectrum and it is unique to each type
of atom For hydrogen the spectrum can be simply written as
H 2 2
1 1 1
f i
Rn n
where RH = 10973732 times 107 m-1 For other atoms the spectrum cannot be expressed as
a formula
Orbital Angular Momentum Electrons in atoms also have quantized values of
angular momentum The orbital quantum number ℓ specifies the value of this
quantized angular momentum through the formula
1L
33
If we want to know the value of the angular momentum along some direction in space
say the z direction the answer is not L but rather
zL m
where mℓ is another quantum number which runs from
1 1m
This quantum number is important when an atom sits in a magnetic field
Spin Angular Momentum It has been found experimentally that electrons and other
charged particles also carry internal angular momentum which we call spin
Electrons have an intrinsic spin angular momentum s along a specified axis that is
extremely quantized it can only take on 2 possible values
1
2zs s
We thus say that electrons have spin s = 12 or that electrons are rdquospin one-half parti-
cles Since s is an angular momentum it obeys the same rule as that for orbital angular
momentum namely that if its value along some axis is s then its total magnitude is
given by
31
2S s s
I know this seems weird but quantum mechanics is weird The only excuse for this
bizarre way of looking at the world is that it predicts what happens in experiments
Exclusion Principle The answer to the question of how many electrons (or any other
spin one-half particle) can be in one particular quantum state was discovered by
Wolfgang Pauli and is called the exclusion principle
ldquoNo two electrons can ever be in the same quantum state therefore no two electrons
in the same atom can have the same set of quantum numbersrdquo
This is the reason that we have atoms with different properties instead of every atom
simple having all of its electrons in the ground state All of the variety we see around
us in the world is the result of chemical differences and these differences would not
exist unless electrons obeyed this important principle The entire structure of the
periodic table (see pages 1377-1379 in Serway) is an expression of this principle
34
Serway Chapter 44
Nuclear Properties
The nucleus is composed of protons with charge +e and mass mp = 1007226 u and of
neutrons with zero charge and mass mn = 1008665 u where u is one atomic mass unit
-271 u=1660540 10 kg
The atomic number Z counts the number of protons in a nucleus while the neutron
number N counts the number of neutrons The mass number A is the sum of the two
A N Z
Protons and nuetrons have about the same mass and some times called baryons (heavy
ones) or nucleons (partcles in the nucleus) A is sometimes called the baryon
number The chemical elements are distinguished by Z ie hydrogen has Z = 1 iron
has Z = 26 uranium has Z = 92 etc But for a given Z there might be several nuclei
with different numbers of neutrons These nuclear siblings are called isotopes
These numbers are used to label nuclei according to the pattern
5626 ie FeA
Z X
denotes the isotope of iron with N = 56 ndash 26 = 30 neutrons
The nucleus is roughly spherical with a radius given approximately by
1 3 150 0where 12 10 mr r A r
Nuclear Stability
Because the positively charged protons electrically repel each other with an enormous
force at distances as small as 10-15 m there must be some really strong force that that
overcomes electrical repulsion to hold protons and neutrons together This force is
called with some lack of imagination the strong force It is a very short-range force
(it only acts over a distance of about 2 x 10-15 m) and attracts protons to protons
neutrons to neutrons and neutrons to protons But in spite of this strong nuclear force
the coulomb repulsion of the protons is still present so anything that might keep the
protons from being right next to each other would help keep the nucleus from
35
exploding This role is played by the neutrons and for nuclei with Z le 20 the stable
nuclei roughly have N = Z
For Z gt 20 the coulomb repulsion force is more powerful and more neutrons are
needed to dilute it up to about N = 15Z around Z = 80 For Z gt 83 no amount of
neutrons can help and these nuclei are unstable (radioactive) There is an ldquoisland of
(relative) stabilityrdquo around Z=90-92 (thorium and uranium respectively) These
elements have one or two isotopes with half-lifes of billions of years so there are
substantial amounts of such elements on Earth This fact makes it possible to have
practical fission devices
Radioactive Decay
There are three types of radioactive decay
Alpha decay the nucleus kicks out a helium nucleus (N = 2 Z = 2 A = 4) So if the
original nucleus is called X and the new nucleus is called Y then the decay would look
like this
4 42 2X Y+ HeA A
Z Z
Beta decay the nucleus either kicks out an electron (endash) or its positively-charged
antimatter twin the positron (e+) plus either an electron neutrino v or an electron
anti-neutrino v
1X Y+e electron decayA AZ Z v
1X Y+e positron decayA AZ Z v
A neutrino is a particle with no charge hardly any mass (much less than the electron
mass) and interacts so weakly with matter that most neutrinos upon encountering the
planet earth just pass right through it as if it werenrsquot there
Gamma decay the nucleons in the nucleus X are in an excited energy state X
(perhaps as a result of having undergone alpha or beta decay) and they drop down to a
lower energy state shedding the energy as a high frequency photon
X XA AZ Z
36
This process is exactly analogous to the way that the electrons in atoms emit photons
Decay Rate and Half Life
There is no way to predict exactly when an unstable or excited nucleus will decay but
there is an average rate at which this decay occurs called the decay constant λ The
meaning of this constant is that if there are a large number N of nuclei in a sample
then the number of decays per second that will be observed (called the decay rate R)
is R = λN In mathematical language
dMR N
dt
This simple differential equation has for its solution
0tN t N e
where N0 is the number of nuclei in the sample at time t = 0
The half-life is the time it takes for half of the nuclei in the sample to decay and is
related to the decay constant by
1 2
ln 2 0693T
Disintegration Energy
When a nucleus decays it is making a transition to an overall state of lower energy
which means according to Einsteinrsquos famous formula E = mc2 that the sum of the
masses after the decay must be less than the mass before with the lost mass appearing
as kinetic energy among the decay products For example in alpha decay this kinetic
energy called the disintegration energy Q is given by
2X YQ M M M c
37
Serway Chapter 45
Nuclear energy
There are two ways to extract energy from the nucleus fission and fusion For nuclei
with Z greater than 26 breaking the nucleus apart into pieces leads to a lower
2mc energy than the original nucleus so energy can be extracted by fission For
nuclei with Z less than 26 a lower energy is achieved by combining nuclei so energy
can be extracted by fusion
Fission
Since neutrons have no charge they are not repelled from nuclei as protons are for
this reason their behavior is the key to understanding how fission works
Fast neutrons mostly bounce off other nuclei with each collision slowing the neutron
down This elastic energy loss is most effective if the other nuclei have low mass (like
hydrogen) and these materials are called moderators because of their ability to slow
down fast neutrons
The reason that slowing neutrons is important is that slow neutrons are much more
likely to be absorbed by a nucleus which then leads to nuclear reactions of various
kinds For a few very large nuclei like uranium-235 and some plutonium isotopes
absorption of a slow neutron causes the nucleus to split into two large fragments plus
2 or 3 fast neutrons (energy is released in the process too) Thus one slow neutron
can produce energy plus 2 or 3 more neutrons which if moderated from fast to slow
can split 2 or 3 more nuclei leading to a runaway chain reaction If the reaction is
allowed to proceed unchecked it produces a large explosion If a neutron absorbing
material is added to the mix (like the cadmium in reactor control rods) it is possible to
keep the reaction under control and to extract the released energy as heat to drive
steam turbines and produce electricity
Fusion
Fusion involves mashing two nuclei together and since they are both charged and repel
each other this reaction is much harder to make go The nuclei must have enough energy to
overcome the coulomb repulsion which is why this reaction requires a high temperature
(hundreds of millions of degrees K) like that in the sun (which is burning hydrogen to
helium via fusion) or like that in the center of the fission explosion that is used to detonate
38
a hydrogen bomb
This reaction is of interest for power production in spite of this difficult temperature
requirement because of the abundance of fusion fuel on the planet There are about
012 g of deuterium ( 21H ) in every gallon of water on earth and it only costs about 4
cents to extract it The fusion energy available from this minuscule amount of
deuterium would run a 1000-Megawatt power plant for 10 seconds Water is so
abundant on earth that if fusion were to work we would have an essentially
inexhaustible source of energy
So why donrsquot we have fusion power plants Well the fuel is cheap but the match is
incredibly expensive The only way we know to control this difficult high-temperature
reaction is with large and expensive pieces of equipment involving either large
magnetic fields and complex high-power electromagnetic antennas or with gigantic
(football-fieldsized) laser facilities involving more than a hundred of the highest-
energy lasers ever built Power plants based on these current methods for controlling
fusion are unattractive to the fiscally-minded people who run the electric power
industry Hopefully better designs will be discovered as experiments continue
39
c actinium 68 Er erbium 101 Md mendelevium 104 Rf rutherfordium l aluminum 63 Eu europium 80 Hg mercury 62 Sm samarium m americium 100 Fm fermium 42 Mo molybdenum 21 Sc scandium b antimony 9 F fluorine 60 Nd neodymium 106 Sg seaborgium r argon 87 Fr francium 10 Ne neon 34 Se selenium s arsenic 64 Gd gadolinium 93 Np neptunium 14 Si silicon t astatine 31 Ga gallium 28 Ni nickel 47 Ag silver a barium 32 Ge germanium 41 Nb niobium 11 Na sodium k berkelium 79 Au gold 7 N nitrogen 38 Sr strontium e beryllium 72 Hf hafnium 102 No nobelium 16 S sulfur i bismuth 108 Hs hassium 76 Os osmium 73 Ta tantalum h bohrium 2 He helium 8 O oxygen 43 Tc technetium
boron 67 Ho holmium 46 Pd palladium 52 Te tellurium r bromine 1 H hydrogen 15 P phosphorus 65 Tb terbium d cadmium 49 In indium 78 Pt platinum 81 Tl thallium a calcium 53 I iodine 94 Pu plutonium 90 Th thorium f californium 77 Ir iridium 84 Po polonium 69 Tm thulium
carbon 26 Fe iron 19 K potassium 50 Sn tin e cerium 36 Kr krypton 59 Pr praseodymium 22 Ti titanium s cesium 57 La lanthanum 61 Pm promethium 74 W tungsten l chlorine 103 Lr lawrencium 91 Pa protactinium 92 U uranium r chromium 82 Pb lead 88 Ra radium 23 V vanadium o cobalt 3 Li lithium 86 Rn radon 54 Xe xenon u copper 71 Lu lutetium 75 Re rhenium 70 Yb ytterbium m curium 12 Mg magnesium 45 Rh rhodium 39 Y yttrium b dubnium 25 Mn manganese 37 Rb rubidium 30 Zn zinc y dysprosium 109 Mt meitnerium 44 Ru ruthenium 40 Zr zirconium s einsteinium
40
UNITS The SI units are given in parentheses Other commonly used units are given in terms of the SI units Symbols conform with the recommendations of the American National Standards Institute (ANSI) and the American Institute of Physics (AIP) Commonly used multiples of SI units
prefix symbol factor tera T 1012 giga G 109
mega M 106 kilo k 103
centi c 10-2 milli m 10-3
micro μ 10-6 nano n 10-9 pico p 10-12
femto f 10-15
length meter (m) angstrom 1 Ǻ = 10-10 m inch 1 in = 254 x 10-2 m foot 1 ft = 03048 m mile 1 mi = 1609 m
mass kilogram (kg) atomic mass unit 1 u = 1661 x 10-27 kgslug 1 slug = 1459 kg
force newton (N = kg bull ms2) dyne 1 dyn = 10-5 N pound 1 lb = 4448 N
pressure Pascal (Pa = kgm bull s2) atmosphere 1 atm = 1013 x 105 Pa poundssquare inch 1 psi = 6895 Pa cm of mercury 1 cm Hg = 1333 Pa bar 1 bar = 1000 x 105 Pa torr 1 torr = 1333 Pa
time second (s) minute 1 min = 60 s hour 1 h =3600 s
frequency hertz (Hz = s-1) radianssecond 1 rads = 12π Hz
energy joule (J = kg bull m2s2) erg 1 erg = 10-7 J electron volt 1 eV = 1602 x 10-19 J calorie 1 cal = 4187 J kilowatt-hour 1 kWh = 36 x 106 J British thermal unit 1 Btu = 1055 J
power watt (W = kg bull m2s3) horsepower 1 hp = 7457 W
charge coulomb (C = A bull s)
electric potential volt (V = kg bull m2 s3 bull A)
current ampere (A)
resistance ohm (Ω = kg bull m2s3 bull A2)
capacitance farad (F = s4 bull A2kg bull m2)
magnetic field tesla (T = kgs2 bull A) gauss 1 G = 10-4 T
magnetic flux weber (Wb = kg m2s2 bull A) maxwell 1 Mx = 10-8 Wb
magnetic inductance henry (H = kg bull m2s2 bullA2)
temperature kelvin (K) degrees Celsius 0degC = 27315 K
angle radian (rad) degree 1 deg = π180 rad revolution 1 rev = 2π rad
41
Some Physical Constantsa Quantity Symbol Valueb Atomic mass unit μ 1660 538 73 (13) x 10-27 kg
931494 013 (37) MeVc2 Avogadros number NA 6022 141 99 (47) x 1023 particlesmol Avogadrorsquos number 6022 x 1023 mol Bohr magneton
2B e
em 9274 008 99 (37) x 10-24 JT
Bohr magneton μB 927 x 10-24 JT Bohr radius
0
2
2e em e k
a 5291 772 083 (19) x 10-11 m
Boltzmanns constant B A
RNk 1380 650 3 (24) x 10-23 JK
Boltzmannrsquos constant kB 1380 x 10-23 JK Compton wavelength
C eh
m c 2426 310 215 (18) x 10-12 m
Deuteron mass md 3343 583 09 (26) x 10-27 kg 2013 553 212 71 (35) u
electron charge e 1602 x 10-19 C Electron mass me 9109 381 88 (72) x 10-31 kg
5485 799 110 (12) x 10-4 u 0510 998 902 (21) MeVc2
electron mass 911 x 10-31 kg Electron volt eV 1602 176 462 (63) x 10-19 J Elementary charge e 1602 176 462 (63) x 10-19 C Gas constant R 8314 472 (15) JKmol Gravitational constant G 6673 (10) x 10-11 Nm2kg2 Neutron mass mn 1674 927 16 (13) x 10-27 kg
1008 664 915 78 (55) u 939565 330 (38) MeVc2
neutron mass 1675 x 10-27 kg Nuclear magneton
2n p
em 5050 783 17 (20) x 10-27 JT
permeability constant μ0 1257 x 10-6 Hm permittivity constant ε0 8854 x 10-12 Fm Plancks constant h
2h
6626 068 76 (52) x 10-34 Js 1054 571 596 (82) x 10-34 Js
Planckrsquos constant h ħ
6626 x 10-34 Js 1055 x 10-34 Js
Proton mass mp 1672621 58 (13) x 10-27 kg 1007 276 466 88 (13) u 938271 998 (38) MeVc2
proton mass 1673 x 10-27 kg Rydberg constant RH 1097 373 156 854 9 (83) x 107 m-1 Speed of light in vacuum c 2997 92458 x 108 ms (exact) speed of light c 300 x 108 ms a
These constants are the values recommended in 1998 by CODATA based on a least-squares adjustment of data from different measurements For a more complete list see P J Mohr and B N Taylor Rev Mod Phys 72351 2000 b The numbers in parentheses for the values above represent the uncertainties of the last two digits
NOTE The ones in red are the ones from Appendix 13 The ones in yellow are the ones that are from the Serway book table but that were already on the Appendix 13 table
42
INDEX
Absolute zero 9
Adiabatic 17
Adiabatic exponent 18
Adiabatic process 17
Alpha decay 46
Amplitude 4
Angle of reflection 25
Angular frequency 3
Angular magnification 30
Angular momentum atomic 43
Angular size 29
Archimedes Principle 1
Atomic spectra 43
Atoms 43
Avogadrorsquos number 10
Beats 8
Bernoullirsquos Equation 2
Beta decay 46
Bifocals 29
Brewsterrsquos angle 34
British Thermal Unit Btu 11
Buoyancy 2
calorie 11
Calorie 11
Camera 28
Carnot cycle 21
Carnot efficiency 22
Celsius scale 9
Ciliary muscle 29
Coefficient of performance 21
Compton effect 39
Constructive interference 7
Continuity equation of 2
Contraction length 35
Convection 14
Curved mirrors 27
Decay constant 46
Decay rate radioactive 46
Decibel scale 5
Degrees of freedom 16
Density 1
Destructive interference 7
Diffraction single slit 33
Diffraction grating 33
Dilation time 35
Disintegration energy 47
Dispersion of light 26
Doppler effect 6
Double slit interference 31
Efficiency 20
Electron interference 39
Emissivity 15
Energy nuclear decay 47
Energy relativistic 36
43
Engineering work 20
Entropy 22
Entropy ideal gas 24
Equipartition of energy 18
Equivalence principle 36
Exclusion principle 44
Expansion thermal 9
Expansion coefficient linear 9
Eye 29
Fahrenheit scale 9
Far point 29
Farsightedness 29
First Law of Thermodynamics 12
Fission 48
Flux volume 2
Free expansion 23
Fringes 31
Fusion 48
Fusion heat of 11
Gamma relativistic 35
Gamma decay 46
General relativity 36
Half-life 46
Heat 11
Heat capacity 11
Heat conduction 13
Heat engine 20
Heat of fusion 11
Heat of vaporization 11
Heat pump 20
Heats of transformation 11
Hydrostatics 1
Ideal Gas Law 10
Images real and virtual 27
Intensity sound 5
Interference 7
Interference two-slit 31
Internal energy 12
Internal energy degrees of freedom 16
Irreversible process 19 23
Isotopes 45
Joule 11
Kelvin scale 9
Kinetic energy relativity 36
Kinetic theory 16
Latent heat 11
Length contraction 35
Linear expansion coefficient 9
Linear polarization 34
Linear superposition 7
Longitudinal wave 3
Loudness 5
Magnification lateral 27
Magnifying glass 30
Malusrsquos law 34
Microscope 30
Momentum relativistic 36
Muscle ciliary 29
44
Musical instruments 8
Musical scale 8
Near point 29
Nearsightedness 29
Nonlinear 7
Nuclear energy 48
Nuclear properties 45
Nuclear stability 45
Octave 8
Optical resolution 33
Orbital quantum number ℓ 43
Orbitals 43
Particle in a Box 41
Particles are waves 39
Pascalrsquos Principle 1
Period 3
Photoelectric effect 38
Photons 38
Pitch 8
Polarization 34
Power sound 5
Presbyopia 29
Pressure 1
Principal quantum number n 43
Principle of equivalence 36
Principle of linear superposition 7
Probability amplitude ψ 41
Processes thermodynamic 13
R-value 14
Radiation thermal 14
Radioactive decay 46
Radioactive decay rate 46
Ray tracing 27
Rayleighrsquos criterion 33
Reading glasses 29
Real image 27
Refraction 25
Refrigerator 20
Relativistic gamma 35
Relativity principles 35
Resolved for light sources 33
Rest energy 36
Reversible process 19 22
Rope wave speed 4
Second Law of Thermodynamics 19
Shock waves 6
Simple magnifier 30
Simultaneity 35
Single slit diffraction 33
Snellrsquos law 25
Sound speed 5
Specific heat 11
Spin Angular momentum 44
Standing waves 7
Stefanrsquos law 14
Strong force 45
Telescope 30
Temperature 9
45
Temperature Scales 9
Thermal conductivity 14
Thermal energy 12
Thermal expansion 9
Thin film interference 31
Thin lenses 28
Time dilation 35
Tone musical 8
Total internal reflection 25
Transverse wave 3
Traveling Waves 3
Two-slit interference 31
Uncertainty principle 40
Vaporization heat of 11
Velocity addition relativity 36
Virtual image 27
Volume flux 2
Wave function 41
Wave Function ψ and Probability 41
Wave speed 3
Wave-particle duality 39
Wavelength 3
Wavenumber 3
Wien Displacement Law 16
Work 11
9
Heat Engines
Because of the vexing difference in sign between work done on and system and work done
by a system we will invent a new work variable Weng Heat engines do work and so the net
W for these engines is negative But in engineering applications hidden minus signs are
regarded as evil so for heat engines we donrsquot talk about W instead we talk about its
magnitude engW W So for heat engines the first law is
int engE Q W
But for heat pumps and refrigerators work is done on the system so we use the usual work
W when we talk about these systems
A heat engine is a machine that absorbs heat converts part of it to work and exhausts the
rest The heat must be absorbed at high temperature and exhausted at low temperature If the
absorbed heat is Qh the exhausted heat is Qc and the work done by the engine is Weng then
eng h cW Q Q
and the efficiency of the engine is defined to be
eng
h
We
Q
A perfect engine would convert the heat hQ completely into work Weng giving an effi-
ciency of e = 1 Energy conservation alone allows a perfect engine but the second law
requires e lt 1
Refrigerators and Heat Pumps
A refrigerator is a machine that absorbs heat at low temperature and exhausts it at high
temperature the ldquobackwardsrdquo heat transfer being driven by the work done on the machine
by some source of power A heat pump is a machine that either works like a refrigerator
keeping a place cold by transferring heat from this cold place to a higher temperature
environment (cooling mode like an air conditioner) or it functions as a heater
transferring heat into a warm place from a cooler one (heating mode like a window unit
that heats a house by extracting thermal energy from the cold outdoors) The coefficient
of performance of a refrigerator or of a heat pump in cooling mode is defined to be
COP cooling modecQ
W
10
For a heat pump in heating mode the coefficient of performance is
COP heating modehQ
W
Note that we donrsquot have to use the engineering work here because in these systems
work is done on the system and W is naturally positive
A perfect refrigerator would take in heat Qc and exhaust the same amount of heat without
doing any work giving an infinite coefficient of performance Energy conservation alone
allows a perfect refrigerator but the second law requires COP lt infin A perfect heat pump
in heating mode would transfer Qh into the house without doing any work and so would
also have an infinite coefficient of performance The second law forbids this too A good
coefficient of performance for a real device would be around 5 or 6
Carnot Cycle
The most efficient of all possible engines is one that uses the Carnot cycle This cycle
employs an ideal gas has no friction and operates very slowly so that the gas can be in
thermal equilibrium at all parts of the cycle This means of course that it canrsquot
possibly be built and even if it could be built it would not run fast enough to be useful
Nevertheless this cycle is very important because it gives an upper bound on the
efficiency of real engines There cannot possibly be an engine that is more efficient
than one based on the Carnot cycle This cycle consists of the following four steps
1 The ideal gas absorbs heat Qh at constant temperature Th while the gas increases its
volume The reason that heat is absorbed is that expansion tends to cool the gas but
thermal contact with the environment at Th keeps the temperature high by heat
conduction into the ideal gas
2 The ideal gas further increases its volume by an adiabatic expansion This expansion
causes the gas to cool so at the end of this part of the cycle the gas is at temperature Tc
3 The gas exhausts heat Qc at constant temperature Tc while the gas decreases in
volume The reason that heat is exhausted is that compression tends to heat the gas
but thermal contact with the environment at Tc keeps the temperature low by heat
conduction out of the ideal gas
4 The gas is adiabatically compressed back to its original volume (the volume it started
with in step 1) This compression heats the gas from Tc up to Th
11
The efficiency of a Carnot engine is given by the very simple formula
1 cC
h
Te
T
where the temperatures must in be Kelvin No real engine can be more efficient than
this
The coefficient of performance of a Carnot refrigerator or heat pump in cooling mode
is given by
COP cooling mode cC
h c
T
T T
and no real refrigerator can have a coefficient of performance greater than this
The coefficient of performance of a Carnot heat pump in heating mode is
COP heating mode hC
h c
T
T T
So why donrsquot we just use these wonderful Carnot engines and have perfect efficiency
To make the reversible steps in the cycle really reversible they would have to occur
infinitely slowly So the price you pay for making a perfect engine is that it takes
forever to get it to do any work
Entropy
The entropy of a system is defined in terms of its molecular makeup and measures
roughly the disorder of the system If the system is packed into a very small volume
then it is quite ordered and the entropy will be low If it occupies a large volume the
entropy is high (To see what this has to do with disorder note that socks in a drawer
occupy a small volume while socks on the bed in the corner by the door and
hanging from the chandelier occupy a large volume) If the system is very cold then
the molecules hardly move and may even reach out to each other and form a crystal
This is a highly ordered state and therefore has low entropy If the system is very hot
with rapidly speeding molecules crashing into the container walls and bouncing off
each other things are disordered and the entropy is high
It is possible to calculate the entropy of a system in terms of its macroscopic thermody-
namic properties ie pressure volume temperature number of moles etc The key to this
calculation is the concept of a reversible process A reversible process is one that is
12
carried out without friction and so slowly that the process can be reversed at any stage by
making an infinitesimal change in the environment of the system The slow expansion of the
gas at Th in the Carnot cycle with tiny amounts of heat being transferred to the environment
is reversible If we turned around and began slowly to compress the gas would just slowly
exhaust heat to the environment in the exact reverse way that it absorbed it during expansion
Most processes however are irreversible For example if a gas-filled box were suddenly
increased in size so that the particles were free to wander into the void created by the sudden
expansion then the gas would eventually fill the new volume uniformly at the same
temperature as before the expansion (The temperature is unchanged in this imaginary
process because the kinetic energy of the molecules would be unaffected by such an
instantaneous expansion of the container walls) This imaginary but highly thought-
stimulating process is called a free expansion and it is impossible to reverse it During the
expansion we didnrsquot push on any of the molecules so reversing this process would mean
making them go back into their original volume without pushing on them they simply will
not cooperate to this extent Another way to see that just pushing them back where they came
from does not reverse the free expansion is to think about what would happen if we just
compressed either adiabatically or isothermally An adiabatic compression back to the
original volume would heat the gas above its original temperature and an isothermal
compression would require that heat be exhausted to the environment But the free expansion
involved neither temperature changes nor heat exchanges so neither of these two processes
is the reverse of the free expansion It is simply impossible to reverse this rapid expansion in
a way that takes us back to the initial state
It is possible to calculate the change of entropy for both reversible and irreversible processes
Letrsquos consider a reversible process first In a reversible process the entropy change is given
by the formula
dQS
T
where dQ is the amount of heat added to the system during a small step of the process
The total energy change during the process may then simply be calculated by integration
f f
f i i i
dQS S S dS
T
13
(Just as in the case of energy we are mainly interested in differences rather than in
absolute magnitudes)
This integration method does not work for irreversible processes For instance in the
rapid free expansion discussed above no heat is added to the system but its disorder
obviously goes up We need to find some other way to calculate the entropy The key
is the fact that the entropy of a system depends only on its current state and not at all
on how it arrived there So to calculate the entropy change in an irreversible process
first find out what the initial and final conditions of the process are Then invent a
reversible process that takes the system from the initial state to the final state Since
the entropy depends only on the state of system and not on the process the entropy
change for the reversible process is the same as that for the irreversible process
Entropy is thus like pressure or temperature if the pressure changes from Pl to to P2
then the pressure difference is P2 - Pl regardless of how the pressure change was
made This seems obvious for pressure but not for entropy only because you arent
comfortable with entropy yet
Entropy of an Ideal Gas
For n moles of an ideal gas the difference in entropy between a state with temperature
T and volume V and some standard state with temperature To and volume Vo is given
by
ln lno o Vo o
T VS T V S T V nC nR
T V
Entropy in a Phase Change
Calculating entropy change in a phase change like melting or boiling is easy It is
QT Where Q is in the latent heat for example the flows in to cause the melting and
T is the temperature of the phase change There is NO integral to do
14
Serway Chapter 35
Angle of Reflection
If plane waves are incident on a reflecting surface with the propagation direction of
the waves making angle θ1 with the normal direction to the surface then the reflected
angle θ1prime relative to the surface normal is simply
1 1
ie the incident angles and reflected angles are the same
Refraction
If plane waves traveling through medium 1 are incident on a plane interface between
medium 1 and medium 2 then the angle of incidence of the incoming wave θ1 and
the angle of refraction of the transmitted wave θ2 are related by Snellrsquos law
2 2
1 1
sin
sin
v
v
where v1 and v2 are the wave speeds in medium 1 and medium 2 The angle of
incidence and the angle of refraction are both measured between the wave propagation
direction and the normal to the interface In terms of indices of refraction in the case
of light waves Snellrsquos law takes the more familiar form
1 1 2 2sin sinn n
where ni = cvi
Total Internal Reflection
If a wave is incident from a medium of low wave speed into a medium of high wave
speed the law of refraction requires that the angle of refraction be greater than the
angle of incidence If the angle of refraction is required to be greater than 90deg then no
refracted wave can exist and total internal reflection occurs The critical incident
angle θc beyond which total internal reflection occurs is given by
2
1
sin c
n
n
15
Dispersion of Light
In addition to the speed of light varying from material to material it also varies with
wavelength within each material This means that the index of refraction is generally a
function of wavelength
c
nv
Since the wave speed is not constant such a medium is dispersive meaning in this
context that refraction actually disperses white light into its various colors because
Snellrsquos law gives a different angle for each wavelength In most materials the
variation with wavelength is quite small but this small effect is responsible for some
of the most spectacular color effects we ever see including rainbows a flashing
crystal chandelier and the colored fire of a diamond solitaire by candlelight
16
Serway Chapter 36
Real and Virtual Images
When light rays are focused at a certain plane producing an image if a sheet of white
paper is placed there we call the image a real image The images produced by film
projectors and overhead projectors are examples of real images
When light rays appear to come from a certain location but no image is produced
when a screen is placed there we say that there is a virtual image at that location For
instance when you look in a mirror it appears that someone is behind the mirror but
a screen placed back there in the dark would show nothing Your image in the mirror
is a virtual image
Ray Tracing
There are lots of rules about how to find the images in optical systems but the best
way to keep things straight is to learn how to draw the principal rays for curved
mirrors and lenses The rays for convex and concave mirrors are shown in Fig 3615
The rays for converging and diverging thin lenses are shown in Fig 3627 You should
memorize the principal rays and know how to use them to locate images
Curved Mirrors
The focal length of a curved mirror with radius of curvature R is given by
2
Rf
The relation between object distance p image distance q and focal length f is
1 1 1
p q f
If the mirror is a diverging mirror f should be negative and if q should turn out to be
negative the image is virtual
Lateral Magnification
The lateral magnification in an optical system is defined by the ratio of the image size
to the object size
17
Image height
Object height
qM
p
for curved mirrors
As usual there are sign conventions here too but this formula just gives the
magnitude It is better to keep keep track of upright images versus inverted images by
means of ray diagrams rather than by memorizing sign conventions
Thin Lenses
The focal length of a thin lens is related to the radii of curvature of the two faces R1
and R2 of the lens by
1 2
1 1 11n
f R R
Note that this formula differs from Eq (3611) in the text by not having a minus sign
between the two R-terms We like this form better because for a simple converging
lens like a magnifying glass we just use positive values of R for both surfaces If one
of the faces is concave producing divergence use a negative value for R And if a
surface is flat use R = infin If the face is flat the radius is infinite
The relation between the image object and focal distances for a thin lens is the same
as that for a curved mirror
1 1 1
q p f
Use a negative focal length if the lens is diverging
The lateral magnification for a thin lens is the same as for a curved mirror
Image height
Object height
qM
p
Camera
The lens system in a camera projects a real image of an object onto the film (or CCD
array in a digital camera) The position of the image is adjusted to be on the film by
moving the lens into or out of the camera
18
Eye
The eye is like a camera in that a real image is formed on the retina Unlike a camera
the image position is adjusted by changing the focal length of the lens This is done by
the ciliary muscle which squeezes the lens changing its shape
Near Point The near point is the closest distance from the eye for which the lens can
focus an image on the retina It is usually 18-25 cm for young persons
Far Point The far point is the greatest distance from the eye for which the lens can
focus an image on the retina For a person with normal vision the far point is at
infinity
Nearsightedness A person is nearsighted if their far point is at some finite distance
less than infinity This condition can be corrected with a lens that takes an object at
infinity and produces a virtual image at the personrsquos far point
Farsightedness and Presbyopia A person is farsighted if their near point is too far
away for comfortable near work like reading or knitting This can be corrected by a
lens which takes an object at a normal near point distance of 18-25 cm and produces a
virtual image at the personrsquos natural near point Presbyopia involves a similar
problem which nearly all people experience as they age The ciliary muscle becomes
too weak and the lens becomes too stiff to allow the eye to provide for both near and
far vision The solution for this problem is either reading glasses or bifocal lenses
Reading classes are just weak magnifying glasses mounted on eyeglass frames
Bifocal lenses are split into upper and lower halves The lower half is a lens which
gives the proper correction for near work and the upper half is a different lens for
proper focusing at infinity
Angular Size
When an object is brought closer to the eye it appears to be larger because the image
on the retina is larger The size of this image is directly proportional to the objectrsquos
angular size which is the angle subtended by the object measured from the center of
the lens of the eye In optical instruments which are to be used with the eye the
angular size of the final image is whatrsquos important because it determines how large
the image will appear to the viewer
19
Simple Magnifier
A simple magnifier is a single converging lens or magnifying glass It takes an object
closer to the eye than a normal near point and produces a virtual image at or beyond
this near point The angular magnification is defined to be the ratio of the angular size
when viewed through the lens to the angular size of the object when viewed at the
normal near point (without aid of the lens)
Microscope
This instrument has two lenses (1) The objective is near the object being viewed and
produces a greatly magnified real image (2) The eyepiece is a simple magnifier which
the viewer uses to closely examine the image from (1)
Telescope
This instrument also has two lenses (1) The objective at the front of the telescope
takes light from a distant object and produces a real inverted image (which is rather
small) near its focal point (2) This small real image is then examined by the eyepiece
functioning as a simple magnifier to produce a virtual image with a larger angular
size
20
Serway Chapter 37
Two-Slit Interference
If light is incident on two closely spaced narrow slits a pattern of light and dark
stripes is produced beyond the slits The bright stripes or fringes are caused by
constructive interference of the two waves coming from the slits Constructive
interference occurs whenever two waves arrive at a location in phase with each other
This occurs when the distance x1 from slit 1 to a point P on the screen and the
distance x2 from slit 2 to point P differ by in integral number of wavelengths
1 2 where 0 1 2x x m m
where λ is the wavelength of the light When x1 and x2 are much larger than the slit
spacing d this condition reduces to
sind m
where θ is the angle between the direction of the incident light and the direction of the
light arriving at the screen
Thin Films
When light is partially reflected and partially transmitted by a thin film of transparent
material it is possible to have interference between the wave reflected from the front
of the film and light reflected from the back of the film (The colored reflections from
the thin film of oil on the water in a rain-soaked parking lot are an example of this
effect) It is difficult to write down formulas that will work in all cases so we will
just review the important principles here
1 If the two reflected waves are in phase with each other the film has enhanced
reflection (constructive interference) but if the two reflected waves are out of phase
with each other reflection is diminished (destructive interference) Phase shifts occur
due to reflection and due to the extra path length through the film of the wave
reflected from the back of the film
2 The phase change due to reflection is determined by the difference in index of
refraction between the two media involved in the reflection If the wave is incident
21
from a medium with a low index of refraction into a medium with a high index of
refraction a phase change of 180deg occurs and the reflected wave is inverted If
incident from high to low no phase shift occurs and the reflected wave is non-
inverted
3 The extra path length through the film of the wave reflected from the back of the
film is equal to 2t where t is the thickness of the film (The incident light is assumed
to be normal to the surface of the film) The number of wavelengths contained in the
extra path length is equal to 2tλn where λn = λn is the wavelength of the light in the
film
4 Rules for reflection from thin films
If one ray is inverted and the other is not then we have
12 constructive
2 nt m
2 destructivent m
If either both rays are inverted or both are non-inverted we have
2 constructivent m
12 destructive
2 nt m
22
Serway Chapter 38
Diffraction Grating
A diffraction grating is simply a fancy version of two-slit interference with the two
slits replaced by thousands of slits Just as in the two-slit case the bright fringes
occur at angles given by
sin md
but in this case d the distance between neighboring slits is made to be very small
The effect of having many slits instead of two is to make each bright fringe highly
localized with wide dark regions between neighboring maxima
Single Slit Diffraction
When light passes through an opening in an opaque screen an interference pattern is
produced beyond the opening To understand why we may replace the single opening
by many small coherent sources of light These many sources interfere with each
other producing a pattern known as a diffraction pattern (Note that many authors do
not distinguish between interference and diffraction treating them as interchangeable
terms) If the opening is a slit of width a then the diffraction pattern far from the slit
will have a bright central maximum with a succession of minima and weaker maxima
on either side The angle between the incident direction and the minima is given by
sin ma
where m = plusmn1 plusmn2
If the opening is circular with diameter D the angle between the incident direction
and the first minimum is given by
sin 122D
Optical Resolution and Rayleighrsquos Criterion
Two point sources can just be resolved (distinguished from each other) if the peak of
the diffraction image of the first source overlies the first minimum of the diffraction
image of the second source For circular holes of the kind usually encountered in
23
optical devices this condition is approximately satisfied when the angular separation
between the two sources as viewed from the optical instrument is greater than or equal
to the critical angle
min 122D
where λ is the wavelength of the light and where D is the diameter of the aperture in
the instrument
Polarization
We say that an electromagnetic wave is polarized if its electric field vector doesnrsquot
change direction in a random fashion The simplest kind of polarization is linear
polarization in which the electric field vector oscillates back and forth along the
same axis in space Polarized light can be produced from normal unpolarized light by
selective absorption (as in Polaroid sunglasses) by passing light through crystals that
have different indices of refraction for different polarizations (double refraction) by
scattering (the blue sky is polarized) and by reflection (glare) Polarization by
reflection occurs when light reflects from a shiny insulating (non-metallic) surface
The amount of polarization is greatest for reflection at Brewsterrsquos angle
2
1
tan P
n
n
where θP is the incidence angle of light from medium 1 onto medium 2 and where n1
and n2 are the indices of refraction for the media (In our everyday experience n1 = 1
since the light comes in through the air and n2 is the index of refraction of the shiny
insulating material producing the glare eg water glass plastic paint etc)
Malusrsquos Law
The intensity of transmitted polarized light through a perfect polarizer is related to the
incident intensity of polarized light by Malusrsquos law
2cosoI I
where Io is the intensity of the incident light and where θ is the angle between the
electric field vector in the incident wave and the transmission axis the polarizer
24
Serway Chapter 39
Principles of Relativity
All of the weirdness of relativity flows from two simple principles
(1) The laws of physics must be the same in all inertial (non-accelerating) reference
frames
(2) The speed of light in vacuum has the same value in all inertial reference frames
Note that (1) seems reasonable but (2) is very odd It says that if two space ships are
approaching each other at nearly the speed of light and a laser pulse is shot from ship
1 toward ship 2 then when the people on ship 2 measure the speed of the pulse as it
goes by it is moving at 3 times 108 ms the same as if the ships were stationary
Simultaneity
If observer 1 sees two events in her own frame as simultaneous at two different
locations a moving observer 2 will see these two events happening at different times
Relativistic gamma
The factor γ (gamma) appears regularly in the formulas of relativity
2
2
1
1 vc
where v is the relative speed between two inertial frames
Time Dilation
If observer 1 sees two events at the same location in space separated by time Δtp in his
own frame then observer 2 moving at speed v relative to observer 1 will see these
two events separated by a longer time Δt
pt t
Moving clocks run slow
Length Contraction
If observer 1 measures the length of an object along the x-direction at some instant of
time in her frame to be Lp then observer 2 moving in the x-direction at speed v will
measure the length L of the object to be shorter
pLL
25
Moving meter sticks are short
Velocity Addition
If a particle is moving at velocity vac relative to frame c and if frame c is moving at
speed vcb with respect to frame b (with both velocities directed along the same line)
what would be the velocity of the particle in frame b In classical physics the answer is
ab ac cbv v v
but if any of these velocities are an appreciable fraction of the speed of light the
answer changes
21ac cb
abac cb
v Vv
v v c
In using this equation make sure that vac has the same sign in the numerator and in the
denominator and that vbc also has the same sign in both places Note that this equation
is different (and in our opinion easier to understand) than the equations in the text
Momentum and Energy
The momentum and total energy of a particle moving at velocity v (its speed is v v )
are given by
22
2 2
22
1 1 vvc c
m mcm E mc
vp v
Rest Energy
When a particle is at rest (v = 0 and γ = 1) then E = mc2 is called the rest energy It
includes the potential energy of the particle so that if we change the potential energy
of a particle we change its rest mass
Kinetic Energy The kinetic energy is the difference between the total energy and the
rest energy
2 2K mc mc
For v ltlt c this reduces to K = mv22
General Relativity
Einsteinrsquos principle of equivalence in the theory of general relativity is In the vicinity
of a point a gravitational field is equivalent to an accelerated frame of reference in
26
the absence of gravitational effects For example there is no difference whatever in
the physics experienced by an astronaut on earth and one who is in interstellar space
far away from any stars or planets but who has just turned on her spaceshiprsquos engines
and is accelerating at 98 ms2
27
Serway Chapter 40
Photons
We now know that even though light behaves like a wave it is better described as
consisting of small packets of energy called photons The energy of a photon is related
to its frequency by
E hf
where h is Planckrsquos constant
346626 10 J sh
Photons also carry momentum given by
hf hp
c
Since the amount of energy in each photon is fixed the difference between dim light
and intense light (of the same wavelength) is that dim light consists of fewer photons
than intense light
Photoelectric Effect
In the photoelectric effect light shining onto a metal surface gives the electrons in the
metal enough energy to escape and be detected It requires a certain amount of energy
(called the work function typically = 3 ndash 5 eV) just to kick the electron out of the
metal so the light must deliver at least this much energy to an electron to produce the
effect It is observed that red light no matter how intense never produces electrons
But ultraviolet light even if quite dim will eject electrons from the metal Since red
light consists of 2 eV photons and ultraviolet light has photons with energies around
4-6 eV the photon idea explains the behavior of the photoelectric effect The
maximum energy that an ejected electron can have is
maxK hf
where f is the photon frequency The minimum frequency that light can have and cause
electrons to be ejected is called the cutoff energy and is found by setting Kmax = 0
cf h
28
Compton Effect
When high-frequency light interacts with free electrons the scattered light does not
have the same wavelength as the incident light contrary to what classical
electromagnetism would predict But the photon picture in which photons have
momentum and energy allows us to treat scattering as a collision between the photon
and the electron In this collision the electron and the scattered photon both have a
different momentum and energy than they did before And since p = hλ for a photon
if the momentum is different the wavelength will also be different
after before 1 cose
h
m c
where the angle θ is the angle between the incoming photon direction and the
direction of the scattered photon
Particles are Waves
Since photons behave like particles sometimes it is not surprising that elementary
particles can behave like waves sometimes The wavelength of a particle with
momentum p is given by
h
p
Wave-particle Duality
Both photons and elementary particles have a dual nature sometimes they behave like
particles and sometimes they behave like waves It is hard for us to comprehend the
nature of such an object by making mental pictures but experiments definitively show
that this is the case Since we have no direct experience with photons and elementary
particles (because their energies are so much smaller than the energies of the everyday
objects we encounter) it is perhaps not surprising that we have a hard time forming a
mental picture of how they behave
Electron Interference
Since an electron is both a particle and a wave just like a photon it should be able to
produce an interference pattern This is observed If an electron beam is shot at two
closely-spaced slits and if the electrons that pass through the slits are detected
downstream it is found that there are some locations where electrons are never detected
29
and others where lots of electrons are detected The pattern is exactly the same as the
one observed for light waves provided that we use the electron wavelength λ = hp in
place of the wavelength of light This pattern is observed even though each electron is
detected as a single dot on the screen Only after many such dots are collected does the
pattern emerge And if we try to understand how this effect could possibly work by
looking closely at each slit to see which one the electron came through the pattern
disappears the act of measurement destroys the interference This means that each
single electron somehow comes through both slits (as a wave would)
Uncertainty Principle
In classical physics we always imagine that the positions and momenta of moving
particles have definite values It might be hard to measure them but surely at each
instant of time a particle should be precisely located at some point in space and have a
similarly precise momentum This turns out not to be true Instead both position and
momentum are required to be uncertain with their uncertainties Δx and Δp satisfying
the Heisenberg uncertainty relation
2x p
So if the particle were known to be precisely at some particular location (so that Δx =
0) we couldnrsquot know anything about its momentum (Δp = infin) and if its momentum
were exactly known we couldnrsquot know its position
There is a similar relation involving the particlersquos energy E and the time interval Δt
over which this energy is measured
2E t
This means that energy is not actually conserved if we are considering very small time
intervals and this brief non-conservation of energy has been observed
30
Serway Chapter 41
Wave Function ψ and Probability
Quantum mechanics does not predict exactly what an electron or a photon will do Instead
it specifies the wave function or probability amplitude ψ of an electron or a photon
This wave function is a complex-valued function of space and time whose squared
magnitude is the probability density P for finding a particle at a particular place in
space at a certain time
2P
where is the complex conjugate of ψ
For example the wave function of an electron with perfectly specified momentum p
would have a wavelength given by
h
p
and its probability amplitude would be proportional to
2i x ipxe e
The corresponding probability density would then be
21ipx ip ipxP e e e
which means that the electron is equally probable to be anywhere along the x axis
This is in accord with the uncertainty principle since we specified the momentum
precisely we canrsquot have any idea about the position of the electron
As another example you have probably seen ldquofuzzy ballrdquo drawings of electron
orbitals in a chemistry book These fuzzy balls are meant to indicate the distribution
of the probability density 2
P in the orbital
Particle in a Box
A simple example in which we can calculate the wave function is the case of a particle of
mass m constrained to be inside a 1-dimensional box between x = 0 and x = L In this
case the general wave function is a linear superposition of wave functions ψn of the form
sin niE tn
n xx t A e
L
31
where A is a positive constant where n = 1 2 3 and where the energy associated
with each of the quantum states ψn is given by
22
28n
hE n
mL
This wave function is zero at x = 0 and x = L which means that the particle will never
be found at the walls of the box The wave function has maximum values in the
interior and at these places the particle is most likely to be found and it also has
places where it is zero and at these places the particle will also never be found as
expressed by the formula
2sinn x
PL
The particle in the box is interfering with itself producing a probability interference
pattern across the box just like the interference patterns we studied with light and
sound
32
Serway Chapter 42
Atoms
Many people picture an atom as a miniature solar system where electrons orbit around
a massive nucleus at the center This picture is misleading because in atoms the wave
nature of electrons dominates The electrons form 3-dimensional standing waves
(called orbitals) centered at the nucleus And if you ask what is it thatrsquos waving the
answer is ψ the probability amplitude (see Chapter 41)
Standing waves on a string can occur only for certain frequencies (the fundamental
and higher harmonics) Similarly atomic orbitals occur only for certain energies For
the hydrogen atom the energies of the orbitals have the a particularly simple form
2
1136 eVnE
n
where n = 1 2 3 is called the principal quantum number For other atoms the
determination of the orbital energies requires numerical calculation by computers
Atomic Spectra
If an electron is somehow given extra extra energy (we say that it is excited) so that it
occupies a higher orbital it will eventually ldquofallrdquo back down to a lower orbital Each
time an electron falls to a lower orbital it loses the difference in energy between the
two orbitals in the form of a photon Since the orbital energies are discrete so are the
energy differences and so are the wavelengths of the emitted light The entire set of
these discrete wavelengths is called the atomic spectrum and it is unique to each type
of atom For hydrogen the spectrum can be simply written as
H 2 2
1 1 1
f i
Rn n
where RH = 10973732 times 107 m-1 For other atoms the spectrum cannot be expressed as
a formula
Orbital Angular Momentum Electrons in atoms also have quantized values of
angular momentum The orbital quantum number ℓ specifies the value of this
quantized angular momentum through the formula
1L
33
If we want to know the value of the angular momentum along some direction in space
say the z direction the answer is not L but rather
zL m
where mℓ is another quantum number which runs from
1 1m
This quantum number is important when an atom sits in a magnetic field
Spin Angular Momentum It has been found experimentally that electrons and other
charged particles also carry internal angular momentum which we call spin
Electrons have an intrinsic spin angular momentum s along a specified axis that is
extremely quantized it can only take on 2 possible values
1
2zs s
We thus say that electrons have spin s = 12 or that electrons are rdquospin one-half parti-
cles Since s is an angular momentum it obeys the same rule as that for orbital angular
momentum namely that if its value along some axis is s then its total magnitude is
given by
31
2S s s
I know this seems weird but quantum mechanics is weird The only excuse for this
bizarre way of looking at the world is that it predicts what happens in experiments
Exclusion Principle The answer to the question of how many electrons (or any other
spin one-half particle) can be in one particular quantum state was discovered by
Wolfgang Pauli and is called the exclusion principle
ldquoNo two electrons can ever be in the same quantum state therefore no two electrons
in the same atom can have the same set of quantum numbersrdquo
This is the reason that we have atoms with different properties instead of every atom
simple having all of its electrons in the ground state All of the variety we see around
us in the world is the result of chemical differences and these differences would not
exist unless electrons obeyed this important principle The entire structure of the
periodic table (see pages 1377-1379 in Serway) is an expression of this principle
34
Serway Chapter 44
Nuclear Properties
The nucleus is composed of protons with charge +e and mass mp = 1007226 u and of
neutrons with zero charge and mass mn = 1008665 u where u is one atomic mass unit
-271 u=1660540 10 kg
The atomic number Z counts the number of protons in a nucleus while the neutron
number N counts the number of neutrons The mass number A is the sum of the two
A N Z
Protons and nuetrons have about the same mass and some times called baryons (heavy
ones) or nucleons (partcles in the nucleus) A is sometimes called the baryon
number The chemical elements are distinguished by Z ie hydrogen has Z = 1 iron
has Z = 26 uranium has Z = 92 etc But for a given Z there might be several nuclei
with different numbers of neutrons These nuclear siblings are called isotopes
These numbers are used to label nuclei according to the pattern
5626 ie FeA
Z X
denotes the isotope of iron with N = 56 ndash 26 = 30 neutrons
The nucleus is roughly spherical with a radius given approximately by
1 3 150 0where 12 10 mr r A r
Nuclear Stability
Because the positively charged protons electrically repel each other with an enormous
force at distances as small as 10-15 m there must be some really strong force that that
overcomes electrical repulsion to hold protons and neutrons together This force is
called with some lack of imagination the strong force It is a very short-range force
(it only acts over a distance of about 2 x 10-15 m) and attracts protons to protons
neutrons to neutrons and neutrons to protons But in spite of this strong nuclear force
the coulomb repulsion of the protons is still present so anything that might keep the
protons from being right next to each other would help keep the nucleus from
35
exploding This role is played by the neutrons and for nuclei with Z le 20 the stable
nuclei roughly have N = Z
For Z gt 20 the coulomb repulsion force is more powerful and more neutrons are
needed to dilute it up to about N = 15Z around Z = 80 For Z gt 83 no amount of
neutrons can help and these nuclei are unstable (radioactive) There is an ldquoisland of
(relative) stabilityrdquo around Z=90-92 (thorium and uranium respectively) These
elements have one or two isotopes with half-lifes of billions of years so there are
substantial amounts of such elements on Earth This fact makes it possible to have
practical fission devices
Radioactive Decay
There are three types of radioactive decay
Alpha decay the nucleus kicks out a helium nucleus (N = 2 Z = 2 A = 4) So if the
original nucleus is called X and the new nucleus is called Y then the decay would look
like this
4 42 2X Y+ HeA A
Z Z
Beta decay the nucleus either kicks out an electron (endash) or its positively-charged
antimatter twin the positron (e+) plus either an electron neutrino v or an electron
anti-neutrino v
1X Y+e electron decayA AZ Z v
1X Y+e positron decayA AZ Z v
A neutrino is a particle with no charge hardly any mass (much less than the electron
mass) and interacts so weakly with matter that most neutrinos upon encountering the
planet earth just pass right through it as if it werenrsquot there
Gamma decay the nucleons in the nucleus X are in an excited energy state X
(perhaps as a result of having undergone alpha or beta decay) and they drop down to a
lower energy state shedding the energy as a high frequency photon
X XA AZ Z
36
This process is exactly analogous to the way that the electrons in atoms emit photons
Decay Rate and Half Life
There is no way to predict exactly when an unstable or excited nucleus will decay but
there is an average rate at which this decay occurs called the decay constant λ The
meaning of this constant is that if there are a large number N of nuclei in a sample
then the number of decays per second that will be observed (called the decay rate R)
is R = λN In mathematical language
dMR N
dt
This simple differential equation has for its solution
0tN t N e
where N0 is the number of nuclei in the sample at time t = 0
The half-life is the time it takes for half of the nuclei in the sample to decay and is
related to the decay constant by
1 2
ln 2 0693T
Disintegration Energy
When a nucleus decays it is making a transition to an overall state of lower energy
which means according to Einsteinrsquos famous formula E = mc2 that the sum of the
masses after the decay must be less than the mass before with the lost mass appearing
as kinetic energy among the decay products For example in alpha decay this kinetic
energy called the disintegration energy Q is given by
2X YQ M M M c
37
Serway Chapter 45
Nuclear energy
There are two ways to extract energy from the nucleus fission and fusion For nuclei
with Z greater than 26 breaking the nucleus apart into pieces leads to a lower
2mc energy than the original nucleus so energy can be extracted by fission For
nuclei with Z less than 26 a lower energy is achieved by combining nuclei so energy
can be extracted by fusion
Fission
Since neutrons have no charge they are not repelled from nuclei as protons are for
this reason their behavior is the key to understanding how fission works
Fast neutrons mostly bounce off other nuclei with each collision slowing the neutron
down This elastic energy loss is most effective if the other nuclei have low mass (like
hydrogen) and these materials are called moderators because of their ability to slow
down fast neutrons
The reason that slowing neutrons is important is that slow neutrons are much more
likely to be absorbed by a nucleus which then leads to nuclear reactions of various
kinds For a few very large nuclei like uranium-235 and some plutonium isotopes
absorption of a slow neutron causes the nucleus to split into two large fragments plus
2 or 3 fast neutrons (energy is released in the process too) Thus one slow neutron
can produce energy plus 2 or 3 more neutrons which if moderated from fast to slow
can split 2 or 3 more nuclei leading to a runaway chain reaction If the reaction is
allowed to proceed unchecked it produces a large explosion If a neutron absorbing
material is added to the mix (like the cadmium in reactor control rods) it is possible to
keep the reaction under control and to extract the released energy as heat to drive
steam turbines and produce electricity
Fusion
Fusion involves mashing two nuclei together and since they are both charged and repel
each other this reaction is much harder to make go The nuclei must have enough energy to
overcome the coulomb repulsion which is why this reaction requires a high temperature
(hundreds of millions of degrees K) like that in the sun (which is burning hydrogen to
helium via fusion) or like that in the center of the fission explosion that is used to detonate
38
a hydrogen bomb
This reaction is of interest for power production in spite of this difficult temperature
requirement because of the abundance of fusion fuel on the planet There are about
012 g of deuterium ( 21H ) in every gallon of water on earth and it only costs about 4
cents to extract it The fusion energy available from this minuscule amount of
deuterium would run a 1000-Megawatt power plant for 10 seconds Water is so
abundant on earth that if fusion were to work we would have an essentially
inexhaustible source of energy
So why donrsquot we have fusion power plants Well the fuel is cheap but the match is
incredibly expensive The only way we know to control this difficult high-temperature
reaction is with large and expensive pieces of equipment involving either large
magnetic fields and complex high-power electromagnetic antennas or with gigantic
(football-fieldsized) laser facilities involving more than a hundred of the highest-
energy lasers ever built Power plants based on these current methods for controlling
fusion are unattractive to the fiscally-minded people who run the electric power
industry Hopefully better designs will be discovered as experiments continue
39
c actinium 68 Er erbium 101 Md mendelevium 104 Rf rutherfordium l aluminum 63 Eu europium 80 Hg mercury 62 Sm samarium m americium 100 Fm fermium 42 Mo molybdenum 21 Sc scandium b antimony 9 F fluorine 60 Nd neodymium 106 Sg seaborgium r argon 87 Fr francium 10 Ne neon 34 Se selenium s arsenic 64 Gd gadolinium 93 Np neptunium 14 Si silicon t astatine 31 Ga gallium 28 Ni nickel 47 Ag silver a barium 32 Ge germanium 41 Nb niobium 11 Na sodium k berkelium 79 Au gold 7 N nitrogen 38 Sr strontium e beryllium 72 Hf hafnium 102 No nobelium 16 S sulfur i bismuth 108 Hs hassium 76 Os osmium 73 Ta tantalum h bohrium 2 He helium 8 O oxygen 43 Tc technetium
boron 67 Ho holmium 46 Pd palladium 52 Te tellurium r bromine 1 H hydrogen 15 P phosphorus 65 Tb terbium d cadmium 49 In indium 78 Pt platinum 81 Tl thallium a calcium 53 I iodine 94 Pu plutonium 90 Th thorium f californium 77 Ir iridium 84 Po polonium 69 Tm thulium
carbon 26 Fe iron 19 K potassium 50 Sn tin e cerium 36 Kr krypton 59 Pr praseodymium 22 Ti titanium s cesium 57 La lanthanum 61 Pm promethium 74 W tungsten l chlorine 103 Lr lawrencium 91 Pa protactinium 92 U uranium r chromium 82 Pb lead 88 Ra radium 23 V vanadium o cobalt 3 Li lithium 86 Rn radon 54 Xe xenon u copper 71 Lu lutetium 75 Re rhenium 70 Yb ytterbium m curium 12 Mg magnesium 45 Rh rhodium 39 Y yttrium b dubnium 25 Mn manganese 37 Rb rubidium 30 Zn zinc y dysprosium 109 Mt meitnerium 44 Ru ruthenium 40 Zr zirconium s einsteinium
40
UNITS The SI units are given in parentheses Other commonly used units are given in terms of the SI units Symbols conform with the recommendations of the American National Standards Institute (ANSI) and the American Institute of Physics (AIP) Commonly used multiples of SI units
prefix symbol factor tera T 1012 giga G 109
mega M 106 kilo k 103
centi c 10-2 milli m 10-3
micro μ 10-6 nano n 10-9 pico p 10-12
femto f 10-15
length meter (m) angstrom 1 Ǻ = 10-10 m inch 1 in = 254 x 10-2 m foot 1 ft = 03048 m mile 1 mi = 1609 m
mass kilogram (kg) atomic mass unit 1 u = 1661 x 10-27 kgslug 1 slug = 1459 kg
force newton (N = kg bull ms2) dyne 1 dyn = 10-5 N pound 1 lb = 4448 N
pressure Pascal (Pa = kgm bull s2) atmosphere 1 atm = 1013 x 105 Pa poundssquare inch 1 psi = 6895 Pa cm of mercury 1 cm Hg = 1333 Pa bar 1 bar = 1000 x 105 Pa torr 1 torr = 1333 Pa
time second (s) minute 1 min = 60 s hour 1 h =3600 s
frequency hertz (Hz = s-1) radianssecond 1 rads = 12π Hz
energy joule (J = kg bull m2s2) erg 1 erg = 10-7 J electron volt 1 eV = 1602 x 10-19 J calorie 1 cal = 4187 J kilowatt-hour 1 kWh = 36 x 106 J British thermal unit 1 Btu = 1055 J
power watt (W = kg bull m2s3) horsepower 1 hp = 7457 W
charge coulomb (C = A bull s)
electric potential volt (V = kg bull m2 s3 bull A)
current ampere (A)
resistance ohm (Ω = kg bull m2s3 bull A2)
capacitance farad (F = s4 bull A2kg bull m2)
magnetic field tesla (T = kgs2 bull A) gauss 1 G = 10-4 T
magnetic flux weber (Wb = kg m2s2 bull A) maxwell 1 Mx = 10-8 Wb
magnetic inductance henry (H = kg bull m2s2 bullA2)
temperature kelvin (K) degrees Celsius 0degC = 27315 K
angle radian (rad) degree 1 deg = π180 rad revolution 1 rev = 2π rad
41
Some Physical Constantsa Quantity Symbol Valueb Atomic mass unit μ 1660 538 73 (13) x 10-27 kg
931494 013 (37) MeVc2 Avogadros number NA 6022 141 99 (47) x 1023 particlesmol Avogadrorsquos number 6022 x 1023 mol Bohr magneton
2B e
em 9274 008 99 (37) x 10-24 JT
Bohr magneton μB 927 x 10-24 JT Bohr radius
0
2
2e em e k
a 5291 772 083 (19) x 10-11 m
Boltzmanns constant B A
RNk 1380 650 3 (24) x 10-23 JK
Boltzmannrsquos constant kB 1380 x 10-23 JK Compton wavelength
C eh
m c 2426 310 215 (18) x 10-12 m
Deuteron mass md 3343 583 09 (26) x 10-27 kg 2013 553 212 71 (35) u
electron charge e 1602 x 10-19 C Electron mass me 9109 381 88 (72) x 10-31 kg
5485 799 110 (12) x 10-4 u 0510 998 902 (21) MeVc2
electron mass 911 x 10-31 kg Electron volt eV 1602 176 462 (63) x 10-19 J Elementary charge e 1602 176 462 (63) x 10-19 C Gas constant R 8314 472 (15) JKmol Gravitational constant G 6673 (10) x 10-11 Nm2kg2 Neutron mass mn 1674 927 16 (13) x 10-27 kg
1008 664 915 78 (55) u 939565 330 (38) MeVc2
neutron mass 1675 x 10-27 kg Nuclear magneton
2n p
em 5050 783 17 (20) x 10-27 JT
permeability constant μ0 1257 x 10-6 Hm permittivity constant ε0 8854 x 10-12 Fm Plancks constant h
2h
6626 068 76 (52) x 10-34 Js 1054 571 596 (82) x 10-34 Js
Planckrsquos constant h ħ
6626 x 10-34 Js 1055 x 10-34 Js
Proton mass mp 1672621 58 (13) x 10-27 kg 1007 276 466 88 (13) u 938271 998 (38) MeVc2
proton mass 1673 x 10-27 kg Rydberg constant RH 1097 373 156 854 9 (83) x 107 m-1 Speed of light in vacuum c 2997 92458 x 108 ms (exact) speed of light c 300 x 108 ms a
These constants are the values recommended in 1998 by CODATA based on a least-squares adjustment of data from different measurements For a more complete list see P J Mohr and B N Taylor Rev Mod Phys 72351 2000 b The numbers in parentheses for the values above represent the uncertainties of the last two digits
NOTE The ones in red are the ones from Appendix 13 The ones in yellow are the ones that are from the Serway book table but that were already on the Appendix 13 table
42
INDEX
Absolute zero 9
Adiabatic 17
Adiabatic exponent 18
Adiabatic process 17
Alpha decay 46
Amplitude 4
Angle of reflection 25
Angular frequency 3
Angular magnification 30
Angular momentum atomic 43
Angular size 29
Archimedes Principle 1
Atomic spectra 43
Atoms 43
Avogadrorsquos number 10
Beats 8
Bernoullirsquos Equation 2
Beta decay 46
Bifocals 29
Brewsterrsquos angle 34
British Thermal Unit Btu 11
Buoyancy 2
calorie 11
Calorie 11
Camera 28
Carnot cycle 21
Carnot efficiency 22
Celsius scale 9
Ciliary muscle 29
Coefficient of performance 21
Compton effect 39
Constructive interference 7
Continuity equation of 2
Contraction length 35
Convection 14
Curved mirrors 27
Decay constant 46
Decay rate radioactive 46
Decibel scale 5
Degrees of freedom 16
Density 1
Destructive interference 7
Diffraction single slit 33
Diffraction grating 33
Dilation time 35
Disintegration energy 47
Dispersion of light 26
Doppler effect 6
Double slit interference 31
Efficiency 20
Electron interference 39
Emissivity 15
Energy nuclear decay 47
Energy relativistic 36
43
Engineering work 20
Entropy 22
Entropy ideal gas 24
Equipartition of energy 18
Equivalence principle 36
Exclusion principle 44
Expansion thermal 9
Expansion coefficient linear 9
Eye 29
Fahrenheit scale 9
Far point 29
Farsightedness 29
First Law of Thermodynamics 12
Fission 48
Flux volume 2
Free expansion 23
Fringes 31
Fusion 48
Fusion heat of 11
Gamma relativistic 35
Gamma decay 46
General relativity 36
Half-life 46
Heat 11
Heat capacity 11
Heat conduction 13
Heat engine 20
Heat of fusion 11
Heat of vaporization 11
Heat pump 20
Heats of transformation 11
Hydrostatics 1
Ideal Gas Law 10
Images real and virtual 27
Intensity sound 5
Interference 7
Interference two-slit 31
Internal energy 12
Internal energy degrees of freedom 16
Irreversible process 19 23
Isotopes 45
Joule 11
Kelvin scale 9
Kinetic energy relativity 36
Kinetic theory 16
Latent heat 11
Length contraction 35
Linear expansion coefficient 9
Linear polarization 34
Linear superposition 7
Longitudinal wave 3
Loudness 5
Magnification lateral 27
Magnifying glass 30
Malusrsquos law 34
Microscope 30
Momentum relativistic 36
Muscle ciliary 29
44
Musical instruments 8
Musical scale 8
Near point 29
Nearsightedness 29
Nonlinear 7
Nuclear energy 48
Nuclear properties 45
Nuclear stability 45
Octave 8
Optical resolution 33
Orbital quantum number ℓ 43
Orbitals 43
Particle in a Box 41
Particles are waves 39
Pascalrsquos Principle 1
Period 3
Photoelectric effect 38
Photons 38
Pitch 8
Polarization 34
Power sound 5
Presbyopia 29
Pressure 1
Principal quantum number n 43
Principle of equivalence 36
Principle of linear superposition 7
Probability amplitude ψ 41
Processes thermodynamic 13
R-value 14
Radiation thermal 14
Radioactive decay 46
Radioactive decay rate 46
Ray tracing 27
Rayleighrsquos criterion 33
Reading glasses 29
Real image 27
Refraction 25
Refrigerator 20
Relativistic gamma 35
Relativity principles 35
Resolved for light sources 33
Rest energy 36
Reversible process 19 22
Rope wave speed 4
Second Law of Thermodynamics 19
Shock waves 6
Simple magnifier 30
Simultaneity 35
Single slit diffraction 33
Snellrsquos law 25
Sound speed 5
Specific heat 11
Spin Angular momentum 44
Standing waves 7
Stefanrsquos law 14
Strong force 45
Telescope 30
Temperature 9
45
Temperature Scales 9
Thermal conductivity 14
Thermal energy 12
Thermal expansion 9
Thin film interference 31
Thin lenses 28
Time dilation 35
Tone musical 8
Total internal reflection 25
Transverse wave 3
Traveling Waves 3
Two-slit interference 31
Uncertainty principle 40
Vaporization heat of 11
Velocity addition relativity 36
Virtual image 27
Volume flux 2
Wave function 41
Wave Function ψ and Probability 41
Wave speed 3
Wave-particle duality 39
Wavelength 3
Wavenumber 3
Wien Displacement Law 16
Work 11
10
For a heat pump in heating mode the coefficient of performance is
COP heating modehQ
W
Note that we donrsquot have to use the engineering work here because in these systems
work is done on the system and W is naturally positive
A perfect refrigerator would take in heat Qc and exhaust the same amount of heat without
doing any work giving an infinite coefficient of performance Energy conservation alone
allows a perfect refrigerator but the second law requires COP lt infin A perfect heat pump
in heating mode would transfer Qh into the house without doing any work and so would
also have an infinite coefficient of performance The second law forbids this too A good
coefficient of performance for a real device would be around 5 or 6
Carnot Cycle
The most efficient of all possible engines is one that uses the Carnot cycle This cycle
employs an ideal gas has no friction and operates very slowly so that the gas can be in
thermal equilibrium at all parts of the cycle This means of course that it canrsquot
possibly be built and even if it could be built it would not run fast enough to be useful
Nevertheless this cycle is very important because it gives an upper bound on the
efficiency of real engines There cannot possibly be an engine that is more efficient
than one based on the Carnot cycle This cycle consists of the following four steps
1 The ideal gas absorbs heat Qh at constant temperature Th while the gas increases its
volume The reason that heat is absorbed is that expansion tends to cool the gas but
thermal contact with the environment at Th keeps the temperature high by heat
conduction into the ideal gas
2 The ideal gas further increases its volume by an adiabatic expansion This expansion
causes the gas to cool so at the end of this part of the cycle the gas is at temperature Tc
3 The gas exhausts heat Qc at constant temperature Tc while the gas decreases in
volume The reason that heat is exhausted is that compression tends to heat the gas
but thermal contact with the environment at Tc keeps the temperature low by heat
conduction out of the ideal gas
4 The gas is adiabatically compressed back to its original volume (the volume it started
with in step 1) This compression heats the gas from Tc up to Th
11
The efficiency of a Carnot engine is given by the very simple formula
1 cC
h
Te
T
where the temperatures must in be Kelvin No real engine can be more efficient than
this
The coefficient of performance of a Carnot refrigerator or heat pump in cooling mode
is given by
COP cooling mode cC
h c
T
T T
and no real refrigerator can have a coefficient of performance greater than this
The coefficient of performance of a Carnot heat pump in heating mode is
COP heating mode hC
h c
T
T T
So why donrsquot we just use these wonderful Carnot engines and have perfect efficiency
To make the reversible steps in the cycle really reversible they would have to occur
infinitely slowly So the price you pay for making a perfect engine is that it takes
forever to get it to do any work
Entropy
The entropy of a system is defined in terms of its molecular makeup and measures
roughly the disorder of the system If the system is packed into a very small volume
then it is quite ordered and the entropy will be low If it occupies a large volume the
entropy is high (To see what this has to do with disorder note that socks in a drawer
occupy a small volume while socks on the bed in the corner by the door and
hanging from the chandelier occupy a large volume) If the system is very cold then
the molecules hardly move and may even reach out to each other and form a crystal
This is a highly ordered state and therefore has low entropy If the system is very hot
with rapidly speeding molecules crashing into the container walls and bouncing off
each other things are disordered and the entropy is high
It is possible to calculate the entropy of a system in terms of its macroscopic thermody-
namic properties ie pressure volume temperature number of moles etc The key to this
calculation is the concept of a reversible process A reversible process is one that is
12
carried out without friction and so slowly that the process can be reversed at any stage by
making an infinitesimal change in the environment of the system The slow expansion of the
gas at Th in the Carnot cycle with tiny amounts of heat being transferred to the environment
is reversible If we turned around and began slowly to compress the gas would just slowly
exhaust heat to the environment in the exact reverse way that it absorbed it during expansion
Most processes however are irreversible For example if a gas-filled box were suddenly
increased in size so that the particles were free to wander into the void created by the sudden
expansion then the gas would eventually fill the new volume uniformly at the same
temperature as before the expansion (The temperature is unchanged in this imaginary
process because the kinetic energy of the molecules would be unaffected by such an
instantaneous expansion of the container walls) This imaginary but highly thought-
stimulating process is called a free expansion and it is impossible to reverse it During the
expansion we didnrsquot push on any of the molecules so reversing this process would mean
making them go back into their original volume without pushing on them they simply will
not cooperate to this extent Another way to see that just pushing them back where they came
from does not reverse the free expansion is to think about what would happen if we just
compressed either adiabatically or isothermally An adiabatic compression back to the
original volume would heat the gas above its original temperature and an isothermal
compression would require that heat be exhausted to the environment But the free expansion
involved neither temperature changes nor heat exchanges so neither of these two processes
is the reverse of the free expansion It is simply impossible to reverse this rapid expansion in
a way that takes us back to the initial state
It is possible to calculate the change of entropy for both reversible and irreversible processes
Letrsquos consider a reversible process first In a reversible process the entropy change is given
by the formula
dQS
T
where dQ is the amount of heat added to the system during a small step of the process
The total energy change during the process may then simply be calculated by integration
f f
f i i i
dQS S S dS
T
13
(Just as in the case of energy we are mainly interested in differences rather than in
absolute magnitudes)
This integration method does not work for irreversible processes For instance in the
rapid free expansion discussed above no heat is added to the system but its disorder
obviously goes up We need to find some other way to calculate the entropy The key
is the fact that the entropy of a system depends only on its current state and not at all
on how it arrived there So to calculate the entropy change in an irreversible process
first find out what the initial and final conditions of the process are Then invent a
reversible process that takes the system from the initial state to the final state Since
the entropy depends only on the state of system and not on the process the entropy
change for the reversible process is the same as that for the irreversible process
Entropy is thus like pressure or temperature if the pressure changes from Pl to to P2
then the pressure difference is P2 - Pl regardless of how the pressure change was
made This seems obvious for pressure but not for entropy only because you arent
comfortable with entropy yet
Entropy of an Ideal Gas
For n moles of an ideal gas the difference in entropy between a state with temperature
T and volume V and some standard state with temperature To and volume Vo is given
by
ln lno o Vo o
T VS T V S T V nC nR
T V
Entropy in a Phase Change
Calculating entropy change in a phase change like melting or boiling is easy It is
QT Where Q is in the latent heat for example the flows in to cause the melting and
T is the temperature of the phase change There is NO integral to do
14
Serway Chapter 35
Angle of Reflection
If plane waves are incident on a reflecting surface with the propagation direction of
the waves making angle θ1 with the normal direction to the surface then the reflected
angle θ1prime relative to the surface normal is simply
1 1
ie the incident angles and reflected angles are the same
Refraction
If plane waves traveling through medium 1 are incident on a plane interface between
medium 1 and medium 2 then the angle of incidence of the incoming wave θ1 and
the angle of refraction of the transmitted wave θ2 are related by Snellrsquos law
2 2
1 1
sin
sin
v
v
where v1 and v2 are the wave speeds in medium 1 and medium 2 The angle of
incidence and the angle of refraction are both measured between the wave propagation
direction and the normal to the interface In terms of indices of refraction in the case
of light waves Snellrsquos law takes the more familiar form
1 1 2 2sin sinn n
where ni = cvi
Total Internal Reflection
If a wave is incident from a medium of low wave speed into a medium of high wave
speed the law of refraction requires that the angle of refraction be greater than the
angle of incidence If the angle of refraction is required to be greater than 90deg then no
refracted wave can exist and total internal reflection occurs The critical incident
angle θc beyond which total internal reflection occurs is given by
2
1
sin c
n
n
15
Dispersion of Light
In addition to the speed of light varying from material to material it also varies with
wavelength within each material This means that the index of refraction is generally a
function of wavelength
c
nv
Since the wave speed is not constant such a medium is dispersive meaning in this
context that refraction actually disperses white light into its various colors because
Snellrsquos law gives a different angle for each wavelength In most materials the
variation with wavelength is quite small but this small effect is responsible for some
of the most spectacular color effects we ever see including rainbows a flashing
crystal chandelier and the colored fire of a diamond solitaire by candlelight
16
Serway Chapter 36
Real and Virtual Images
When light rays are focused at a certain plane producing an image if a sheet of white
paper is placed there we call the image a real image The images produced by film
projectors and overhead projectors are examples of real images
When light rays appear to come from a certain location but no image is produced
when a screen is placed there we say that there is a virtual image at that location For
instance when you look in a mirror it appears that someone is behind the mirror but
a screen placed back there in the dark would show nothing Your image in the mirror
is a virtual image
Ray Tracing
There are lots of rules about how to find the images in optical systems but the best
way to keep things straight is to learn how to draw the principal rays for curved
mirrors and lenses The rays for convex and concave mirrors are shown in Fig 3615
The rays for converging and diverging thin lenses are shown in Fig 3627 You should
memorize the principal rays and know how to use them to locate images
Curved Mirrors
The focal length of a curved mirror with radius of curvature R is given by
2
Rf
The relation between object distance p image distance q and focal length f is
1 1 1
p q f
If the mirror is a diverging mirror f should be negative and if q should turn out to be
negative the image is virtual
Lateral Magnification
The lateral magnification in an optical system is defined by the ratio of the image size
to the object size
17
Image height
Object height
qM
p
for curved mirrors
As usual there are sign conventions here too but this formula just gives the
magnitude It is better to keep keep track of upright images versus inverted images by
means of ray diagrams rather than by memorizing sign conventions
Thin Lenses
The focal length of a thin lens is related to the radii of curvature of the two faces R1
and R2 of the lens by
1 2
1 1 11n
f R R
Note that this formula differs from Eq (3611) in the text by not having a minus sign
between the two R-terms We like this form better because for a simple converging
lens like a magnifying glass we just use positive values of R for both surfaces If one
of the faces is concave producing divergence use a negative value for R And if a
surface is flat use R = infin If the face is flat the radius is infinite
The relation between the image object and focal distances for a thin lens is the same
as that for a curved mirror
1 1 1
q p f
Use a negative focal length if the lens is diverging
The lateral magnification for a thin lens is the same as for a curved mirror
Image height
Object height
qM
p
Camera
The lens system in a camera projects a real image of an object onto the film (or CCD
array in a digital camera) The position of the image is adjusted to be on the film by
moving the lens into or out of the camera
18
Eye
The eye is like a camera in that a real image is formed on the retina Unlike a camera
the image position is adjusted by changing the focal length of the lens This is done by
the ciliary muscle which squeezes the lens changing its shape
Near Point The near point is the closest distance from the eye for which the lens can
focus an image on the retina It is usually 18-25 cm for young persons
Far Point The far point is the greatest distance from the eye for which the lens can
focus an image on the retina For a person with normal vision the far point is at
infinity
Nearsightedness A person is nearsighted if their far point is at some finite distance
less than infinity This condition can be corrected with a lens that takes an object at
infinity and produces a virtual image at the personrsquos far point
Farsightedness and Presbyopia A person is farsighted if their near point is too far
away for comfortable near work like reading or knitting This can be corrected by a
lens which takes an object at a normal near point distance of 18-25 cm and produces a
virtual image at the personrsquos natural near point Presbyopia involves a similar
problem which nearly all people experience as they age The ciliary muscle becomes
too weak and the lens becomes too stiff to allow the eye to provide for both near and
far vision The solution for this problem is either reading glasses or bifocal lenses
Reading classes are just weak magnifying glasses mounted on eyeglass frames
Bifocal lenses are split into upper and lower halves The lower half is a lens which
gives the proper correction for near work and the upper half is a different lens for
proper focusing at infinity
Angular Size
When an object is brought closer to the eye it appears to be larger because the image
on the retina is larger The size of this image is directly proportional to the objectrsquos
angular size which is the angle subtended by the object measured from the center of
the lens of the eye In optical instruments which are to be used with the eye the
angular size of the final image is whatrsquos important because it determines how large
the image will appear to the viewer
19
Simple Magnifier
A simple magnifier is a single converging lens or magnifying glass It takes an object
closer to the eye than a normal near point and produces a virtual image at or beyond
this near point The angular magnification is defined to be the ratio of the angular size
when viewed through the lens to the angular size of the object when viewed at the
normal near point (without aid of the lens)
Microscope
This instrument has two lenses (1) The objective is near the object being viewed and
produces a greatly magnified real image (2) The eyepiece is a simple magnifier which
the viewer uses to closely examine the image from (1)
Telescope
This instrument also has two lenses (1) The objective at the front of the telescope
takes light from a distant object and produces a real inverted image (which is rather
small) near its focal point (2) This small real image is then examined by the eyepiece
functioning as a simple magnifier to produce a virtual image with a larger angular
size
20
Serway Chapter 37
Two-Slit Interference
If light is incident on two closely spaced narrow slits a pattern of light and dark
stripes is produced beyond the slits The bright stripes or fringes are caused by
constructive interference of the two waves coming from the slits Constructive
interference occurs whenever two waves arrive at a location in phase with each other
This occurs when the distance x1 from slit 1 to a point P on the screen and the
distance x2 from slit 2 to point P differ by in integral number of wavelengths
1 2 where 0 1 2x x m m
where λ is the wavelength of the light When x1 and x2 are much larger than the slit
spacing d this condition reduces to
sind m
where θ is the angle between the direction of the incident light and the direction of the
light arriving at the screen
Thin Films
When light is partially reflected and partially transmitted by a thin film of transparent
material it is possible to have interference between the wave reflected from the front
of the film and light reflected from the back of the film (The colored reflections from
the thin film of oil on the water in a rain-soaked parking lot are an example of this
effect) It is difficult to write down formulas that will work in all cases so we will
just review the important principles here
1 If the two reflected waves are in phase with each other the film has enhanced
reflection (constructive interference) but if the two reflected waves are out of phase
with each other reflection is diminished (destructive interference) Phase shifts occur
due to reflection and due to the extra path length through the film of the wave
reflected from the back of the film
2 The phase change due to reflection is determined by the difference in index of
refraction between the two media involved in the reflection If the wave is incident
21
from a medium with a low index of refraction into a medium with a high index of
refraction a phase change of 180deg occurs and the reflected wave is inverted If
incident from high to low no phase shift occurs and the reflected wave is non-
inverted
3 The extra path length through the film of the wave reflected from the back of the
film is equal to 2t where t is the thickness of the film (The incident light is assumed
to be normal to the surface of the film) The number of wavelengths contained in the
extra path length is equal to 2tλn where λn = λn is the wavelength of the light in the
film
4 Rules for reflection from thin films
If one ray is inverted and the other is not then we have
12 constructive
2 nt m
2 destructivent m
If either both rays are inverted or both are non-inverted we have
2 constructivent m
12 destructive
2 nt m
22
Serway Chapter 38
Diffraction Grating
A diffraction grating is simply a fancy version of two-slit interference with the two
slits replaced by thousands of slits Just as in the two-slit case the bright fringes
occur at angles given by
sin md
but in this case d the distance between neighboring slits is made to be very small
The effect of having many slits instead of two is to make each bright fringe highly
localized with wide dark regions between neighboring maxima
Single Slit Diffraction
When light passes through an opening in an opaque screen an interference pattern is
produced beyond the opening To understand why we may replace the single opening
by many small coherent sources of light These many sources interfere with each
other producing a pattern known as a diffraction pattern (Note that many authors do
not distinguish between interference and diffraction treating them as interchangeable
terms) If the opening is a slit of width a then the diffraction pattern far from the slit
will have a bright central maximum with a succession of minima and weaker maxima
on either side The angle between the incident direction and the minima is given by
sin ma
where m = plusmn1 plusmn2
If the opening is circular with diameter D the angle between the incident direction
and the first minimum is given by
sin 122D
Optical Resolution and Rayleighrsquos Criterion
Two point sources can just be resolved (distinguished from each other) if the peak of
the diffraction image of the first source overlies the first minimum of the diffraction
image of the second source For circular holes of the kind usually encountered in
23
optical devices this condition is approximately satisfied when the angular separation
between the two sources as viewed from the optical instrument is greater than or equal
to the critical angle
min 122D
where λ is the wavelength of the light and where D is the diameter of the aperture in
the instrument
Polarization
We say that an electromagnetic wave is polarized if its electric field vector doesnrsquot
change direction in a random fashion The simplest kind of polarization is linear
polarization in which the electric field vector oscillates back and forth along the
same axis in space Polarized light can be produced from normal unpolarized light by
selective absorption (as in Polaroid sunglasses) by passing light through crystals that
have different indices of refraction for different polarizations (double refraction) by
scattering (the blue sky is polarized) and by reflection (glare) Polarization by
reflection occurs when light reflects from a shiny insulating (non-metallic) surface
The amount of polarization is greatest for reflection at Brewsterrsquos angle
2
1
tan P
n
n
where θP is the incidence angle of light from medium 1 onto medium 2 and where n1
and n2 are the indices of refraction for the media (In our everyday experience n1 = 1
since the light comes in through the air and n2 is the index of refraction of the shiny
insulating material producing the glare eg water glass plastic paint etc)
Malusrsquos Law
The intensity of transmitted polarized light through a perfect polarizer is related to the
incident intensity of polarized light by Malusrsquos law
2cosoI I
where Io is the intensity of the incident light and where θ is the angle between the
electric field vector in the incident wave and the transmission axis the polarizer
24
Serway Chapter 39
Principles of Relativity
All of the weirdness of relativity flows from two simple principles
(1) The laws of physics must be the same in all inertial (non-accelerating) reference
frames
(2) The speed of light in vacuum has the same value in all inertial reference frames
Note that (1) seems reasonable but (2) is very odd It says that if two space ships are
approaching each other at nearly the speed of light and a laser pulse is shot from ship
1 toward ship 2 then when the people on ship 2 measure the speed of the pulse as it
goes by it is moving at 3 times 108 ms the same as if the ships were stationary
Simultaneity
If observer 1 sees two events in her own frame as simultaneous at two different
locations a moving observer 2 will see these two events happening at different times
Relativistic gamma
The factor γ (gamma) appears regularly in the formulas of relativity
2
2
1
1 vc
where v is the relative speed between two inertial frames
Time Dilation
If observer 1 sees two events at the same location in space separated by time Δtp in his
own frame then observer 2 moving at speed v relative to observer 1 will see these
two events separated by a longer time Δt
pt t
Moving clocks run slow
Length Contraction
If observer 1 measures the length of an object along the x-direction at some instant of
time in her frame to be Lp then observer 2 moving in the x-direction at speed v will
measure the length L of the object to be shorter
pLL
25
Moving meter sticks are short
Velocity Addition
If a particle is moving at velocity vac relative to frame c and if frame c is moving at
speed vcb with respect to frame b (with both velocities directed along the same line)
what would be the velocity of the particle in frame b In classical physics the answer is
ab ac cbv v v
but if any of these velocities are an appreciable fraction of the speed of light the
answer changes
21ac cb
abac cb
v Vv
v v c
In using this equation make sure that vac has the same sign in the numerator and in the
denominator and that vbc also has the same sign in both places Note that this equation
is different (and in our opinion easier to understand) than the equations in the text
Momentum and Energy
The momentum and total energy of a particle moving at velocity v (its speed is v v )
are given by
22
2 2
22
1 1 vvc c
m mcm E mc
vp v
Rest Energy
When a particle is at rest (v = 0 and γ = 1) then E = mc2 is called the rest energy It
includes the potential energy of the particle so that if we change the potential energy
of a particle we change its rest mass
Kinetic Energy The kinetic energy is the difference between the total energy and the
rest energy
2 2K mc mc
For v ltlt c this reduces to K = mv22
General Relativity
Einsteinrsquos principle of equivalence in the theory of general relativity is In the vicinity
of a point a gravitational field is equivalent to an accelerated frame of reference in
26
the absence of gravitational effects For example there is no difference whatever in
the physics experienced by an astronaut on earth and one who is in interstellar space
far away from any stars or planets but who has just turned on her spaceshiprsquos engines
and is accelerating at 98 ms2
27
Serway Chapter 40
Photons
We now know that even though light behaves like a wave it is better described as
consisting of small packets of energy called photons The energy of a photon is related
to its frequency by
E hf
where h is Planckrsquos constant
346626 10 J sh
Photons also carry momentum given by
hf hp
c
Since the amount of energy in each photon is fixed the difference between dim light
and intense light (of the same wavelength) is that dim light consists of fewer photons
than intense light
Photoelectric Effect
In the photoelectric effect light shining onto a metal surface gives the electrons in the
metal enough energy to escape and be detected It requires a certain amount of energy
(called the work function typically = 3 ndash 5 eV) just to kick the electron out of the
metal so the light must deliver at least this much energy to an electron to produce the
effect It is observed that red light no matter how intense never produces electrons
But ultraviolet light even if quite dim will eject electrons from the metal Since red
light consists of 2 eV photons and ultraviolet light has photons with energies around
4-6 eV the photon idea explains the behavior of the photoelectric effect The
maximum energy that an ejected electron can have is
maxK hf
where f is the photon frequency The minimum frequency that light can have and cause
electrons to be ejected is called the cutoff energy and is found by setting Kmax = 0
cf h
28
Compton Effect
When high-frequency light interacts with free electrons the scattered light does not
have the same wavelength as the incident light contrary to what classical
electromagnetism would predict But the photon picture in which photons have
momentum and energy allows us to treat scattering as a collision between the photon
and the electron In this collision the electron and the scattered photon both have a
different momentum and energy than they did before And since p = hλ for a photon
if the momentum is different the wavelength will also be different
after before 1 cose
h
m c
where the angle θ is the angle between the incoming photon direction and the
direction of the scattered photon
Particles are Waves
Since photons behave like particles sometimes it is not surprising that elementary
particles can behave like waves sometimes The wavelength of a particle with
momentum p is given by
h
p
Wave-particle Duality
Both photons and elementary particles have a dual nature sometimes they behave like
particles and sometimes they behave like waves It is hard for us to comprehend the
nature of such an object by making mental pictures but experiments definitively show
that this is the case Since we have no direct experience with photons and elementary
particles (because their energies are so much smaller than the energies of the everyday
objects we encounter) it is perhaps not surprising that we have a hard time forming a
mental picture of how they behave
Electron Interference
Since an electron is both a particle and a wave just like a photon it should be able to
produce an interference pattern This is observed If an electron beam is shot at two
closely-spaced slits and if the electrons that pass through the slits are detected
downstream it is found that there are some locations where electrons are never detected
29
and others where lots of electrons are detected The pattern is exactly the same as the
one observed for light waves provided that we use the electron wavelength λ = hp in
place of the wavelength of light This pattern is observed even though each electron is
detected as a single dot on the screen Only after many such dots are collected does the
pattern emerge And if we try to understand how this effect could possibly work by
looking closely at each slit to see which one the electron came through the pattern
disappears the act of measurement destroys the interference This means that each
single electron somehow comes through both slits (as a wave would)
Uncertainty Principle
In classical physics we always imagine that the positions and momenta of moving
particles have definite values It might be hard to measure them but surely at each
instant of time a particle should be precisely located at some point in space and have a
similarly precise momentum This turns out not to be true Instead both position and
momentum are required to be uncertain with their uncertainties Δx and Δp satisfying
the Heisenberg uncertainty relation
2x p
So if the particle were known to be precisely at some particular location (so that Δx =
0) we couldnrsquot know anything about its momentum (Δp = infin) and if its momentum
were exactly known we couldnrsquot know its position
There is a similar relation involving the particlersquos energy E and the time interval Δt
over which this energy is measured
2E t
This means that energy is not actually conserved if we are considering very small time
intervals and this brief non-conservation of energy has been observed
30
Serway Chapter 41
Wave Function ψ and Probability
Quantum mechanics does not predict exactly what an electron or a photon will do Instead
it specifies the wave function or probability amplitude ψ of an electron or a photon
This wave function is a complex-valued function of space and time whose squared
magnitude is the probability density P for finding a particle at a particular place in
space at a certain time
2P
where is the complex conjugate of ψ
For example the wave function of an electron with perfectly specified momentum p
would have a wavelength given by
h
p
and its probability amplitude would be proportional to
2i x ipxe e
The corresponding probability density would then be
21ipx ip ipxP e e e
which means that the electron is equally probable to be anywhere along the x axis
This is in accord with the uncertainty principle since we specified the momentum
precisely we canrsquot have any idea about the position of the electron
As another example you have probably seen ldquofuzzy ballrdquo drawings of electron
orbitals in a chemistry book These fuzzy balls are meant to indicate the distribution
of the probability density 2
P in the orbital
Particle in a Box
A simple example in which we can calculate the wave function is the case of a particle of
mass m constrained to be inside a 1-dimensional box between x = 0 and x = L In this
case the general wave function is a linear superposition of wave functions ψn of the form
sin niE tn
n xx t A e
L
31
where A is a positive constant where n = 1 2 3 and where the energy associated
with each of the quantum states ψn is given by
22
28n
hE n
mL
This wave function is zero at x = 0 and x = L which means that the particle will never
be found at the walls of the box The wave function has maximum values in the
interior and at these places the particle is most likely to be found and it also has
places where it is zero and at these places the particle will also never be found as
expressed by the formula
2sinn x
PL
The particle in the box is interfering with itself producing a probability interference
pattern across the box just like the interference patterns we studied with light and
sound
32
Serway Chapter 42
Atoms
Many people picture an atom as a miniature solar system where electrons orbit around
a massive nucleus at the center This picture is misleading because in atoms the wave
nature of electrons dominates The electrons form 3-dimensional standing waves
(called orbitals) centered at the nucleus And if you ask what is it thatrsquos waving the
answer is ψ the probability amplitude (see Chapter 41)
Standing waves on a string can occur only for certain frequencies (the fundamental
and higher harmonics) Similarly atomic orbitals occur only for certain energies For
the hydrogen atom the energies of the orbitals have the a particularly simple form
2
1136 eVnE
n
where n = 1 2 3 is called the principal quantum number For other atoms the
determination of the orbital energies requires numerical calculation by computers
Atomic Spectra
If an electron is somehow given extra extra energy (we say that it is excited) so that it
occupies a higher orbital it will eventually ldquofallrdquo back down to a lower orbital Each
time an electron falls to a lower orbital it loses the difference in energy between the
two orbitals in the form of a photon Since the orbital energies are discrete so are the
energy differences and so are the wavelengths of the emitted light The entire set of
these discrete wavelengths is called the atomic spectrum and it is unique to each type
of atom For hydrogen the spectrum can be simply written as
H 2 2
1 1 1
f i
Rn n
where RH = 10973732 times 107 m-1 For other atoms the spectrum cannot be expressed as
a formula
Orbital Angular Momentum Electrons in atoms also have quantized values of
angular momentum The orbital quantum number ℓ specifies the value of this
quantized angular momentum through the formula
1L
33
If we want to know the value of the angular momentum along some direction in space
say the z direction the answer is not L but rather
zL m
where mℓ is another quantum number which runs from
1 1m
This quantum number is important when an atom sits in a magnetic field
Spin Angular Momentum It has been found experimentally that electrons and other
charged particles also carry internal angular momentum which we call spin
Electrons have an intrinsic spin angular momentum s along a specified axis that is
extremely quantized it can only take on 2 possible values
1
2zs s
We thus say that electrons have spin s = 12 or that electrons are rdquospin one-half parti-
cles Since s is an angular momentum it obeys the same rule as that for orbital angular
momentum namely that if its value along some axis is s then its total magnitude is
given by
31
2S s s
I know this seems weird but quantum mechanics is weird The only excuse for this
bizarre way of looking at the world is that it predicts what happens in experiments
Exclusion Principle The answer to the question of how many electrons (or any other
spin one-half particle) can be in one particular quantum state was discovered by
Wolfgang Pauli and is called the exclusion principle
ldquoNo two electrons can ever be in the same quantum state therefore no two electrons
in the same atom can have the same set of quantum numbersrdquo
This is the reason that we have atoms with different properties instead of every atom
simple having all of its electrons in the ground state All of the variety we see around
us in the world is the result of chemical differences and these differences would not
exist unless electrons obeyed this important principle The entire structure of the
periodic table (see pages 1377-1379 in Serway) is an expression of this principle
34
Serway Chapter 44
Nuclear Properties
The nucleus is composed of protons with charge +e and mass mp = 1007226 u and of
neutrons with zero charge and mass mn = 1008665 u where u is one atomic mass unit
-271 u=1660540 10 kg
The atomic number Z counts the number of protons in a nucleus while the neutron
number N counts the number of neutrons The mass number A is the sum of the two
A N Z
Protons and nuetrons have about the same mass and some times called baryons (heavy
ones) or nucleons (partcles in the nucleus) A is sometimes called the baryon
number The chemical elements are distinguished by Z ie hydrogen has Z = 1 iron
has Z = 26 uranium has Z = 92 etc But for a given Z there might be several nuclei
with different numbers of neutrons These nuclear siblings are called isotopes
These numbers are used to label nuclei according to the pattern
5626 ie FeA
Z X
denotes the isotope of iron with N = 56 ndash 26 = 30 neutrons
The nucleus is roughly spherical with a radius given approximately by
1 3 150 0where 12 10 mr r A r
Nuclear Stability
Because the positively charged protons electrically repel each other with an enormous
force at distances as small as 10-15 m there must be some really strong force that that
overcomes electrical repulsion to hold protons and neutrons together This force is
called with some lack of imagination the strong force It is a very short-range force
(it only acts over a distance of about 2 x 10-15 m) and attracts protons to protons
neutrons to neutrons and neutrons to protons But in spite of this strong nuclear force
the coulomb repulsion of the protons is still present so anything that might keep the
protons from being right next to each other would help keep the nucleus from
35
exploding This role is played by the neutrons and for nuclei with Z le 20 the stable
nuclei roughly have N = Z
For Z gt 20 the coulomb repulsion force is more powerful and more neutrons are
needed to dilute it up to about N = 15Z around Z = 80 For Z gt 83 no amount of
neutrons can help and these nuclei are unstable (radioactive) There is an ldquoisland of
(relative) stabilityrdquo around Z=90-92 (thorium and uranium respectively) These
elements have one or two isotopes with half-lifes of billions of years so there are
substantial amounts of such elements on Earth This fact makes it possible to have
practical fission devices
Radioactive Decay
There are three types of radioactive decay
Alpha decay the nucleus kicks out a helium nucleus (N = 2 Z = 2 A = 4) So if the
original nucleus is called X and the new nucleus is called Y then the decay would look
like this
4 42 2X Y+ HeA A
Z Z
Beta decay the nucleus either kicks out an electron (endash) or its positively-charged
antimatter twin the positron (e+) plus either an electron neutrino v or an electron
anti-neutrino v
1X Y+e electron decayA AZ Z v
1X Y+e positron decayA AZ Z v
A neutrino is a particle with no charge hardly any mass (much less than the electron
mass) and interacts so weakly with matter that most neutrinos upon encountering the
planet earth just pass right through it as if it werenrsquot there
Gamma decay the nucleons in the nucleus X are in an excited energy state X
(perhaps as a result of having undergone alpha or beta decay) and they drop down to a
lower energy state shedding the energy as a high frequency photon
X XA AZ Z
36
This process is exactly analogous to the way that the electrons in atoms emit photons
Decay Rate and Half Life
There is no way to predict exactly when an unstable or excited nucleus will decay but
there is an average rate at which this decay occurs called the decay constant λ The
meaning of this constant is that if there are a large number N of nuclei in a sample
then the number of decays per second that will be observed (called the decay rate R)
is R = λN In mathematical language
dMR N
dt
This simple differential equation has for its solution
0tN t N e
where N0 is the number of nuclei in the sample at time t = 0
The half-life is the time it takes for half of the nuclei in the sample to decay and is
related to the decay constant by
1 2
ln 2 0693T
Disintegration Energy
When a nucleus decays it is making a transition to an overall state of lower energy
which means according to Einsteinrsquos famous formula E = mc2 that the sum of the
masses after the decay must be less than the mass before with the lost mass appearing
as kinetic energy among the decay products For example in alpha decay this kinetic
energy called the disintegration energy Q is given by
2X YQ M M M c
37
Serway Chapter 45
Nuclear energy
There are two ways to extract energy from the nucleus fission and fusion For nuclei
with Z greater than 26 breaking the nucleus apart into pieces leads to a lower
2mc energy than the original nucleus so energy can be extracted by fission For
nuclei with Z less than 26 a lower energy is achieved by combining nuclei so energy
can be extracted by fusion
Fission
Since neutrons have no charge they are not repelled from nuclei as protons are for
this reason their behavior is the key to understanding how fission works
Fast neutrons mostly bounce off other nuclei with each collision slowing the neutron
down This elastic energy loss is most effective if the other nuclei have low mass (like
hydrogen) and these materials are called moderators because of their ability to slow
down fast neutrons
The reason that slowing neutrons is important is that slow neutrons are much more
likely to be absorbed by a nucleus which then leads to nuclear reactions of various
kinds For a few very large nuclei like uranium-235 and some plutonium isotopes
absorption of a slow neutron causes the nucleus to split into two large fragments plus
2 or 3 fast neutrons (energy is released in the process too) Thus one slow neutron
can produce energy plus 2 or 3 more neutrons which if moderated from fast to slow
can split 2 or 3 more nuclei leading to a runaway chain reaction If the reaction is
allowed to proceed unchecked it produces a large explosion If a neutron absorbing
material is added to the mix (like the cadmium in reactor control rods) it is possible to
keep the reaction under control and to extract the released energy as heat to drive
steam turbines and produce electricity
Fusion
Fusion involves mashing two nuclei together and since they are both charged and repel
each other this reaction is much harder to make go The nuclei must have enough energy to
overcome the coulomb repulsion which is why this reaction requires a high temperature
(hundreds of millions of degrees K) like that in the sun (which is burning hydrogen to
helium via fusion) or like that in the center of the fission explosion that is used to detonate
38
a hydrogen bomb
This reaction is of interest for power production in spite of this difficult temperature
requirement because of the abundance of fusion fuel on the planet There are about
012 g of deuterium ( 21H ) in every gallon of water on earth and it only costs about 4
cents to extract it The fusion energy available from this minuscule amount of
deuterium would run a 1000-Megawatt power plant for 10 seconds Water is so
abundant on earth that if fusion were to work we would have an essentially
inexhaustible source of energy
So why donrsquot we have fusion power plants Well the fuel is cheap but the match is
incredibly expensive The only way we know to control this difficult high-temperature
reaction is with large and expensive pieces of equipment involving either large
magnetic fields and complex high-power electromagnetic antennas or with gigantic
(football-fieldsized) laser facilities involving more than a hundred of the highest-
energy lasers ever built Power plants based on these current methods for controlling
fusion are unattractive to the fiscally-minded people who run the electric power
industry Hopefully better designs will be discovered as experiments continue
39
c actinium 68 Er erbium 101 Md mendelevium 104 Rf rutherfordium l aluminum 63 Eu europium 80 Hg mercury 62 Sm samarium m americium 100 Fm fermium 42 Mo molybdenum 21 Sc scandium b antimony 9 F fluorine 60 Nd neodymium 106 Sg seaborgium r argon 87 Fr francium 10 Ne neon 34 Se selenium s arsenic 64 Gd gadolinium 93 Np neptunium 14 Si silicon t astatine 31 Ga gallium 28 Ni nickel 47 Ag silver a barium 32 Ge germanium 41 Nb niobium 11 Na sodium k berkelium 79 Au gold 7 N nitrogen 38 Sr strontium e beryllium 72 Hf hafnium 102 No nobelium 16 S sulfur i bismuth 108 Hs hassium 76 Os osmium 73 Ta tantalum h bohrium 2 He helium 8 O oxygen 43 Tc technetium
boron 67 Ho holmium 46 Pd palladium 52 Te tellurium r bromine 1 H hydrogen 15 P phosphorus 65 Tb terbium d cadmium 49 In indium 78 Pt platinum 81 Tl thallium a calcium 53 I iodine 94 Pu plutonium 90 Th thorium f californium 77 Ir iridium 84 Po polonium 69 Tm thulium
carbon 26 Fe iron 19 K potassium 50 Sn tin e cerium 36 Kr krypton 59 Pr praseodymium 22 Ti titanium s cesium 57 La lanthanum 61 Pm promethium 74 W tungsten l chlorine 103 Lr lawrencium 91 Pa protactinium 92 U uranium r chromium 82 Pb lead 88 Ra radium 23 V vanadium o cobalt 3 Li lithium 86 Rn radon 54 Xe xenon u copper 71 Lu lutetium 75 Re rhenium 70 Yb ytterbium m curium 12 Mg magnesium 45 Rh rhodium 39 Y yttrium b dubnium 25 Mn manganese 37 Rb rubidium 30 Zn zinc y dysprosium 109 Mt meitnerium 44 Ru ruthenium 40 Zr zirconium s einsteinium
40
UNITS The SI units are given in parentheses Other commonly used units are given in terms of the SI units Symbols conform with the recommendations of the American National Standards Institute (ANSI) and the American Institute of Physics (AIP) Commonly used multiples of SI units
prefix symbol factor tera T 1012 giga G 109
mega M 106 kilo k 103
centi c 10-2 milli m 10-3
micro μ 10-6 nano n 10-9 pico p 10-12
femto f 10-15
length meter (m) angstrom 1 Ǻ = 10-10 m inch 1 in = 254 x 10-2 m foot 1 ft = 03048 m mile 1 mi = 1609 m
mass kilogram (kg) atomic mass unit 1 u = 1661 x 10-27 kgslug 1 slug = 1459 kg
force newton (N = kg bull ms2) dyne 1 dyn = 10-5 N pound 1 lb = 4448 N
pressure Pascal (Pa = kgm bull s2) atmosphere 1 atm = 1013 x 105 Pa poundssquare inch 1 psi = 6895 Pa cm of mercury 1 cm Hg = 1333 Pa bar 1 bar = 1000 x 105 Pa torr 1 torr = 1333 Pa
time second (s) minute 1 min = 60 s hour 1 h =3600 s
frequency hertz (Hz = s-1) radianssecond 1 rads = 12π Hz
energy joule (J = kg bull m2s2) erg 1 erg = 10-7 J electron volt 1 eV = 1602 x 10-19 J calorie 1 cal = 4187 J kilowatt-hour 1 kWh = 36 x 106 J British thermal unit 1 Btu = 1055 J
power watt (W = kg bull m2s3) horsepower 1 hp = 7457 W
charge coulomb (C = A bull s)
electric potential volt (V = kg bull m2 s3 bull A)
current ampere (A)
resistance ohm (Ω = kg bull m2s3 bull A2)
capacitance farad (F = s4 bull A2kg bull m2)
magnetic field tesla (T = kgs2 bull A) gauss 1 G = 10-4 T
magnetic flux weber (Wb = kg m2s2 bull A) maxwell 1 Mx = 10-8 Wb
magnetic inductance henry (H = kg bull m2s2 bullA2)
temperature kelvin (K) degrees Celsius 0degC = 27315 K
angle radian (rad) degree 1 deg = π180 rad revolution 1 rev = 2π rad
41
Some Physical Constantsa Quantity Symbol Valueb Atomic mass unit μ 1660 538 73 (13) x 10-27 kg
931494 013 (37) MeVc2 Avogadros number NA 6022 141 99 (47) x 1023 particlesmol Avogadrorsquos number 6022 x 1023 mol Bohr magneton
2B e
em 9274 008 99 (37) x 10-24 JT
Bohr magneton μB 927 x 10-24 JT Bohr radius
0
2
2e em e k
a 5291 772 083 (19) x 10-11 m
Boltzmanns constant B A
RNk 1380 650 3 (24) x 10-23 JK
Boltzmannrsquos constant kB 1380 x 10-23 JK Compton wavelength
C eh
m c 2426 310 215 (18) x 10-12 m
Deuteron mass md 3343 583 09 (26) x 10-27 kg 2013 553 212 71 (35) u
electron charge e 1602 x 10-19 C Electron mass me 9109 381 88 (72) x 10-31 kg
5485 799 110 (12) x 10-4 u 0510 998 902 (21) MeVc2
electron mass 911 x 10-31 kg Electron volt eV 1602 176 462 (63) x 10-19 J Elementary charge e 1602 176 462 (63) x 10-19 C Gas constant R 8314 472 (15) JKmol Gravitational constant G 6673 (10) x 10-11 Nm2kg2 Neutron mass mn 1674 927 16 (13) x 10-27 kg
1008 664 915 78 (55) u 939565 330 (38) MeVc2
neutron mass 1675 x 10-27 kg Nuclear magneton
2n p
em 5050 783 17 (20) x 10-27 JT
permeability constant μ0 1257 x 10-6 Hm permittivity constant ε0 8854 x 10-12 Fm Plancks constant h
2h
6626 068 76 (52) x 10-34 Js 1054 571 596 (82) x 10-34 Js
Planckrsquos constant h ħ
6626 x 10-34 Js 1055 x 10-34 Js
Proton mass mp 1672621 58 (13) x 10-27 kg 1007 276 466 88 (13) u 938271 998 (38) MeVc2
proton mass 1673 x 10-27 kg Rydberg constant RH 1097 373 156 854 9 (83) x 107 m-1 Speed of light in vacuum c 2997 92458 x 108 ms (exact) speed of light c 300 x 108 ms a
These constants are the values recommended in 1998 by CODATA based on a least-squares adjustment of data from different measurements For a more complete list see P J Mohr and B N Taylor Rev Mod Phys 72351 2000 b The numbers in parentheses for the values above represent the uncertainties of the last two digits
NOTE The ones in red are the ones from Appendix 13 The ones in yellow are the ones that are from the Serway book table but that were already on the Appendix 13 table
42
INDEX
Absolute zero 9
Adiabatic 17
Adiabatic exponent 18
Adiabatic process 17
Alpha decay 46
Amplitude 4
Angle of reflection 25
Angular frequency 3
Angular magnification 30
Angular momentum atomic 43
Angular size 29
Archimedes Principle 1
Atomic spectra 43
Atoms 43
Avogadrorsquos number 10
Beats 8
Bernoullirsquos Equation 2
Beta decay 46
Bifocals 29
Brewsterrsquos angle 34
British Thermal Unit Btu 11
Buoyancy 2
calorie 11
Calorie 11
Camera 28
Carnot cycle 21
Carnot efficiency 22
Celsius scale 9
Ciliary muscle 29
Coefficient of performance 21
Compton effect 39
Constructive interference 7
Continuity equation of 2
Contraction length 35
Convection 14
Curved mirrors 27
Decay constant 46
Decay rate radioactive 46
Decibel scale 5
Degrees of freedom 16
Density 1
Destructive interference 7
Diffraction single slit 33
Diffraction grating 33
Dilation time 35
Disintegration energy 47
Dispersion of light 26
Doppler effect 6
Double slit interference 31
Efficiency 20
Electron interference 39
Emissivity 15
Energy nuclear decay 47
Energy relativistic 36
43
Engineering work 20
Entropy 22
Entropy ideal gas 24
Equipartition of energy 18
Equivalence principle 36
Exclusion principle 44
Expansion thermal 9
Expansion coefficient linear 9
Eye 29
Fahrenheit scale 9
Far point 29
Farsightedness 29
First Law of Thermodynamics 12
Fission 48
Flux volume 2
Free expansion 23
Fringes 31
Fusion 48
Fusion heat of 11
Gamma relativistic 35
Gamma decay 46
General relativity 36
Half-life 46
Heat 11
Heat capacity 11
Heat conduction 13
Heat engine 20
Heat of fusion 11
Heat of vaporization 11
Heat pump 20
Heats of transformation 11
Hydrostatics 1
Ideal Gas Law 10
Images real and virtual 27
Intensity sound 5
Interference 7
Interference two-slit 31
Internal energy 12
Internal energy degrees of freedom 16
Irreversible process 19 23
Isotopes 45
Joule 11
Kelvin scale 9
Kinetic energy relativity 36
Kinetic theory 16
Latent heat 11
Length contraction 35
Linear expansion coefficient 9
Linear polarization 34
Linear superposition 7
Longitudinal wave 3
Loudness 5
Magnification lateral 27
Magnifying glass 30
Malusrsquos law 34
Microscope 30
Momentum relativistic 36
Muscle ciliary 29
44
Musical instruments 8
Musical scale 8
Near point 29
Nearsightedness 29
Nonlinear 7
Nuclear energy 48
Nuclear properties 45
Nuclear stability 45
Octave 8
Optical resolution 33
Orbital quantum number ℓ 43
Orbitals 43
Particle in a Box 41
Particles are waves 39
Pascalrsquos Principle 1
Period 3
Photoelectric effect 38
Photons 38
Pitch 8
Polarization 34
Power sound 5
Presbyopia 29
Pressure 1
Principal quantum number n 43
Principle of equivalence 36
Principle of linear superposition 7
Probability amplitude ψ 41
Processes thermodynamic 13
R-value 14
Radiation thermal 14
Radioactive decay 46
Radioactive decay rate 46
Ray tracing 27
Rayleighrsquos criterion 33
Reading glasses 29
Real image 27
Refraction 25
Refrigerator 20
Relativistic gamma 35
Relativity principles 35
Resolved for light sources 33
Rest energy 36
Reversible process 19 22
Rope wave speed 4
Second Law of Thermodynamics 19
Shock waves 6
Simple magnifier 30
Simultaneity 35
Single slit diffraction 33
Snellrsquos law 25
Sound speed 5
Specific heat 11
Spin Angular momentum 44
Standing waves 7
Stefanrsquos law 14
Strong force 45
Telescope 30
Temperature 9
45
Temperature Scales 9
Thermal conductivity 14
Thermal energy 12
Thermal expansion 9
Thin film interference 31
Thin lenses 28
Time dilation 35
Tone musical 8
Total internal reflection 25
Transverse wave 3
Traveling Waves 3
Two-slit interference 31
Uncertainty principle 40
Vaporization heat of 11
Velocity addition relativity 36
Virtual image 27
Volume flux 2
Wave function 41
Wave Function ψ and Probability 41
Wave speed 3
Wave-particle duality 39
Wavelength 3
Wavenumber 3
Wien Displacement Law 16
Work 11
11
The efficiency of a Carnot engine is given by the very simple formula
1 cC
h
Te
T
where the temperatures must in be Kelvin No real engine can be more efficient than
this
The coefficient of performance of a Carnot refrigerator or heat pump in cooling mode
is given by
COP cooling mode cC
h c
T
T T
and no real refrigerator can have a coefficient of performance greater than this
The coefficient of performance of a Carnot heat pump in heating mode is
COP heating mode hC
h c
T
T T
So why donrsquot we just use these wonderful Carnot engines and have perfect efficiency
To make the reversible steps in the cycle really reversible they would have to occur
infinitely slowly So the price you pay for making a perfect engine is that it takes
forever to get it to do any work
Entropy
The entropy of a system is defined in terms of its molecular makeup and measures
roughly the disorder of the system If the system is packed into a very small volume
then it is quite ordered and the entropy will be low If it occupies a large volume the
entropy is high (To see what this has to do with disorder note that socks in a drawer
occupy a small volume while socks on the bed in the corner by the door and
hanging from the chandelier occupy a large volume) If the system is very cold then
the molecules hardly move and may even reach out to each other and form a crystal
This is a highly ordered state and therefore has low entropy If the system is very hot
with rapidly speeding molecules crashing into the container walls and bouncing off
each other things are disordered and the entropy is high
It is possible to calculate the entropy of a system in terms of its macroscopic thermody-
namic properties ie pressure volume temperature number of moles etc The key to this
calculation is the concept of a reversible process A reversible process is one that is
12
carried out without friction and so slowly that the process can be reversed at any stage by
making an infinitesimal change in the environment of the system The slow expansion of the
gas at Th in the Carnot cycle with tiny amounts of heat being transferred to the environment
is reversible If we turned around and began slowly to compress the gas would just slowly
exhaust heat to the environment in the exact reverse way that it absorbed it during expansion
Most processes however are irreversible For example if a gas-filled box were suddenly
increased in size so that the particles were free to wander into the void created by the sudden
expansion then the gas would eventually fill the new volume uniformly at the same
temperature as before the expansion (The temperature is unchanged in this imaginary
process because the kinetic energy of the molecules would be unaffected by such an
instantaneous expansion of the container walls) This imaginary but highly thought-
stimulating process is called a free expansion and it is impossible to reverse it During the
expansion we didnrsquot push on any of the molecules so reversing this process would mean
making them go back into their original volume without pushing on them they simply will
not cooperate to this extent Another way to see that just pushing them back where they came
from does not reverse the free expansion is to think about what would happen if we just
compressed either adiabatically or isothermally An adiabatic compression back to the
original volume would heat the gas above its original temperature and an isothermal
compression would require that heat be exhausted to the environment But the free expansion
involved neither temperature changes nor heat exchanges so neither of these two processes
is the reverse of the free expansion It is simply impossible to reverse this rapid expansion in
a way that takes us back to the initial state
It is possible to calculate the change of entropy for both reversible and irreversible processes
Letrsquos consider a reversible process first In a reversible process the entropy change is given
by the formula
dQS
T
where dQ is the amount of heat added to the system during a small step of the process
The total energy change during the process may then simply be calculated by integration
f f
f i i i
dQS S S dS
T
13
(Just as in the case of energy we are mainly interested in differences rather than in
absolute magnitudes)
This integration method does not work for irreversible processes For instance in the
rapid free expansion discussed above no heat is added to the system but its disorder
obviously goes up We need to find some other way to calculate the entropy The key
is the fact that the entropy of a system depends only on its current state and not at all
on how it arrived there So to calculate the entropy change in an irreversible process
first find out what the initial and final conditions of the process are Then invent a
reversible process that takes the system from the initial state to the final state Since
the entropy depends only on the state of system and not on the process the entropy
change for the reversible process is the same as that for the irreversible process
Entropy is thus like pressure or temperature if the pressure changes from Pl to to P2
then the pressure difference is P2 - Pl regardless of how the pressure change was
made This seems obvious for pressure but not for entropy only because you arent
comfortable with entropy yet
Entropy of an Ideal Gas
For n moles of an ideal gas the difference in entropy between a state with temperature
T and volume V and some standard state with temperature To and volume Vo is given
by
ln lno o Vo o
T VS T V S T V nC nR
T V
Entropy in a Phase Change
Calculating entropy change in a phase change like melting or boiling is easy It is
QT Where Q is in the latent heat for example the flows in to cause the melting and
T is the temperature of the phase change There is NO integral to do
14
Serway Chapter 35
Angle of Reflection
If plane waves are incident on a reflecting surface with the propagation direction of
the waves making angle θ1 with the normal direction to the surface then the reflected
angle θ1prime relative to the surface normal is simply
1 1
ie the incident angles and reflected angles are the same
Refraction
If plane waves traveling through medium 1 are incident on a plane interface between
medium 1 and medium 2 then the angle of incidence of the incoming wave θ1 and
the angle of refraction of the transmitted wave θ2 are related by Snellrsquos law
2 2
1 1
sin
sin
v
v
where v1 and v2 are the wave speeds in medium 1 and medium 2 The angle of
incidence and the angle of refraction are both measured between the wave propagation
direction and the normal to the interface In terms of indices of refraction in the case
of light waves Snellrsquos law takes the more familiar form
1 1 2 2sin sinn n
where ni = cvi
Total Internal Reflection
If a wave is incident from a medium of low wave speed into a medium of high wave
speed the law of refraction requires that the angle of refraction be greater than the
angle of incidence If the angle of refraction is required to be greater than 90deg then no
refracted wave can exist and total internal reflection occurs The critical incident
angle θc beyond which total internal reflection occurs is given by
2
1
sin c
n
n
15
Dispersion of Light
In addition to the speed of light varying from material to material it also varies with
wavelength within each material This means that the index of refraction is generally a
function of wavelength
c
nv
Since the wave speed is not constant such a medium is dispersive meaning in this
context that refraction actually disperses white light into its various colors because
Snellrsquos law gives a different angle for each wavelength In most materials the
variation with wavelength is quite small but this small effect is responsible for some
of the most spectacular color effects we ever see including rainbows a flashing
crystal chandelier and the colored fire of a diamond solitaire by candlelight
16
Serway Chapter 36
Real and Virtual Images
When light rays are focused at a certain plane producing an image if a sheet of white
paper is placed there we call the image a real image The images produced by film
projectors and overhead projectors are examples of real images
When light rays appear to come from a certain location but no image is produced
when a screen is placed there we say that there is a virtual image at that location For
instance when you look in a mirror it appears that someone is behind the mirror but
a screen placed back there in the dark would show nothing Your image in the mirror
is a virtual image
Ray Tracing
There are lots of rules about how to find the images in optical systems but the best
way to keep things straight is to learn how to draw the principal rays for curved
mirrors and lenses The rays for convex and concave mirrors are shown in Fig 3615
The rays for converging and diverging thin lenses are shown in Fig 3627 You should
memorize the principal rays and know how to use them to locate images
Curved Mirrors
The focal length of a curved mirror with radius of curvature R is given by
2
Rf
The relation between object distance p image distance q and focal length f is
1 1 1
p q f
If the mirror is a diverging mirror f should be negative and if q should turn out to be
negative the image is virtual
Lateral Magnification
The lateral magnification in an optical system is defined by the ratio of the image size
to the object size
17
Image height
Object height
qM
p
for curved mirrors
As usual there are sign conventions here too but this formula just gives the
magnitude It is better to keep keep track of upright images versus inverted images by
means of ray diagrams rather than by memorizing sign conventions
Thin Lenses
The focal length of a thin lens is related to the radii of curvature of the two faces R1
and R2 of the lens by
1 2
1 1 11n
f R R
Note that this formula differs from Eq (3611) in the text by not having a minus sign
between the two R-terms We like this form better because for a simple converging
lens like a magnifying glass we just use positive values of R for both surfaces If one
of the faces is concave producing divergence use a negative value for R And if a
surface is flat use R = infin If the face is flat the radius is infinite
The relation between the image object and focal distances for a thin lens is the same
as that for a curved mirror
1 1 1
q p f
Use a negative focal length if the lens is diverging
The lateral magnification for a thin lens is the same as for a curved mirror
Image height
Object height
qM
p
Camera
The lens system in a camera projects a real image of an object onto the film (or CCD
array in a digital camera) The position of the image is adjusted to be on the film by
moving the lens into or out of the camera
18
Eye
The eye is like a camera in that a real image is formed on the retina Unlike a camera
the image position is adjusted by changing the focal length of the lens This is done by
the ciliary muscle which squeezes the lens changing its shape
Near Point The near point is the closest distance from the eye for which the lens can
focus an image on the retina It is usually 18-25 cm for young persons
Far Point The far point is the greatest distance from the eye for which the lens can
focus an image on the retina For a person with normal vision the far point is at
infinity
Nearsightedness A person is nearsighted if their far point is at some finite distance
less than infinity This condition can be corrected with a lens that takes an object at
infinity and produces a virtual image at the personrsquos far point
Farsightedness and Presbyopia A person is farsighted if their near point is too far
away for comfortable near work like reading or knitting This can be corrected by a
lens which takes an object at a normal near point distance of 18-25 cm and produces a
virtual image at the personrsquos natural near point Presbyopia involves a similar
problem which nearly all people experience as they age The ciliary muscle becomes
too weak and the lens becomes too stiff to allow the eye to provide for both near and
far vision The solution for this problem is either reading glasses or bifocal lenses
Reading classes are just weak magnifying glasses mounted on eyeglass frames
Bifocal lenses are split into upper and lower halves The lower half is a lens which
gives the proper correction for near work and the upper half is a different lens for
proper focusing at infinity
Angular Size
When an object is brought closer to the eye it appears to be larger because the image
on the retina is larger The size of this image is directly proportional to the objectrsquos
angular size which is the angle subtended by the object measured from the center of
the lens of the eye In optical instruments which are to be used with the eye the
angular size of the final image is whatrsquos important because it determines how large
the image will appear to the viewer
19
Simple Magnifier
A simple magnifier is a single converging lens or magnifying glass It takes an object
closer to the eye than a normal near point and produces a virtual image at or beyond
this near point The angular magnification is defined to be the ratio of the angular size
when viewed through the lens to the angular size of the object when viewed at the
normal near point (without aid of the lens)
Microscope
This instrument has two lenses (1) The objective is near the object being viewed and
produces a greatly magnified real image (2) The eyepiece is a simple magnifier which
the viewer uses to closely examine the image from (1)
Telescope
This instrument also has two lenses (1) The objective at the front of the telescope
takes light from a distant object and produces a real inverted image (which is rather
small) near its focal point (2) This small real image is then examined by the eyepiece
functioning as a simple magnifier to produce a virtual image with a larger angular
size
20
Serway Chapter 37
Two-Slit Interference
If light is incident on two closely spaced narrow slits a pattern of light and dark
stripes is produced beyond the slits The bright stripes or fringes are caused by
constructive interference of the two waves coming from the slits Constructive
interference occurs whenever two waves arrive at a location in phase with each other
This occurs when the distance x1 from slit 1 to a point P on the screen and the
distance x2 from slit 2 to point P differ by in integral number of wavelengths
1 2 where 0 1 2x x m m
where λ is the wavelength of the light When x1 and x2 are much larger than the slit
spacing d this condition reduces to
sind m
where θ is the angle between the direction of the incident light and the direction of the
light arriving at the screen
Thin Films
When light is partially reflected and partially transmitted by a thin film of transparent
material it is possible to have interference between the wave reflected from the front
of the film and light reflected from the back of the film (The colored reflections from
the thin film of oil on the water in a rain-soaked parking lot are an example of this
effect) It is difficult to write down formulas that will work in all cases so we will
just review the important principles here
1 If the two reflected waves are in phase with each other the film has enhanced
reflection (constructive interference) but if the two reflected waves are out of phase
with each other reflection is diminished (destructive interference) Phase shifts occur
due to reflection and due to the extra path length through the film of the wave
reflected from the back of the film
2 The phase change due to reflection is determined by the difference in index of
refraction between the two media involved in the reflection If the wave is incident
21
from a medium with a low index of refraction into a medium with a high index of
refraction a phase change of 180deg occurs and the reflected wave is inverted If
incident from high to low no phase shift occurs and the reflected wave is non-
inverted
3 The extra path length through the film of the wave reflected from the back of the
film is equal to 2t where t is the thickness of the film (The incident light is assumed
to be normal to the surface of the film) The number of wavelengths contained in the
extra path length is equal to 2tλn where λn = λn is the wavelength of the light in the
film
4 Rules for reflection from thin films
If one ray is inverted and the other is not then we have
12 constructive
2 nt m
2 destructivent m
If either both rays are inverted or both are non-inverted we have
2 constructivent m
12 destructive
2 nt m
22
Serway Chapter 38
Diffraction Grating
A diffraction grating is simply a fancy version of two-slit interference with the two
slits replaced by thousands of slits Just as in the two-slit case the bright fringes
occur at angles given by
sin md
but in this case d the distance between neighboring slits is made to be very small
The effect of having many slits instead of two is to make each bright fringe highly
localized with wide dark regions between neighboring maxima
Single Slit Diffraction
When light passes through an opening in an opaque screen an interference pattern is
produced beyond the opening To understand why we may replace the single opening
by many small coherent sources of light These many sources interfere with each
other producing a pattern known as a diffraction pattern (Note that many authors do
not distinguish between interference and diffraction treating them as interchangeable
terms) If the opening is a slit of width a then the diffraction pattern far from the slit
will have a bright central maximum with a succession of minima and weaker maxima
on either side The angle between the incident direction and the minima is given by
sin ma
where m = plusmn1 plusmn2
If the opening is circular with diameter D the angle between the incident direction
and the first minimum is given by
sin 122D
Optical Resolution and Rayleighrsquos Criterion
Two point sources can just be resolved (distinguished from each other) if the peak of
the diffraction image of the first source overlies the first minimum of the diffraction
image of the second source For circular holes of the kind usually encountered in
23
optical devices this condition is approximately satisfied when the angular separation
between the two sources as viewed from the optical instrument is greater than or equal
to the critical angle
min 122D
where λ is the wavelength of the light and where D is the diameter of the aperture in
the instrument
Polarization
We say that an electromagnetic wave is polarized if its electric field vector doesnrsquot
change direction in a random fashion The simplest kind of polarization is linear
polarization in which the electric field vector oscillates back and forth along the
same axis in space Polarized light can be produced from normal unpolarized light by
selective absorption (as in Polaroid sunglasses) by passing light through crystals that
have different indices of refraction for different polarizations (double refraction) by
scattering (the blue sky is polarized) and by reflection (glare) Polarization by
reflection occurs when light reflects from a shiny insulating (non-metallic) surface
The amount of polarization is greatest for reflection at Brewsterrsquos angle
2
1
tan P
n
n
where θP is the incidence angle of light from medium 1 onto medium 2 and where n1
and n2 are the indices of refraction for the media (In our everyday experience n1 = 1
since the light comes in through the air and n2 is the index of refraction of the shiny
insulating material producing the glare eg water glass plastic paint etc)
Malusrsquos Law
The intensity of transmitted polarized light through a perfect polarizer is related to the
incident intensity of polarized light by Malusrsquos law
2cosoI I
where Io is the intensity of the incident light and where θ is the angle between the
electric field vector in the incident wave and the transmission axis the polarizer
24
Serway Chapter 39
Principles of Relativity
All of the weirdness of relativity flows from two simple principles
(1) The laws of physics must be the same in all inertial (non-accelerating) reference
frames
(2) The speed of light in vacuum has the same value in all inertial reference frames
Note that (1) seems reasonable but (2) is very odd It says that if two space ships are
approaching each other at nearly the speed of light and a laser pulse is shot from ship
1 toward ship 2 then when the people on ship 2 measure the speed of the pulse as it
goes by it is moving at 3 times 108 ms the same as if the ships were stationary
Simultaneity
If observer 1 sees two events in her own frame as simultaneous at two different
locations a moving observer 2 will see these two events happening at different times
Relativistic gamma
The factor γ (gamma) appears regularly in the formulas of relativity
2
2
1
1 vc
where v is the relative speed between two inertial frames
Time Dilation
If observer 1 sees two events at the same location in space separated by time Δtp in his
own frame then observer 2 moving at speed v relative to observer 1 will see these
two events separated by a longer time Δt
pt t
Moving clocks run slow
Length Contraction
If observer 1 measures the length of an object along the x-direction at some instant of
time in her frame to be Lp then observer 2 moving in the x-direction at speed v will
measure the length L of the object to be shorter
pLL
25
Moving meter sticks are short
Velocity Addition
If a particle is moving at velocity vac relative to frame c and if frame c is moving at
speed vcb with respect to frame b (with both velocities directed along the same line)
what would be the velocity of the particle in frame b In classical physics the answer is
ab ac cbv v v
but if any of these velocities are an appreciable fraction of the speed of light the
answer changes
21ac cb
abac cb
v Vv
v v c
In using this equation make sure that vac has the same sign in the numerator and in the
denominator and that vbc also has the same sign in both places Note that this equation
is different (and in our opinion easier to understand) than the equations in the text
Momentum and Energy
The momentum and total energy of a particle moving at velocity v (its speed is v v )
are given by
22
2 2
22
1 1 vvc c
m mcm E mc
vp v
Rest Energy
When a particle is at rest (v = 0 and γ = 1) then E = mc2 is called the rest energy It
includes the potential energy of the particle so that if we change the potential energy
of a particle we change its rest mass
Kinetic Energy The kinetic energy is the difference between the total energy and the
rest energy
2 2K mc mc
For v ltlt c this reduces to K = mv22
General Relativity
Einsteinrsquos principle of equivalence in the theory of general relativity is In the vicinity
of a point a gravitational field is equivalent to an accelerated frame of reference in
26
the absence of gravitational effects For example there is no difference whatever in
the physics experienced by an astronaut on earth and one who is in interstellar space
far away from any stars or planets but who has just turned on her spaceshiprsquos engines
and is accelerating at 98 ms2
27
Serway Chapter 40
Photons
We now know that even though light behaves like a wave it is better described as
consisting of small packets of energy called photons The energy of a photon is related
to its frequency by
E hf
where h is Planckrsquos constant
346626 10 J sh
Photons also carry momentum given by
hf hp
c
Since the amount of energy in each photon is fixed the difference between dim light
and intense light (of the same wavelength) is that dim light consists of fewer photons
than intense light
Photoelectric Effect
In the photoelectric effect light shining onto a metal surface gives the electrons in the
metal enough energy to escape and be detected It requires a certain amount of energy
(called the work function typically = 3 ndash 5 eV) just to kick the electron out of the
metal so the light must deliver at least this much energy to an electron to produce the
effect It is observed that red light no matter how intense never produces electrons
But ultraviolet light even if quite dim will eject electrons from the metal Since red
light consists of 2 eV photons and ultraviolet light has photons with energies around
4-6 eV the photon idea explains the behavior of the photoelectric effect The
maximum energy that an ejected electron can have is
maxK hf
where f is the photon frequency The minimum frequency that light can have and cause
electrons to be ejected is called the cutoff energy and is found by setting Kmax = 0
cf h
28
Compton Effect
When high-frequency light interacts with free electrons the scattered light does not
have the same wavelength as the incident light contrary to what classical
electromagnetism would predict But the photon picture in which photons have
momentum and energy allows us to treat scattering as a collision between the photon
and the electron In this collision the electron and the scattered photon both have a
different momentum and energy than they did before And since p = hλ for a photon
if the momentum is different the wavelength will also be different
after before 1 cose
h
m c
where the angle θ is the angle between the incoming photon direction and the
direction of the scattered photon
Particles are Waves
Since photons behave like particles sometimes it is not surprising that elementary
particles can behave like waves sometimes The wavelength of a particle with
momentum p is given by
h
p
Wave-particle Duality
Both photons and elementary particles have a dual nature sometimes they behave like
particles and sometimes they behave like waves It is hard for us to comprehend the
nature of such an object by making mental pictures but experiments definitively show
that this is the case Since we have no direct experience with photons and elementary
particles (because their energies are so much smaller than the energies of the everyday
objects we encounter) it is perhaps not surprising that we have a hard time forming a
mental picture of how they behave
Electron Interference
Since an electron is both a particle and a wave just like a photon it should be able to
produce an interference pattern This is observed If an electron beam is shot at two
closely-spaced slits and if the electrons that pass through the slits are detected
downstream it is found that there are some locations where electrons are never detected
29
and others where lots of electrons are detected The pattern is exactly the same as the
one observed for light waves provided that we use the electron wavelength λ = hp in
place of the wavelength of light This pattern is observed even though each electron is
detected as a single dot on the screen Only after many such dots are collected does the
pattern emerge And if we try to understand how this effect could possibly work by
looking closely at each slit to see which one the electron came through the pattern
disappears the act of measurement destroys the interference This means that each
single electron somehow comes through both slits (as a wave would)
Uncertainty Principle
In classical physics we always imagine that the positions and momenta of moving
particles have definite values It might be hard to measure them but surely at each
instant of time a particle should be precisely located at some point in space and have a
similarly precise momentum This turns out not to be true Instead both position and
momentum are required to be uncertain with their uncertainties Δx and Δp satisfying
the Heisenberg uncertainty relation
2x p
So if the particle were known to be precisely at some particular location (so that Δx =
0) we couldnrsquot know anything about its momentum (Δp = infin) and if its momentum
were exactly known we couldnrsquot know its position
There is a similar relation involving the particlersquos energy E and the time interval Δt
over which this energy is measured
2E t
This means that energy is not actually conserved if we are considering very small time
intervals and this brief non-conservation of energy has been observed
30
Serway Chapter 41
Wave Function ψ and Probability
Quantum mechanics does not predict exactly what an electron or a photon will do Instead
it specifies the wave function or probability amplitude ψ of an electron or a photon
This wave function is a complex-valued function of space and time whose squared
magnitude is the probability density P for finding a particle at a particular place in
space at a certain time
2P
where is the complex conjugate of ψ
For example the wave function of an electron with perfectly specified momentum p
would have a wavelength given by
h
p
and its probability amplitude would be proportional to
2i x ipxe e
The corresponding probability density would then be
21ipx ip ipxP e e e
which means that the electron is equally probable to be anywhere along the x axis
This is in accord with the uncertainty principle since we specified the momentum
precisely we canrsquot have any idea about the position of the electron
As another example you have probably seen ldquofuzzy ballrdquo drawings of electron
orbitals in a chemistry book These fuzzy balls are meant to indicate the distribution
of the probability density 2
P in the orbital
Particle in a Box
A simple example in which we can calculate the wave function is the case of a particle of
mass m constrained to be inside a 1-dimensional box between x = 0 and x = L In this
case the general wave function is a linear superposition of wave functions ψn of the form
sin niE tn
n xx t A e
L
31
where A is a positive constant where n = 1 2 3 and where the energy associated
with each of the quantum states ψn is given by
22
28n
hE n
mL
This wave function is zero at x = 0 and x = L which means that the particle will never
be found at the walls of the box The wave function has maximum values in the
interior and at these places the particle is most likely to be found and it also has
places where it is zero and at these places the particle will also never be found as
expressed by the formula
2sinn x
PL
The particle in the box is interfering with itself producing a probability interference
pattern across the box just like the interference patterns we studied with light and
sound
32
Serway Chapter 42
Atoms
Many people picture an atom as a miniature solar system where electrons orbit around
a massive nucleus at the center This picture is misleading because in atoms the wave
nature of electrons dominates The electrons form 3-dimensional standing waves
(called orbitals) centered at the nucleus And if you ask what is it thatrsquos waving the
answer is ψ the probability amplitude (see Chapter 41)
Standing waves on a string can occur only for certain frequencies (the fundamental
and higher harmonics) Similarly atomic orbitals occur only for certain energies For
the hydrogen atom the energies of the orbitals have the a particularly simple form
2
1136 eVnE
n
where n = 1 2 3 is called the principal quantum number For other atoms the
determination of the orbital energies requires numerical calculation by computers
Atomic Spectra
If an electron is somehow given extra extra energy (we say that it is excited) so that it
occupies a higher orbital it will eventually ldquofallrdquo back down to a lower orbital Each
time an electron falls to a lower orbital it loses the difference in energy between the
two orbitals in the form of a photon Since the orbital energies are discrete so are the
energy differences and so are the wavelengths of the emitted light The entire set of
these discrete wavelengths is called the atomic spectrum and it is unique to each type
of atom For hydrogen the spectrum can be simply written as
H 2 2
1 1 1
f i
Rn n
where RH = 10973732 times 107 m-1 For other atoms the spectrum cannot be expressed as
a formula
Orbital Angular Momentum Electrons in atoms also have quantized values of
angular momentum The orbital quantum number ℓ specifies the value of this
quantized angular momentum through the formula
1L
33
If we want to know the value of the angular momentum along some direction in space
say the z direction the answer is not L but rather
zL m
where mℓ is another quantum number which runs from
1 1m
This quantum number is important when an atom sits in a magnetic field
Spin Angular Momentum It has been found experimentally that electrons and other
charged particles also carry internal angular momentum which we call spin
Electrons have an intrinsic spin angular momentum s along a specified axis that is
extremely quantized it can only take on 2 possible values
1
2zs s
We thus say that electrons have spin s = 12 or that electrons are rdquospin one-half parti-
cles Since s is an angular momentum it obeys the same rule as that for orbital angular
momentum namely that if its value along some axis is s then its total magnitude is
given by
31
2S s s
I know this seems weird but quantum mechanics is weird The only excuse for this
bizarre way of looking at the world is that it predicts what happens in experiments
Exclusion Principle The answer to the question of how many electrons (or any other
spin one-half particle) can be in one particular quantum state was discovered by
Wolfgang Pauli and is called the exclusion principle
ldquoNo two electrons can ever be in the same quantum state therefore no two electrons
in the same atom can have the same set of quantum numbersrdquo
This is the reason that we have atoms with different properties instead of every atom
simple having all of its electrons in the ground state All of the variety we see around
us in the world is the result of chemical differences and these differences would not
exist unless electrons obeyed this important principle The entire structure of the
periodic table (see pages 1377-1379 in Serway) is an expression of this principle
34
Serway Chapter 44
Nuclear Properties
The nucleus is composed of protons with charge +e and mass mp = 1007226 u and of
neutrons with zero charge and mass mn = 1008665 u where u is one atomic mass unit
-271 u=1660540 10 kg
The atomic number Z counts the number of protons in a nucleus while the neutron
number N counts the number of neutrons The mass number A is the sum of the two
A N Z
Protons and nuetrons have about the same mass and some times called baryons (heavy
ones) or nucleons (partcles in the nucleus) A is sometimes called the baryon
number The chemical elements are distinguished by Z ie hydrogen has Z = 1 iron
has Z = 26 uranium has Z = 92 etc But for a given Z there might be several nuclei
with different numbers of neutrons These nuclear siblings are called isotopes
These numbers are used to label nuclei according to the pattern
5626 ie FeA
Z X
denotes the isotope of iron with N = 56 ndash 26 = 30 neutrons
The nucleus is roughly spherical with a radius given approximately by
1 3 150 0where 12 10 mr r A r
Nuclear Stability
Because the positively charged protons electrically repel each other with an enormous
force at distances as small as 10-15 m there must be some really strong force that that
overcomes electrical repulsion to hold protons and neutrons together This force is
called with some lack of imagination the strong force It is a very short-range force
(it only acts over a distance of about 2 x 10-15 m) and attracts protons to protons
neutrons to neutrons and neutrons to protons But in spite of this strong nuclear force
the coulomb repulsion of the protons is still present so anything that might keep the
protons from being right next to each other would help keep the nucleus from
35
exploding This role is played by the neutrons and for nuclei with Z le 20 the stable
nuclei roughly have N = Z
For Z gt 20 the coulomb repulsion force is more powerful and more neutrons are
needed to dilute it up to about N = 15Z around Z = 80 For Z gt 83 no amount of
neutrons can help and these nuclei are unstable (radioactive) There is an ldquoisland of
(relative) stabilityrdquo around Z=90-92 (thorium and uranium respectively) These
elements have one or two isotopes with half-lifes of billions of years so there are
substantial amounts of such elements on Earth This fact makes it possible to have
practical fission devices
Radioactive Decay
There are three types of radioactive decay
Alpha decay the nucleus kicks out a helium nucleus (N = 2 Z = 2 A = 4) So if the
original nucleus is called X and the new nucleus is called Y then the decay would look
like this
4 42 2X Y+ HeA A
Z Z
Beta decay the nucleus either kicks out an electron (endash) or its positively-charged
antimatter twin the positron (e+) plus either an electron neutrino v or an electron
anti-neutrino v
1X Y+e electron decayA AZ Z v
1X Y+e positron decayA AZ Z v
A neutrino is a particle with no charge hardly any mass (much less than the electron
mass) and interacts so weakly with matter that most neutrinos upon encountering the
planet earth just pass right through it as if it werenrsquot there
Gamma decay the nucleons in the nucleus X are in an excited energy state X
(perhaps as a result of having undergone alpha or beta decay) and they drop down to a
lower energy state shedding the energy as a high frequency photon
X XA AZ Z
36
This process is exactly analogous to the way that the electrons in atoms emit photons
Decay Rate and Half Life
There is no way to predict exactly when an unstable or excited nucleus will decay but
there is an average rate at which this decay occurs called the decay constant λ The
meaning of this constant is that if there are a large number N of nuclei in a sample
then the number of decays per second that will be observed (called the decay rate R)
is R = λN In mathematical language
dMR N
dt
This simple differential equation has for its solution
0tN t N e
where N0 is the number of nuclei in the sample at time t = 0
The half-life is the time it takes for half of the nuclei in the sample to decay and is
related to the decay constant by
1 2
ln 2 0693T
Disintegration Energy
When a nucleus decays it is making a transition to an overall state of lower energy
which means according to Einsteinrsquos famous formula E = mc2 that the sum of the
masses after the decay must be less than the mass before with the lost mass appearing
as kinetic energy among the decay products For example in alpha decay this kinetic
energy called the disintegration energy Q is given by
2X YQ M M M c
37
Serway Chapter 45
Nuclear energy
There are two ways to extract energy from the nucleus fission and fusion For nuclei
with Z greater than 26 breaking the nucleus apart into pieces leads to a lower
2mc energy than the original nucleus so energy can be extracted by fission For
nuclei with Z less than 26 a lower energy is achieved by combining nuclei so energy
can be extracted by fusion
Fission
Since neutrons have no charge they are not repelled from nuclei as protons are for
this reason their behavior is the key to understanding how fission works
Fast neutrons mostly bounce off other nuclei with each collision slowing the neutron
down This elastic energy loss is most effective if the other nuclei have low mass (like
hydrogen) and these materials are called moderators because of their ability to slow
down fast neutrons
The reason that slowing neutrons is important is that slow neutrons are much more
likely to be absorbed by a nucleus which then leads to nuclear reactions of various
kinds For a few very large nuclei like uranium-235 and some plutonium isotopes
absorption of a slow neutron causes the nucleus to split into two large fragments plus
2 or 3 fast neutrons (energy is released in the process too) Thus one slow neutron
can produce energy plus 2 or 3 more neutrons which if moderated from fast to slow
can split 2 or 3 more nuclei leading to a runaway chain reaction If the reaction is
allowed to proceed unchecked it produces a large explosion If a neutron absorbing
material is added to the mix (like the cadmium in reactor control rods) it is possible to
keep the reaction under control and to extract the released energy as heat to drive
steam turbines and produce electricity
Fusion
Fusion involves mashing two nuclei together and since they are both charged and repel
each other this reaction is much harder to make go The nuclei must have enough energy to
overcome the coulomb repulsion which is why this reaction requires a high temperature
(hundreds of millions of degrees K) like that in the sun (which is burning hydrogen to
helium via fusion) or like that in the center of the fission explosion that is used to detonate
38
a hydrogen bomb
This reaction is of interest for power production in spite of this difficult temperature
requirement because of the abundance of fusion fuel on the planet There are about
012 g of deuterium ( 21H ) in every gallon of water on earth and it only costs about 4
cents to extract it The fusion energy available from this minuscule amount of
deuterium would run a 1000-Megawatt power plant for 10 seconds Water is so
abundant on earth that if fusion were to work we would have an essentially
inexhaustible source of energy
So why donrsquot we have fusion power plants Well the fuel is cheap but the match is
incredibly expensive The only way we know to control this difficult high-temperature
reaction is with large and expensive pieces of equipment involving either large
magnetic fields and complex high-power electromagnetic antennas or with gigantic
(football-fieldsized) laser facilities involving more than a hundred of the highest-
energy lasers ever built Power plants based on these current methods for controlling
fusion are unattractive to the fiscally-minded people who run the electric power
industry Hopefully better designs will be discovered as experiments continue
39
c actinium 68 Er erbium 101 Md mendelevium 104 Rf rutherfordium l aluminum 63 Eu europium 80 Hg mercury 62 Sm samarium m americium 100 Fm fermium 42 Mo molybdenum 21 Sc scandium b antimony 9 F fluorine 60 Nd neodymium 106 Sg seaborgium r argon 87 Fr francium 10 Ne neon 34 Se selenium s arsenic 64 Gd gadolinium 93 Np neptunium 14 Si silicon t astatine 31 Ga gallium 28 Ni nickel 47 Ag silver a barium 32 Ge germanium 41 Nb niobium 11 Na sodium k berkelium 79 Au gold 7 N nitrogen 38 Sr strontium e beryllium 72 Hf hafnium 102 No nobelium 16 S sulfur i bismuth 108 Hs hassium 76 Os osmium 73 Ta tantalum h bohrium 2 He helium 8 O oxygen 43 Tc technetium
boron 67 Ho holmium 46 Pd palladium 52 Te tellurium r bromine 1 H hydrogen 15 P phosphorus 65 Tb terbium d cadmium 49 In indium 78 Pt platinum 81 Tl thallium a calcium 53 I iodine 94 Pu plutonium 90 Th thorium f californium 77 Ir iridium 84 Po polonium 69 Tm thulium
carbon 26 Fe iron 19 K potassium 50 Sn tin e cerium 36 Kr krypton 59 Pr praseodymium 22 Ti titanium s cesium 57 La lanthanum 61 Pm promethium 74 W tungsten l chlorine 103 Lr lawrencium 91 Pa protactinium 92 U uranium r chromium 82 Pb lead 88 Ra radium 23 V vanadium o cobalt 3 Li lithium 86 Rn radon 54 Xe xenon u copper 71 Lu lutetium 75 Re rhenium 70 Yb ytterbium m curium 12 Mg magnesium 45 Rh rhodium 39 Y yttrium b dubnium 25 Mn manganese 37 Rb rubidium 30 Zn zinc y dysprosium 109 Mt meitnerium 44 Ru ruthenium 40 Zr zirconium s einsteinium
40
UNITS The SI units are given in parentheses Other commonly used units are given in terms of the SI units Symbols conform with the recommendations of the American National Standards Institute (ANSI) and the American Institute of Physics (AIP) Commonly used multiples of SI units
prefix symbol factor tera T 1012 giga G 109
mega M 106 kilo k 103
centi c 10-2 milli m 10-3
micro μ 10-6 nano n 10-9 pico p 10-12
femto f 10-15
length meter (m) angstrom 1 Ǻ = 10-10 m inch 1 in = 254 x 10-2 m foot 1 ft = 03048 m mile 1 mi = 1609 m
mass kilogram (kg) atomic mass unit 1 u = 1661 x 10-27 kgslug 1 slug = 1459 kg
force newton (N = kg bull ms2) dyne 1 dyn = 10-5 N pound 1 lb = 4448 N
pressure Pascal (Pa = kgm bull s2) atmosphere 1 atm = 1013 x 105 Pa poundssquare inch 1 psi = 6895 Pa cm of mercury 1 cm Hg = 1333 Pa bar 1 bar = 1000 x 105 Pa torr 1 torr = 1333 Pa
time second (s) minute 1 min = 60 s hour 1 h =3600 s
frequency hertz (Hz = s-1) radianssecond 1 rads = 12π Hz
energy joule (J = kg bull m2s2) erg 1 erg = 10-7 J electron volt 1 eV = 1602 x 10-19 J calorie 1 cal = 4187 J kilowatt-hour 1 kWh = 36 x 106 J British thermal unit 1 Btu = 1055 J
power watt (W = kg bull m2s3) horsepower 1 hp = 7457 W
charge coulomb (C = A bull s)
electric potential volt (V = kg bull m2 s3 bull A)
current ampere (A)
resistance ohm (Ω = kg bull m2s3 bull A2)
capacitance farad (F = s4 bull A2kg bull m2)
magnetic field tesla (T = kgs2 bull A) gauss 1 G = 10-4 T
magnetic flux weber (Wb = kg m2s2 bull A) maxwell 1 Mx = 10-8 Wb
magnetic inductance henry (H = kg bull m2s2 bullA2)
temperature kelvin (K) degrees Celsius 0degC = 27315 K
angle radian (rad) degree 1 deg = π180 rad revolution 1 rev = 2π rad
41
Some Physical Constantsa Quantity Symbol Valueb Atomic mass unit μ 1660 538 73 (13) x 10-27 kg
931494 013 (37) MeVc2 Avogadros number NA 6022 141 99 (47) x 1023 particlesmol Avogadrorsquos number 6022 x 1023 mol Bohr magneton
2B e
em 9274 008 99 (37) x 10-24 JT
Bohr magneton μB 927 x 10-24 JT Bohr radius
0
2
2e em e k
a 5291 772 083 (19) x 10-11 m
Boltzmanns constant B A
RNk 1380 650 3 (24) x 10-23 JK
Boltzmannrsquos constant kB 1380 x 10-23 JK Compton wavelength
C eh
m c 2426 310 215 (18) x 10-12 m
Deuteron mass md 3343 583 09 (26) x 10-27 kg 2013 553 212 71 (35) u
electron charge e 1602 x 10-19 C Electron mass me 9109 381 88 (72) x 10-31 kg
5485 799 110 (12) x 10-4 u 0510 998 902 (21) MeVc2
electron mass 911 x 10-31 kg Electron volt eV 1602 176 462 (63) x 10-19 J Elementary charge e 1602 176 462 (63) x 10-19 C Gas constant R 8314 472 (15) JKmol Gravitational constant G 6673 (10) x 10-11 Nm2kg2 Neutron mass mn 1674 927 16 (13) x 10-27 kg
1008 664 915 78 (55) u 939565 330 (38) MeVc2
neutron mass 1675 x 10-27 kg Nuclear magneton
2n p
em 5050 783 17 (20) x 10-27 JT
permeability constant μ0 1257 x 10-6 Hm permittivity constant ε0 8854 x 10-12 Fm Plancks constant h
2h
6626 068 76 (52) x 10-34 Js 1054 571 596 (82) x 10-34 Js
Planckrsquos constant h ħ
6626 x 10-34 Js 1055 x 10-34 Js
Proton mass mp 1672621 58 (13) x 10-27 kg 1007 276 466 88 (13) u 938271 998 (38) MeVc2
proton mass 1673 x 10-27 kg Rydberg constant RH 1097 373 156 854 9 (83) x 107 m-1 Speed of light in vacuum c 2997 92458 x 108 ms (exact) speed of light c 300 x 108 ms a
These constants are the values recommended in 1998 by CODATA based on a least-squares adjustment of data from different measurements For a more complete list see P J Mohr and B N Taylor Rev Mod Phys 72351 2000 b The numbers in parentheses for the values above represent the uncertainties of the last two digits
NOTE The ones in red are the ones from Appendix 13 The ones in yellow are the ones that are from the Serway book table but that were already on the Appendix 13 table
42
INDEX
Absolute zero 9
Adiabatic 17
Adiabatic exponent 18
Adiabatic process 17
Alpha decay 46
Amplitude 4
Angle of reflection 25
Angular frequency 3
Angular magnification 30
Angular momentum atomic 43
Angular size 29
Archimedes Principle 1
Atomic spectra 43
Atoms 43
Avogadrorsquos number 10
Beats 8
Bernoullirsquos Equation 2
Beta decay 46
Bifocals 29
Brewsterrsquos angle 34
British Thermal Unit Btu 11
Buoyancy 2
calorie 11
Calorie 11
Camera 28
Carnot cycle 21
Carnot efficiency 22
Celsius scale 9
Ciliary muscle 29
Coefficient of performance 21
Compton effect 39
Constructive interference 7
Continuity equation of 2
Contraction length 35
Convection 14
Curved mirrors 27
Decay constant 46
Decay rate radioactive 46
Decibel scale 5
Degrees of freedom 16
Density 1
Destructive interference 7
Diffraction single slit 33
Diffraction grating 33
Dilation time 35
Disintegration energy 47
Dispersion of light 26
Doppler effect 6
Double slit interference 31
Efficiency 20
Electron interference 39
Emissivity 15
Energy nuclear decay 47
Energy relativistic 36
43
Engineering work 20
Entropy 22
Entropy ideal gas 24
Equipartition of energy 18
Equivalence principle 36
Exclusion principle 44
Expansion thermal 9
Expansion coefficient linear 9
Eye 29
Fahrenheit scale 9
Far point 29
Farsightedness 29
First Law of Thermodynamics 12
Fission 48
Flux volume 2
Free expansion 23
Fringes 31
Fusion 48
Fusion heat of 11
Gamma relativistic 35
Gamma decay 46
General relativity 36
Half-life 46
Heat 11
Heat capacity 11
Heat conduction 13
Heat engine 20
Heat of fusion 11
Heat of vaporization 11
Heat pump 20
Heats of transformation 11
Hydrostatics 1
Ideal Gas Law 10
Images real and virtual 27
Intensity sound 5
Interference 7
Interference two-slit 31
Internal energy 12
Internal energy degrees of freedom 16
Irreversible process 19 23
Isotopes 45
Joule 11
Kelvin scale 9
Kinetic energy relativity 36
Kinetic theory 16
Latent heat 11
Length contraction 35
Linear expansion coefficient 9
Linear polarization 34
Linear superposition 7
Longitudinal wave 3
Loudness 5
Magnification lateral 27
Magnifying glass 30
Malusrsquos law 34
Microscope 30
Momentum relativistic 36
Muscle ciliary 29
44
Musical instruments 8
Musical scale 8
Near point 29
Nearsightedness 29
Nonlinear 7
Nuclear energy 48
Nuclear properties 45
Nuclear stability 45
Octave 8
Optical resolution 33
Orbital quantum number ℓ 43
Orbitals 43
Particle in a Box 41
Particles are waves 39
Pascalrsquos Principle 1
Period 3
Photoelectric effect 38
Photons 38
Pitch 8
Polarization 34
Power sound 5
Presbyopia 29
Pressure 1
Principal quantum number n 43
Principle of equivalence 36
Principle of linear superposition 7
Probability amplitude ψ 41
Processes thermodynamic 13
R-value 14
Radiation thermal 14
Radioactive decay 46
Radioactive decay rate 46
Ray tracing 27
Rayleighrsquos criterion 33
Reading glasses 29
Real image 27
Refraction 25
Refrigerator 20
Relativistic gamma 35
Relativity principles 35
Resolved for light sources 33
Rest energy 36
Reversible process 19 22
Rope wave speed 4
Second Law of Thermodynamics 19
Shock waves 6
Simple magnifier 30
Simultaneity 35
Single slit diffraction 33
Snellrsquos law 25
Sound speed 5
Specific heat 11
Spin Angular momentum 44
Standing waves 7
Stefanrsquos law 14
Strong force 45
Telescope 30
Temperature 9
45
Temperature Scales 9
Thermal conductivity 14
Thermal energy 12
Thermal expansion 9
Thin film interference 31
Thin lenses 28
Time dilation 35
Tone musical 8
Total internal reflection 25
Transverse wave 3
Traveling Waves 3
Two-slit interference 31
Uncertainty principle 40
Vaporization heat of 11
Velocity addition relativity 36
Virtual image 27
Volume flux 2
Wave function 41
Wave Function ψ and Probability 41
Wave speed 3
Wave-particle duality 39
Wavelength 3
Wavenumber 3
Wien Displacement Law 16
Work 11
12
carried out without friction and so slowly that the process can be reversed at any stage by
making an infinitesimal change in the environment of the system The slow expansion of the
gas at Th in the Carnot cycle with tiny amounts of heat being transferred to the environment
is reversible If we turned around and began slowly to compress the gas would just slowly
exhaust heat to the environment in the exact reverse way that it absorbed it during expansion
Most processes however are irreversible For example if a gas-filled box were suddenly
increased in size so that the particles were free to wander into the void created by the sudden
expansion then the gas would eventually fill the new volume uniformly at the same
temperature as before the expansion (The temperature is unchanged in this imaginary
process because the kinetic energy of the molecules would be unaffected by such an
instantaneous expansion of the container walls) This imaginary but highly thought-
stimulating process is called a free expansion and it is impossible to reverse it During the
expansion we didnrsquot push on any of the molecules so reversing this process would mean
making them go back into their original volume without pushing on them they simply will
not cooperate to this extent Another way to see that just pushing them back where they came
from does not reverse the free expansion is to think about what would happen if we just
compressed either adiabatically or isothermally An adiabatic compression back to the
original volume would heat the gas above its original temperature and an isothermal
compression would require that heat be exhausted to the environment But the free expansion
involved neither temperature changes nor heat exchanges so neither of these two processes
is the reverse of the free expansion It is simply impossible to reverse this rapid expansion in
a way that takes us back to the initial state
It is possible to calculate the change of entropy for both reversible and irreversible processes
Letrsquos consider a reversible process first In a reversible process the entropy change is given
by the formula
dQS
T
where dQ is the amount of heat added to the system during a small step of the process
The total energy change during the process may then simply be calculated by integration
f f
f i i i
dQS S S dS
T
13
(Just as in the case of energy we are mainly interested in differences rather than in
absolute magnitudes)
This integration method does not work for irreversible processes For instance in the
rapid free expansion discussed above no heat is added to the system but its disorder
obviously goes up We need to find some other way to calculate the entropy The key
is the fact that the entropy of a system depends only on its current state and not at all
on how it arrived there So to calculate the entropy change in an irreversible process
first find out what the initial and final conditions of the process are Then invent a
reversible process that takes the system from the initial state to the final state Since
the entropy depends only on the state of system and not on the process the entropy
change for the reversible process is the same as that for the irreversible process
Entropy is thus like pressure or temperature if the pressure changes from Pl to to P2
then the pressure difference is P2 - Pl regardless of how the pressure change was
made This seems obvious for pressure but not for entropy only because you arent
comfortable with entropy yet
Entropy of an Ideal Gas
For n moles of an ideal gas the difference in entropy between a state with temperature
T and volume V and some standard state with temperature To and volume Vo is given
by
ln lno o Vo o
T VS T V S T V nC nR
T V
Entropy in a Phase Change
Calculating entropy change in a phase change like melting or boiling is easy It is
QT Where Q is in the latent heat for example the flows in to cause the melting and
T is the temperature of the phase change There is NO integral to do
14
Serway Chapter 35
Angle of Reflection
If plane waves are incident on a reflecting surface with the propagation direction of
the waves making angle θ1 with the normal direction to the surface then the reflected
angle θ1prime relative to the surface normal is simply
1 1
ie the incident angles and reflected angles are the same
Refraction
If plane waves traveling through medium 1 are incident on a plane interface between
medium 1 and medium 2 then the angle of incidence of the incoming wave θ1 and
the angle of refraction of the transmitted wave θ2 are related by Snellrsquos law
2 2
1 1
sin
sin
v
v
where v1 and v2 are the wave speeds in medium 1 and medium 2 The angle of
incidence and the angle of refraction are both measured between the wave propagation
direction and the normal to the interface In terms of indices of refraction in the case
of light waves Snellrsquos law takes the more familiar form
1 1 2 2sin sinn n
where ni = cvi
Total Internal Reflection
If a wave is incident from a medium of low wave speed into a medium of high wave
speed the law of refraction requires that the angle of refraction be greater than the
angle of incidence If the angle of refraction is required to be greater than 90deg then no
refracted wave can exist and total internal reflection occurs The critical incident
angle θc beyond which total internal reflection occurs is given by
2
1
sin c
n
n
15
Dispersion of Light
In addition to the speed of light varying from material to material it also varies with
wavelength within each material This means that the index of refraction is generally a
function of wavelength
c
nv
Since the wave speed is not constant such a medium is dispersive meaning in this
context that refraction actually disperses white light into its various colors because
Snellrsquos law gives a different angle for each wavelength In most materials the
variation with wavelength is quite small but this small effect is responsible for some
of the most spectacular color effects we ever see including rainbows a flashing
crystal chandelier and the colored fire of a diamond solitaire by candlelight
16
Serway Chapter 36
Real and Virtual Images
When light rays are focused at a certain plane producing an image if a sheet of white
paper is placed there we call the image a real image The images produced by film
projectors and overhead projectors are examples of real images
When light rays appear to come from a certain location but no image is produced
when a screen is placed there we say that there is a virtual image at that location For
instance when you look in a mirror it appears that someone is behind the mirror but
a screen placed back there in the dark would show nothing Your image in the mirror
is a virtual image
Ray Tracing
There are lots of rules about how to find the images in optical systems but the best
way to keep things straight is to learn how to draw the principal rays for curved
mirrors and lenses The rays for convex and concave mirrors are shown in Fig 3615
The rays for converging and diverging thin lenses are shown in Fig 3627 You should
memorize the principal rays and know how to use them to locate images
Curved Mirrors
The focal length of a curved mirror with radius of curvature R is given by
2
Rf
The relation between object distance p image distance q and focal length f is
1 1 1
p q f
If the mirror is a diverging mirror f should be negative and if q should turn out to be
negative the image is virtual
Lateral Magnification
The lateral magnification in an optical system is defined by the ratio of the image size
to the object size
17
Image height
Object height
qM
p
for curved mirrors
As usual there are sign conventions here too but this formula just gives the
magnitude It is better to keep keep track of upright images versus inverted images by
means of ray diagrams rather than by memorizing sign conventions
Thin Lenses
The focal length of a thin lens is related to the radii of curvature of the two faces R1
and R2 of the lens by
1 2
1 1 11n
f R R
Note that this formula differs from Eq (3611) in the text by not having a minus sign
between the two R-terms We like this form better because for a simple converging
lens like a magnifying glass we just use positive values of R for both surfaces If one
of the faces is concave producing divergence use a negative value for R And if a
surface is flat use R = infin If the face is flat the radius is infinite
The relation between the image object and focal distances for a thin lens is the same
as that for a curved mirror
1 1 1
q p f
Use a negative focal length if the lens is diverging
The lateral magnification for a thin lens is the same as for a curved mirror
Image height
Object height
qM
p
Camera
The lens system in a camera projects a real image of an object onto the film (or CCD
array in a digital camera) The position of the image is adjusted to be on the film by
moving the lens into or out of the camera
18
Eye
The eye is like a camera in that a real image is formed on the retina Unlike a camera
the image position is adjusted by changing the focal length of the lens This is done by
the ciliary muscle which squeezes the lens changing its shape
Near Point The near point is the closest distance from the eye for which the lens can
focus an image on the retina It is usually 18-25 cm for young persons
Far Point The far point is the greatest distance from the eye for which the lens can
focus an image on the retina For a person with normal vision the far point is at
infinity
Nearsightedness A person is nearsighted if their far point is at some finite distance
less than infinity This condition can be corrected with a lens that takes an object at
infinity and produces a virtual image at the personrsquos far point
Farsightedness and Presbyopia A person is farsighted if their near point is too far
away for comfortable near work like reading or knitting This can be corrected by a
lens which takes an object at a normal near point distance of 18-25 cm and produces a
virtual image at the personrsquos natural near point Presbyopia involves a similar
problem which nearly all people experience as they age The ciliary muscle becomes
too weak and the lens becomes too stiff to allow the eye to provide for both near and
far vision The solution for this problem is either reading glasses or bifocal lenses
Reading classes are just weak magnifying glasses mounted on eyeglass frames
Bifocal lenses are split into upper and lower halves The lower half is a lens which
gives the proper correction for near work and the upper half is a different lens for
proper focusing at infinity
Angular Size
When an object is brought closer to the eye it appears to be larger because the image
on the retina is larger The size of this image is directly proportional to the objectrsquos
angular size which is the angle subtended by the object measured from the center of
the lens of the eye In optical instruments which are to be used with the eye the
angular size of the final image is whatrsquos important because it determines how large
the image will appear to the viewer
19
Simple Magnifier
A simple magnifier is a single converging lens or magnifying glass It takes an object
closer to the eye than a normal near point and produces a virtual image at or beyond
this near point The angular magnification is defined to be the ratio of the angular size
when viewed through the lens to the angular size of the object when viewed at the
normal near point (without aid of the lens)
Microscope
This instrument has two lenses (1) The objective is near the object being viewed and
produces a greatly magnified real image (2) The eyepiece is a simple magnifier which
the viewer uses to closely examine the image from (1)
Telescope
This instrument also has two lenses (1) The objective at the front of the telescope
takes light from a distant object and produces a real inverted image (which is rather
small) near its focal point (2) This small real image is then examined by the eyepiece
functioning as a simple magnifier to produce a virtual image with a larger angular
size
20
Serway Chapter 37
Two-Slit Interference
If light is incident on two closely spaced narrow slits a pattern of light and dark
stripes is produced beyond the slits The bright stripes or fringes are caused by
constructive interference of the two waves coming from the slits Constructive
interference occurs whenever two waves arrive at a location in phase with each other
This occurs when the distance x1 from slit 1 to a point P on the screen and the
distance x2 from slit 2 to point P differ by in integral number of wavelengths
1 2 where 0 1 2x x m m
where λ is the wavelength of the light When x1 and x2 are much larger than the slit
spacing d this condition reduces to
sind m
where θ is the angle between the direction of the incident light and the direction of the
light arriving at the screen
Thin Films
When light is partially reflected and partially transmitted by a thin film of transparent
material it is possible to have interference between the wave reflected from the front
of the film and light reflected from the back of the film (The colored reflections from
the thin film of oil on the water in a rain-soaked parking lot are an example of this
effect) It is difficult to write down formulas that will work in all cases so we will
just review the important principles here
1 If the two reflected waves are in phase with each other the film has enhanced
reflection (constructive interference) but if the two reflected waves are out of phase
with each other reflection is diminished (destructive interference) Phase shifts occur
due to reflection and due to the extra path length through the film of the wave
reflected from the back of the film
2 The phase change due to reflection is determined by the difference in index of
refraction between the two media involved in the reflection If the wave is incident
21
from a medium with a low index of refraction into a medium with a high index of
refraction a phase change of 180deg occurs and the reflected wave is inverted If
incident from high to low no phase shift occurs and the reflected wave is non-
inverted
3 The extra path length through the film of the wave reflected from the back of the
film is equal to 2t where t is the thickness of the film (The incident light is assumed
to be normal to the surface of the film) The number of wavelengths contained in the
extra path length is equal to 2tλn where λn = λn is the wavelength of the light in the
film
4 Rules for reflection from thin films
If one ray is inverted and the other is not then we have
12 constructive
2 nt m
2 destructivent m
If either both rays are inverted or both are non-inverted we have
2 constructivent m
12 destructive
2 nt m
22
Serway Chapter 38
Diffraction Grating
A diffraction grating is simply a fancy version of two-slit interference with the two
slits replaced by thousands of slits Just as in the two-slit case the bright fringes
occur at angles given by
sin md
but in this case d the distance between neighboring slits is made to be very small
The effect of having many slits instead of two is to make each bright fringe highly
localized with wide dark regions between neighboring maxima
Single Slit Diffraction
When light passes through an opening in an opaque screen an interference pattern is
produced beyond the opening To understand why we may replace the single opening
by many small coherent sources of light These many sources interfere with each
other producing a pattern known as a diffraction pattern (Note that many authors do
not distinguish between interference and diffraction treating them as interchangeable
terms) If the opening is a slit of width a then the diffraction pattern far from the slit
will have a bright central maximum with a succession of minima and weaker maxima
on either side The angle between the incident direction and the minima is given by
sin ma
where m = plusmn1 plusmn2
If the opening is circular with diameter D the angle between the incident direction
and the first minimum is given by
sin 122D
Optical Resolution and Rayleighrsquos Criterion
Two point sources can just be resolved (distinguished from each other) if the peak of
the diffraction image of the first source overlies the first minimum of the diffraction
image of the second source For circular holes of the kind usually encountered in
23
optical devices this condition is approximately satisfied when the angular separation
between the two sources as viewed from the optical instrument is greater than or equal
to the critical angle
min 122D
where λ is the wavelength of the light and where D is the diameter of the aperture in
the instrument
Polarization
We say that an electromagnetic wave is polarized if its electric field vector doesnrsquot
change direction in a random fashion The simplest kind of polarization is linear
polarization in which the electric field vector oscillates back and forth along the
same axis in space Polarized light can be produced from normal unpolarized light by
selective absorption (as in Polaroid sunglasses) by passing light through crystals that
have different indices of refraction for different polarizations (double refraction) by
scattering (the blue sky is polarized) and by reflection (glare) Polarization by
reflection occurs when light reflects from a shiny insulating (non-metallic) surface
The amount of polarization is greatest for reflection at Brewsterrsquos angle
2
1
tan P
n
n
where θP is the incidence angle of light from medium 1 onto medium 2 and where n1
and n2 are the indices of refraction for the media (In our everyday experience n1 = 1
since the light comes in through the air and n2 is the index of refraction of the shiny
insulating material producing the glare eg water glass plastic paint etc)
Malusrsquos Law
The intensity of transmitted polarized light through a perfect polarizer is related to the
incident intensity of polarized light by Malusrsquos law
2cosoI I
where Io is the intensity of the incident light and where θ is the angle between the
electric field vector in the incident wave and the transmission axis the polarizer
24
Serway Chapter 39
Principles of Relativity
All of the weirdness of relativity flows from two simple principles
(1) The laws of physics must be the same in all inertial (non-accelerating) reference
frames
(2) The speed of light in vacuum has the same value in all inertial reference frames
Note that (1) seems reasonable but (2) is very odd It says that if two space ships are
approaching each other at nearly the speed of light and a laser pulse is shot from ship
1 toward ship 2 then when the people on ship 2 measure the speed of the pulse as it
goes by it is moving at 3 times 108 ms the same as if the ships were stationary
Simultaneity
If observer 1 sees two events in her own frame as simultaneous at two different
locations a moving observer 2 will see these two events happening at different times
Relativistic gamma
The factor γ (gamma) appears regularly in the formulas of relativity
2
2
1
1 vc
where v is the relative speed between two inertial frames
Time Dilation
If observer 1 sees two events at the same location in space separated by time Δtp in his
own frame then observer 2 moving at speed v relative to observer 1 will see these
two events separated by a longer time Δt
pt t
Moving clocks run slow
Length Contraction
If observer 1 measures the length of an object along the x-direction at some instant of
time in her frame to be Lp then observer 2 moving in the x-direction at speed v will
measure the length L of the object to be shorter
pLL
25
Moving meter sticks are short
Velocity Addition
If a particle is moving at velocity vac relative to frame c and if frame c is moving at
speed vcb with respect to frame b (with both velocities directed along the same line)
what would be the velocity of the particle in frame b In classical physics the answer is
ab ac cbv v v
but if any of these velocities are an appreciable fraction of the speed of light the
answer changes
21ac cb
abac cb
v Vv
v v c
In using this equation make sure that vac has the same sign in the numerator and in the
denominator and that vbc also has the same sign in both places Note that this equation
is different (and in our opinion easier to understand) than the equations in the text
Momentum and Energy
The momentum and total energy of a particle moving at velocity v (its speed is v v )
are given by
22
2 2
22
1 1 vvc c
m mcm E mc
vp v
Rest Energy
When a particle is at rest (v = 0 and γ = 1) then E = mc2 is called the rest energy It
includes the potential energy of the particle so that if we change the potential energy
of a particle we change its rest mass
Kinetic Energy The kinetic energy is the difference between the total energy and the
rest energy
2 2K mc mc
For v ltlt c this reduces to K = mv22
General Relativity
Einsteinrsquos principle of equivalence in the theory of general relativity is In the vicinity
of a point a gravitational field is equivalent to an accelerated frame of reference in
26
the absence of gravitational effects For example there is no difference whatever in
the physics experienced by an astronaut on earth and one who is in interstellar space
far away from any stars or planets but who has just turned on her spaceshiprsquos engines
and is accelerating at 98 ms2
27
Serway Chapter 40
Photons
We now know that even though light behaves like a wave it is better described as
consisting of small packets of energy called photons The energy of a photon is related
to its frequency by
E hf
where h is Planckrsquos constant
346626 10 J sh
Photons also carry momentum given by
hf hp
c
Since the amount of energy in each photon is fixed the difference between dim light
and intense light (of the same wavelength) is that dim light consists of fewer photons
than intense light
Photoelectric Effect
In the photoelectric effect light shining onto a metal surface gives the electrons in the
metal enough energy to escape and be detected It requires a certain amount of energy
(called the work function typically = 3 ndash 5 eV) just to kick the electron out of the
metal so the light must deliver at least this much energy to an electron to produce the
effect It is observed that red light no matter how intense never produces electrons
But ultraviolet light even if quite dim will eject electrons from the metal Since red
light consists of 2 eV photons and ultraviolet light has photons with energies around
4-6 eV the photon idea explains the behavior of the photoelectric effect The
maximum energy that an ejected electron can have is
maxK hf
where f is the photon frequency The minimum frequency that light can have and cause
electrons to be ejected is called the cutoff energy and is found by setting Kmax = 0
cf h
28
Compton Effect
When high-frequency light interacts with free electrons the scattered light does not
have the same wavelength as the incident light contrary to what classical
electromagnetism would predict But the photon picture in which photons have
momentum and energy allows us to treat scattering as a collision between the photon
and the electron In this collision the electron and the scattered photon both have a
different momentum and energy than they did before And since p = hλ for a photon
if the momentum is different the wavelength will also be different
after before 1 cose
h
m c
where the angle θ is the angle between the incoming photon direction and the
direction of the scattered photon
Particles are Waves
Since photons behave like particles sometimes it is not surprising that elementary
particles can behave like waves sometimes The wavelength of a particle with
momentum p is given by
h
p
Wave-particle Duality
Both photons and elementary particles have a dual nature sometimes they behave like
particles and sometimes they behave like waves It is hard for us to comprehend the
nature of such an object by making mental pictures but experiments definitively show
that this is the case Since we have no direct experience with photons and elementary
particles (because their energies are so much smaller than the energies of the everyday
objects we encounter) it is perhaps not surprising that we have a hard time forming a
mental picture of how they behave
Electron Interference
Since an electron is both a particle and a wave just like a photon it should be able to
produce an interference pattern This is observed If an electron beam is shot at two
closely-spaced slits and if the electrons that pass through the slits are detected
downstream it is found that there are some locations where electrons are never detected
29
and others where lots of electrons are detected The pattern is exactly the same as the
one observed for light waves provided that we use the electron wavelength λ = hp in
place of the wavelength of light This pattern is observed even though each electron is
detected as a single dot on the screen Only after many such dots are collected does the
pattern emerge And if we try to understand how this effect could possibly work by
looking closely at each slit to see which one the electron came through the pattern
disappears the act of measurement destroys the interference This means that each
single electron somehow comes through both slits (as a wave would)
Uncertainty Principle
In classical physics we always imagine that the positions and momenta of moving
particles have definite values It might be hard to measure them but surely at each
instant of time a particle should be precisely located at some point in space and have a
similarly precise momentum This turns out not to be true Instead both position and
momentum are required to be uncertain with their uncertainties Δx and Δp satisfying
the Heisenberg uncertainty relation
2x p
So if the particle were known to be precisely at some particular location (so that Δx =
0) we couldnrsquot know anything about its momentum (Δp = infin) and if its momentum
were exactly known we couldnrsquot know its position
There is a similar relation involving the particlersquos energy E and the time interval Δt
over which this energy is measured
2E t
This means that energy is not actually conserved if we are considering very small time
intervals and this brief non-conservation of energy has been observed
30
Serway Chapter 41
Wave Function ψ and Probability
Quantum mechanics does not predict exactly what an electron or a photon will do Instead
it specifies the wave function or probability amplitude ψ of an electron or a photon
This wave function is a complex-valued function of space and time whose squared
magnitude is the probability density P for finding a particle at a particular place in
space at a certain time
2P
where is the complex conjugate of ψ
For example the wave function of an electron with perfectly specified momentum p
would have a wavelength given by
h
p
and its probability amplitude would be proportional to
2i x ipxe e
The corresponding probability density would then be
21ipx ip ipxP e e e
which means that the electron is equally probable to be anywhere along the x axis
This is in accord with the uncertainty principle since we specified the momentum
precisely we canrsquot have any idea about the position of the electron
As another example you have probably seen ldquofuzzy ballrdquo drawings of electron
orbitals in a chemistry book These fuzzy balls are meant to indicate the distribution
of the probability density 2
P in the orbital
Particle in a Box
A simple example in which we can calculate the wave function is the case of a particle of
mass m constrained to be inside a 1-dimensional box between x = 0 and x = L In this
case the general wave function is a linear superposition of wave functions ψn of the form
sin niE tn
n xx t A e
L
31
where A is a positive constant where n = 1 2 3 and where the energy associated
with each of the quantum states ψn is given by
22
28n
hE n
mL
This wave function is zero at x = 0 and x = L which means that the particle will never
be found at the walls of the box The wave function has maximum values in the
interior and at these places the particle is most likely to be found and it also has
places where it is zero and at these places the particle will also never be found as
expressed by the formula
2sinn x
PL
The particle in the box is interfering with itself producing a probability interference
pattern across the box just like the interference patterns we studied with light and
sound
32
Serway Chapter 42
Atoms
Many people picture an atom as a miniature solar system where electrons orbit around
a massive nucleus at the center This picture is misleading because in atoms the wave
nature of electrons dominates The electrons form 3-dimensional standing waves
(called orbitals) centered at the nucleus And if you ask what is it thatrsquos waving the
answer is ψ the probability amplitude (see Chapter 41)
Standing waves on a string can occur only for certain frequencies (the fundamental
and higher harmonics) Similarly atomic orbitals occur only for certain energies For
the hydrogen atom the energies of the orbitals have the a particularly simple form
2
1136 eVnE
n
where n = 1 2 3 is called the principal quantum number For other atoms the
determination of the orbital energies requires numerical calculation by computers
Atomic Spectra
If an electron is somehow given extra extra energy (we say that it is excited) so that it
occupies a higher orbital it will eventually ldquofallrdquo back down to a lower orbital Each
time an electron falls to a lower orbital it loses the difference in energy between the
two orbitals in the form of a photon Since the orbital energies are discrete so are the
energy differences and so are the wavelengths of the emitted light The entire set of
these discrete wavelengths is called the atomic spectrum and it is unique to each type
of atom For hydrogen the spectrum can be simply written as
H 2 2
1 1 1
f i
Rn n
where RH = 10973732 times 107 m-1 For other atoms the spectrum cannot be expressed as
a formula
Orbital Angular Momentum Electrons in atoms also have quantized values of
angular momentum The orbital quantum number ℓ specifies the value of this
quantized angular momentum through the formula
1L
33
If we want to know the value of the angular momentum along some direction in space
say the z direction the answer is not L but rather
zL m
where mℓ is another quantum number which runs from
1 1m
This quantum number is important when an atom sits in a magnetic field
Spin Angular Momentum It has been found experimentally that electrons and other
charged particles also carry internal angular momentum which we call spin
Electrons have an intrinsic spin angular momentum s along a specified axis that is
extremely quantized it can only take on 2 possible values
1
2zs s
We thus say that electrons have spin s = 12 or that electrons are rdquospin one-half parti-
cles Since s is an angular momentum it obeys the same rule as that for orbital angular
momentum namely that if its value along some axis is s then its total magnitude is
given by
31
2S s s
I know this seems weird but quantum mechanics is weird The only excuse for this
bizarre way of looking at the world is that it predicts what happens in experiments
Exclusion Principle The answer to the question of how many electrons (or any other
spin one-half particle) can be in one particular quantum state was discovered by
Wolfgang Pauli and is called the exclusion principle
ldquoNo two electrons can ever be in the same quantum state therefore no two electrons
in the same atom can have the same set of quantum numbersrdquo
This is the reason that we have atoms with different properties instead of every atom
simple having all of its electrons in the ground state All of the variety we see around
us in the world is the result of chemical differences and these differences would not
exist unless electrons obeyed this important principle The entire structure of the
periodic table (see pages 1377-1379 in Serway) is an expression of this principle
34
Serway Chapter 44
Nuclear Properties
The nucleus is composed of protons with charge +e and mass mp = 1007226 u and of
neutrons with zero charge and mass mn = 1008665 u where u is one atomic mass unit
-271 u=1660540 10 kg
The atomic number Z counts the number of protons in a nucleus while the neutron
number N counts the number of neutrons The mass number A is the sum of the two
A N Z
Protons and nuetrons have about the same mass and some times called baryons (heavy
ones) or nucleons (partcles in the nucleus) A is sometimes called the baryon
number The chemical elements are distinguished by Z ie hydrogen has Z = 1 iron
has Z = 26 uranium has Z = 92 etc But for a given Z there might be several nuclei
with different numbers of neutrons These nuclear siblings are called isotopes
These numbers are used to label nuclei according to the pattern
5626 ie FeA
Z X
denotes the isotope of iron with N = 56 ndash 26 = 30 neutrons
The nucleus is roughly spherical with a radius given approximately by
1 3 150 0where 12 10 mr r A r
Nuclear Stability
Because the positively charged protons electrically repel each other with an enormous
force at distances as small as 10-15 m there must be some really strong force that that
overcomes electrical repulsion to hold protons and neutrons together This force is
called with some lack of imagination the strong force It is a very short-range force
(it only acts over a distance of about 2 x 10-15 m) and attracts protons to protons
neutrons to neutrons and neutrons to protons But in spite of this strong nuclear force
the coulomb repulsion of the protons is still present so anything that might keep the
protons from being right next to each other would help keep the nucleus from
35
exploding This role is played by the neutrons and for nuclei with Z le 20 the stable
nuclei roughly have N = Z
For Z gt 20 the coulomb repulsion force is more powerful and more neutrons are
needed to dilute it up to about N = 15Z around Z = 80 For Z gt 83 no amount of
neutrons can help and these nuclei are unstable (radioactive) There is an ldquoisland of
(relative) stabilityrdquo around Z=90-92 (thorium and uranium respectively) These
elements have one or two isotopes with half-lifes of billions of years so there are
substantial amounts of such elements on Earth This fact makes it possible to have
practical fission devices
Radioactive Decay
There are three types of radioactive decay
Alpha decay the nucleus kicks out a helium nucleus (N = 2 Z = 2 A = 4) So if the
original nucleus is called X and the new nucleus is called Y then the decay would look
like this
4 42 2X Y+ HeA A
Z Z
Beta decay the nucleus either kicks out an electron (endash) or its positively-charged
antimatter twin the positron (e+) plus either an electron neutrino v or an electron
anti-neutrino v
1X Y+e electron decayA AZ Z v
1X Y+e positron decayA AZ Z v
A neutrino is a particle with no charge hardly any mass (much less than the electron
mass) and interacts so weakly with matter that most neutrinos upon encountering the
planet earth just pass right through it as if it werenrsquot there
Gamma decay the nucleons in the nucleus X are in an excited energy state X
(perhaps as a result of having undergone alpha or beta decay) and they drop down to a
lower energy state shedding the energy as a high frequency photon
X XA AZ Z
36
This process is exactly analogous to the way that the electrons in atoms emit photons
Decay Rate and Half Life
There is no way to predict exactly when an unstable or excited nucleus will decay but
there is an average rate at which this decay occurs called the decay constant λ The
meaning of this constant is that if there are a large number N of nuclei in a sample
then the number of decays per second that will be observed (called the decay rate R)
is R = λN In mathematical language
dMR N
dt
This simple differential equation has for its solution
0tN t N e
where N0 is the number of nuclei in the sample at time t = 0
The half-life is the time it takes for half of the nuclei in the sample to decay and is
related to the decay constant by
1 2
ln 2 0693T
Disintegration Energy
When a nucleus decays it is making a transition to an overall state of lower energy
which means according to Einsteinrsquos famous formula E = mc2 that the sum of the
masses after the decay must be less than the mass before with the lost mass appearing
as kinetic energy among the decay products For example in alpha decay this kinetic
energy called the disintegration energy Q is given by
2X YQ M M M c
37
Serway Chapter 45
Nuclear energy
There are two ways to extract energy from the nucleus fission and fusion For nuclei
with Z greater than 26 breaking the nucleus apart into pieces leads to a lower
2mc energy than the original nucleus so energy can be extracted by fission For
nuclei with Z less than 26 a lower energy is achieved by combining nuclei so energy
can be extracted by fusion
Fission
Since neutrons have no charge they are not repelled from nuclei as protons are for
this reason their behavior is the key to understanding how fission works
Fast neutrons mostly bounce off other nuclei with each collision slowing the neutron
down This elastic energy loss is most effective if the other nuclei have low mass (like
hydrogen) and these materials are called moderators because of their ability to slow
down fast neutrons
The reason that slowing neutrons is important is that slow neutrons are much more
likely to be absorbed by a nucleus which then leads to nuclear reactions of various
kinds For a few very large nuclei like uranium-235 and some plutonium isotopes
absorption of a slow neutron causes the nucleus to split into two large fragments plus
2 or 3 fast neutrons (energy is released in the process too) Thus one slow neutron
can produce energy plus 2 or 3 more neutrons which if moderated from fast to slow
can split 2 or 3 more nuclei leading to a runaway chain reaction If the reaction is
allowed to proceed unchecked it produces a large explosion If a neutron absorbing
material is added to the mix (like the cadmium in reactor control rods) it is possible to
keep the reaction under control and to extract the released energy as heat to drive
steam turbines and produce electricity
Fusion
Fusion involves mashing two nuclei together and since they are both charged and repel
each other this reaction is much harder to make go The nuclei must have enough energy to
overcome the coulomb repulsion which is why this reaction requires a high temperature
(hundreds of millions of degrees K) like that in the sun (which is burning hydrogen to
helium via fusion) or like that in the center of the fission explosion that is used to detonate
38
a hydrogen bomb
This reaction is of interest for power production in spite of this difficult temperature
requirement because of the abundance of fusion fuel on the planet There are about
012 g of deuterium ( 21H ) in every gallon of water on earth and it only costs about 4
cents to extract it The fusion energy available from this minuscule amount of
deuterium would run a 1000-Megawatt power plant for 10 seconds Water is so
abundant on earth that if fusion were to work we would have an essentially
inexhaustible source of energy
So why donrsquot we have fusion power plants Well the fuel is cheap but the match is
incredibly expensive The only way we know to control this difficult high-temperature
reaction is with large and expensive pieces of equipment involving either large
magnetic fields and complex high-power electromagnetic antennas or with gigantic
(football-fieldsized) laser facilities involving more than a hundred of the highest-
energy lasers ever built Power plants based on these current methods for controlling
fusion are unattractive to the fiscally-minded people who run the electric power
industry Hopefully better designs will be discovered as experiments continue
39
c actinium 68 Er erbium 101 Md mendelevium 104 Rf rutherfordium l aluminum 63 Eu europium 80 Hg mercury 62 Sm samarium m americium 100 Fm fermium 42 Mo molybdenum 21 Sc scandium b antimony 9 F fluorine 60 Nd neodymium 106 Sg seaborgium r argon 87 Fr francium 10 Ne neon 34 Se selenium s arsenic 64 Gd gadolinium 93 Np neptunium 14 Si silicon t astatine 31 Ga gallium 28 Ni nickel 47 Ag silver a barium 32 Ge germanium 41 Nb niobium 11 Na sodium k berkelium 79 Au gold 7 N nitrogen 38 Sr strontium e beryllium 72 Hf hafnium 102 No nobelium 16 S sulfur i bismuth 108 Hs hassium 76 Os osmium 73 Ta tantalum h bohrium 2 He helium 8 O oxygen 43 Tc technetium
boron 67 Ho holmium 46 Pd palladium 52 Te tellurium r bromine 1 H hydrogen 15 P phosphorus 65 Tb terbium d cadmium 49 In indium 78 Pt platinum 81 Tl thallium a calcium 53 I iodine 94 Pu plutonium 90 Th thorium f californium 77 Ir iridium 84 Po polonium 69 Tm thulium
carbon 26 Fe iron 19 K potassium 50 Sn tin e cerium 36 Kr krypton 59 Pr praseodymium 22 Ti titanium s cesium 57 La lanthanum 61 Pm promethium 74 W tungsten l chlorine 103 Lr lawrencium 91 Pa protactinium 92 U uranium r chromium 82 Pb lead 88 Ra radium 23 V vanadium o cobalt 3 Li lithium 86 Rn radon 54 Xe xenon u copper 71 Lu lutetium 75 Re rhenium 70 Yb ytterbium m curium 12 Mg magnesium 45 Rh rhodium 39 Y yttrium b dubnium 25 Mn manganese 37 Rb rubidium 30 Zn zinc y dysprosium 109 Mt meitnerium 44 Ru ruthenium 40 Zr zirconium s einsteinium
40
UNITS The SI units are given in parentheses Other commonly used units are given in terms of the SI units Symbols conform with the recommendations of the American National Standards Institute (ANSI) and the American Institute of Physics (AIP) Commonly used multiples of SI units
prefix symbol factor tera T 1012 giga G 109
mega M 106 kilo k 103
centi c 10-2 milli m 10-3
micro μ 10-6 nano n 10-9 pico p 10-12
femto f 10-15
length meter (m) angstrom 1 Ǻ = 10-10 m inch 1 in = 254 x 10-2 m foot 1 ft = 03048 m mile 1 mi = 1609 m
mass kilogram (kg) atomic mass unit 1 u = 1661 x 10-27 kgslug 1 slug = 1459 kg
force newton (N = kg bull ms2) dyne 1 dyn = 10-5 N pound 1 lb = 4448 N
pressure Pascal (Pa = kgm bull s2) atmosphere 1 atm = 1013 x 105 Pa poundssquare inch 1 psi = 6895 Pa cm of mercury 1 cm Hg = 1333 Pa bar 1 bar = 1000 x 105 Pa torr 1 torr = 1333 Pa
time second (s) minute 1 min = 60 s hour 1 h =3600 s
frequency hertz (Hz = s-1) radianssecond 1 rads = 12π Hz
energy joule (J = kg bull m2s2) erg 1 erg = 10-7 J electron volt 1 eV = 1602 x 10-19 J calorie 1 cal = 4187 J kilowatt-hour 1 kWh = 36 x 106 J British thermal unit 1 Btu = 1055 J
power watt (W = kg bull m2s3) horsepower 1 hp = 7457 W
charge coulomb (C = A bull s)
electric potential volt (V = kg bull m2 s3 bull A)
current ampere (A)
resistance ohm (Ω = kg bull m2s3 bull A2)
capacitance farad (F = s4 bull A2kg bull m2)
magnetic field tesla (T = kgs2 bull A) gauss 1 G = 10-4 T
magnetic flux weber (Wb = kg m2s2 bull A) maxwell 1 Mx = 10-8 Wb
magnetic inductance henry (H = kg bull m2s2 bullA2)
temperature kelvin (K) degrees Celsius 0degC = 27315 K
angle radian (rad) degree 1 deg = π180 rad revolution 1 rev = 2π rad
41
Some Physical Constantsa Quantity Symbol Valueb Atomic mass unit μ 1660 538 73 (13) x 10-27 kg
931494 013 (37) MeVc2 Avogadros number NA 6022 141 99 (47) x 1023 particlesmol Avogadrorsquos number 6022 x 1023 mol Bohr magneton
2B e
em 9274 008 99 (37) x 10-24 JT
Bohr magneton μB 927 x 10-24 JT Bohr radius
0
2
2e em e k
a 5291 772 083 (19) x 10-11 m
Boltzmanns constant B A
RNk 1380 650 3 (24) x 10-23 JK
Boltzmannrsquos constant kB 1380 x 10-23 JK Compton wavelength
C eh
m c 2426 310 215 (18) x 10-12 m
Deuteron mass md 3343 583 09 (26) x 10-27 kg 2013 553 212 71 (35) u
electron charge e 1602 x 10-19 C Electron mass me 9109 381 88 (72) x 10-31 kg
5485 799 110 (12) x 10-4 u 0510 998 902 (21) MeVc2
electron mass 911 x 10-31 kg Electron volt eV 1602 176 462 (63) x 10-19 J Elementary charge e 1602 176 462 (63) x 10-19 C Gas constant R 8314 472 (15) JKmol Gravitational constant G 6673 (10) x 10-11 Nm2kg2 Neutron mass mn 1674 927 16 (13) x 10-27 kg
1008 664 915 78 (55) u 939565 330 (38) MeVc2
neutron mass 1675 x 10-27 kg Nuclear magneton
2n p
em 5050 783 17 (20) x 10-27 JT
permeability constant μ0 1257 x 10-6 Hm permittivity constant ε0 8854 x 10-12 Fm Plancks constant h
2h
6626 068 76 (52) x 10-34 Js 1054 571 596 (82) x 10-34 Js
Planckrsquos constant h ħ
6626 x 10-34 Js 1055 x 10-34 Js
Proton mass mp 1672621 58 (13) x 10-27 kg 1007 276 466 88 (13) u 938271 998 (38) MeVc2
proton mass 1673 x 10-27 kg Rydberg constant RH 1097 373 156 854 9 (83) x 107 m-1 Speed of light in vacuum c 2997 92458 x 108 ms (exact) speed of light c 300 x 108 ms a
These constants are the values recommended in 1998 by CODATA based on a least-squares adjustment of data from different measurements For a more complete list see P J Mohr and B N Taylor Rev Mod Phys 72351 2000 b The numbers in parentheses for the values above represent the uncertainties of the last two digits
NOTE The ones in red are the ones from Appendix 13 The ones in yellow are the ones that are from the Serway book table but that were already on the Appendix 13 table
42
INDEX
Absolute zero 9
Adiabatic 17
Adiabatic exponent 18
Adiabatic process 17
Alpha decay 46
Amplitude 4
Angle of reflection 25
Angular frequency 3
Angular magnification 30
Angular momentum atomic 43
Angular size 29
Archimedes Principle 1
Atomic spectra 43
Atoms 43
Avogadrorsquos number 10
Beats 8
Bernoullirsquos Equation 2
Beta decay 46
Bifocals 29
Brewsterrsquos angle 34
British Thermal Unit Btu 11
Buoyancy 2
calorie 11
Calorie 11
Camera 28
Carnot cycle 21
Carnot efficiency 22
Celsius scale 9
Ciliary muscle 29
Coefficient of performance 21
Compton effect 39
Constructive interference 7
Continuity equation of 2
Contraction length 35
Convection 14
Curved mirrors 27
Decay constant 46
Decay rate radioactive 46
Decibel scale 5
Degrees of freedom 16
Density 1
Destructive interference 7
Diffraction single slit 33
Diffraction grating 33
Dilation time 35
Disintegration energy 47
Dispersion of light 26
Doppler effect 6
Double slit interference 31
Efficiency 20
Electron interference 39
Emissivity 15
Energy nuclear decay 47
Energy relativistic 36
43
Engineering work 20
Entropy 22
Entropy ideal gas 24
Equipartition of energy 18
Equivalence principle 36
Exclusion principle 44
Expansion thermal 9
Expansion coefficient linear 9
Eye 29
Fahrenheit scale 9
Far point 29
Farsightedness 29
First Law of Thermodynamics 12
Fission 48
Flux volume 2
Free expansion 23
Fringes 31
Fusion 48
Fusion heat of 11
Gamma relativistic 35
Gamma decay 46
General relativity 36
Half-life 46
Heat 11
Heat capacity 11
Heat conduction 13
Heat engine 20
Heat of fusion 11
Heat of vaporization 11
Heat pump 20
Heats of transformation 11
Hydrostatics 1
Ideal Gas Law 10
Images real and virtual 27
Intensity sound 5
Interference 7
Interference two-slit 31
Internal energy 12
Internal energy degrees of freedom 16
Irreversible process 19 23
Isotopes 45
Joule 11
Kelvin scale 9
Kinetic energy relativity 36
Kinetic theory 16
Latent heat 11
Length contraction 35
Linear expansion coefficient 9
Linear polarization 34
Linear superposition 7
Longitudinal wave 3
Loudness 5
Magnification lateral 27
Magnifying glass 30
Malusrsquos law 34
Microscope 30
Momentum relativistic 36
Muscle ciliary 29
44
Musical instruments 8
Musical scale 8
Near point 29
Nearsightedness 29
Nonlinear 7
Nuclear energy 48
Nuclear properties 45
Nuclear stability 45
Octave 8
Optical resolution 33
Orbital quantum number ℓ 43
Orbitals 43
Particle in a Box 41
Particles are waves 39
Pascalrsquos Principle 1
Period 3
Photoelectric effect 38
Photons 38
Pitch 8
Polarization 34
Power sound 5
Presbyopia 29
Pressure 1
Principal quantum number n 43
Principle of equivalence 36
Principle of linear superposition 7
Probability amplitude ψ 41
Processes thermodynamic 13
R-value 14
Radiation thermal 14
Radioactive decay 46
Radioactive decay rate 46
Ray tracing 27
Rayleighrsquos criterion 33
Reading glasses 29
Real image 27
Refraction 25
Refrigerator 20
Relativistic gamma 35
Relativity principles 35
Resolved for light sources 33
Rest energy 36
Reversible process 19 22
Rope wave speed 4
Second Law of Thermodynamics 19
Shock waves 6
Simple magnifier 30
Simultaneity 35
Single slit diffraction 33
Snellrsquos law 25
Sound speed 5
Specific heat 11
Spin Angular momentum 44
Standing waves 7
Stefanrsquos law 14
Strong force 45
Telescope 30
Temperature 9
45
Temperature Scales 9
Thermal conductivity 14
Thermal energy 12
Thermal expansion 9
Thin film interference 31
Thin lenses 28
Time dilation 35
Tone musical 8
Total internal reflection 25
Transverse wave 3
Traveling Waves 3
Two-slit interference 31
Uncertainty principle 40
Vaporization heat of 11
Velocity addition relativity 36
Virtual image 27
Volume flux 2
Wave function 41
Wave Function ψ and Probability 41
Wave speed 3
Wave-particle duality 39
Wavelength 3
Wavenumber 3
Wien Displacement Law 16
Work 11
13
(Just as in the case of energy we are mainly interested in differences rather than in
absolute magnitudes)
This integration method does not work for irreversible processes For instance in the
rapid free expansion discussed above no heat is added to the system but its disorder
obviously goes up We need to find some other way to calculate the entropy The key
is the fact that the entropy of a system depends only on its current state and not at all
on how it arrived there So to calculate the entropy change in an irreversible process
first find out what the initial and final conditions of the process are Then invent a
reversible process that takes the system from the initial state to the final state Since
the entropy depends only on the state of system and not on the process the entropy
change for the reversible process is the same as that for the irreversible process
Entropy is thus like pressure or temperature if the pressure changes from Pl to to P2
then the pressure difference is P2 - Pl regardless of how the pressure change was
made This seems obvious for pressure but not for entropy only because you arent
comfortable with entropy yet
Entropy of an Ideal Gas
For n moles of an ideal gas the difference in entropy between a state with temperature
T and volume V and some standard state with temperature To and volume Vo is given
by
ln lno o Vo o
T VS T V S T V nC nR
T V
Entropy in a Phase Change
Calculating entropy change in a phase change like melting or boiling is easy It is
QT Where Q is in the latent heat for example the flows in to cause the melting and
T is the temperature of the phase change There is NO integral to do
14
Serway Chapter 35
Angle of Reflection
If plane waves are incident on a reflecting surface with the propagation direction of
the waves making angle θ1 with the normal direction to the surface then the reflected
angle θ1prime relative to the surface normal is simply
1 1
ie the incident angles and reflected angles are the same
Refraction
If plane waves traveling through medium 1 are incident on a plane interface between
medium 1 and medium 2 then the angle of incidence of the incoming wave θ1 and
the angle of refraction of the transmitted wave θ2 are related by Snellrsquos law
2 2
1 1
sin
sin
v
v
where v1 and v2 are the wave speeds in medium 1 and medium 2 The angle of
incidence and the angle of refraction are both measured between the wave propagation
direction and the normal to the interface In terms of indices of refraction in the case
of light waves Snellrsquos law takes the more familiar form
1 1 2 2sin sinn n
where ni = cvi
Total Internal Reflection
If a wave is incident from a medium of low wave speed into a medium of high wave
speed the law of refraction requires that the angle of refraction be greater than the
angle of incidence If the angle of refraction is required to be greater than 90deg then no
refracted wave can exist and total internal reflection occurs The critical incident
angle θc beyond which total internal reflection occurs is given by
2
1
sin c
n
n
15
Dispersion of Light
In addition to the speed of light varying from material to material it also varies with
wavelength within each material This means that the index of refraction is generally a
function of wavelength
c
nv
Since the wave speed is not constant such a medium is dispersive meaning in this
context that refraction actually disperses white light into its various colors because
Snellrsquos law gives a different angle for each wavelength In most materials the
variation with wavelength is quite small but this small effect is responsible for some
of the most spectacular color effects we ever see including rainbows a flashing
crystal chandelier and the colored fire of a diamond solitaire by candlelight
16
Serway Chapter 36
Real and Virtual Images
When light rays are focused at a certain plane producing an image if a sheet of white
paper is placed there we call the image a real image The images produced by film
projectors and overhead projectors are examples of real images
When light rays appear to come from a certain location but no image is produced
when a screen is placed there we say that there is a virtual image at that location For
instance when you look in a mirror it appears that someone is behind the mirror but
a screen placed back there in the dark would show nothing Your image in the mirror
is a virtual image
Ray Tracing
There are lots of rules about how to find the images in optical systems but the best
way to keep things straight is to learn how to draw the principal rays for curved
mirrors and lenses The rays for convex and concave mirrors are shown in Fig 3615
The rays for converging and diverging thin lenses are shown in Fig 3627 You should
memorize the principal rays and know how to use them to locate images
Curved Mirrors
The focal length of a curved mirror with radius of curvature R is given by
2
Rf
The relation between object distance p image distance q and focal length f is
1 1 1
p q f
If the mirror is a diverging mirror f should be negative and if q should turn out to be
negative the image is virtual
Lateral Magnification
The lateral magnification in an optical system is defined by the ratio of the image size
to the object size
17
Image height
Object height
qM
p
for curved mirrors
As usual there are sign conventions here too but this formula just gives the
magnitude It is better to keep keep track of upright images versus inverted images by
means of ray diagrams rather than by memorizing sign conventions
Thin Lenses
The focal length of a thin lens is related to the radii of curvature of the two faces R1
and R2 of the lens by
1 2
1 1 11n
f R R
Note that this formula differs from Eq (3611) in the text by not having a minus sign
between the two R-terms We like this form better because for a simple converging
lens like a magnifying glass we just use positive values of R for both surfaces If one
of the faces is concave producing divergence use a negative value for R And if a
surface is flat use R = infin If the face is flat the radius is infinite
The relation between the image object and focal distances for a thin lens is the same
as that for a curved mirror
1 1 1
q p f
Use a negative focal length if the lens is diverging
The lateral magnification for a thin lens is the same as for a curved mirror
Image height
Object height
qM
p
Camera
The lens system in a camera projects a real image of an object onto the film (or CCD
array in a digital camera) The position of the image is adjusted to be on the film by
moving the lens into or out of the camera
18
Eye
The eye is like a camera in that a real image is formed on the retina Unlike a camera
the image position is adjusted by changing the focal length of the lens This is done by
the ciliary muscle which squeezes the lens changing its shape
Near Point The near point is the closest distance from the eye for which the lens can
focus an image on the retina It is usually 18-25 cm for young persons
Far Point The far point is the greatest distance from the eye for which the lens can
focus an image on the retina For a person with normal vision the far point is at
infinity
Nearsightedness A person is nearsighted if their far point is at some finite distance
less than infinity This condition can be corrected with a lens that takes an object at
infinity and produces a virtual image at the personrsquos far point
Farsightedness and Presbyopia A person is farsighted if their near point is too far
away for comfortable near work like reading or knitting This can be corrected by a
lens which takes an object at a normal near point distance of 18-25 cm and produces a
virtual image at the personrsquos natural near point Presbyopia involves a similar
problem which nearly all people experience as they age The ciliary muscle becomes
too weak and the lens becomes too stiff to allow the eye to provide for both near and
far vision The solution for this problem is either reading glasses or bifocal lenses
Reading classes are just weak magnifying glasses mounted on eyeglass frames
Bifocal lenses are split into upper and lower halves The lower half is a lens which
gives the proper correction for near work and the upper half is a different lens for
proper focusing at infinity
Angular Size
When an object is brought closer to the eye it appears to be larger because the image
on the retina is larger The size of this image is directly proportional to the objectrsquos
angular size which is the angle subtended by the object measured from the center of
the lens of the eye In optical instruments which are to be used with the eye the
angular size of the final image is whatrsquos important because it determines how large
the image will appear to the viewer
19
Simple Magnifier
A simple magnifier is a single converging lens or magnifying glass It takes an object
closer to the eye than a normal near point and produces a virtual image at or beyond
this near point The angular magnification is defined to be the ratio of the angular size
when viewed through the lens to the angular size of the object when viewed at the
normal near point (without aid of the lens)
Microscope
This instrument has two lenses (1) The objective is near the object being viewed and
produces a greatly magnified real image (2) The eyepiece is a simple magnifier which
the viewer uses to closely examine the image from (1)
Telescope
This instrument also has two lenses (1) The objective at the front of the telescope
takes light from a distant object and produces a real inverted image (which is rather
small) near its focal point (2) This small real image is then examined by the eyepiece
functioning as a simple magnifier to produce a virtual image with a larger angular
size
20
Serway Chapter 37
Two-Slit Interference
If light is incident on two closely spaced narrow slits a pattern of light and dark
stripes is produced beyond the slits The bright stripes or fringes are caused by
constructive interference of the two waves coming from the slits Constructive
interference occurs whenever two waves arrive at a location in phase with each other
This occurs when the distance x1 from slit 1 to a point P on the screen and the
distance x2 from slit 2 to point P differ by in integral number of wavelengths
1 2 where 0 1 2x x m m
where λ is the wavelength of the light When x1 and x2 are much larger than the slit
spacing d this condition reduces to
sind m
where θ is the angle between the direction of the incident light and the direction of the
light arriving at the screen
Thin Films
When light is partially reflected and partially transmitted by a thin film of transparent
material it is possible to have interference between the wave reflected from the front
of the film and light reflected from the back of the film (The colored reflections from
the thin film of oil on the water in a rain-soaked parking lot are an example of this
effect) It is difficult to write down formulas that will work in all cases so we will
just review the important principles here
1 If the two reflected waves are in phase with each other the film has enhanced
reflection (constructive interference) but if the two reflected waves are out of phase
with each other reflection is diminished (destructive interference) Phase shifts occur
due to reflection and due to the extra path length through the film of the wave
reflected from the back of the film
2 The phase change due to reflection is determined by the difference in index of
refraction between the two media involved in the reflection If the wave is incident
21
from a medium with a low index of refraction into a medium with a high index of
refraction a phase change of 180deg occurs and the reflected wave is inverted If
incident from high to low no phase shift occurs and the reflected wave is non-
inverted
3 The extra path length through the film of the wave reflected from the back of the
film is equal to 2t where t is the thickness of the film (The incident light is assumed
to be normal to the surface of the film) The number of wavelengths contained in the
extra path length is equal to 2tλn where λn = λn is the wavelength of the light in the
film
4 Rules for reflection from thin films
If one ray is inverted and the other is not then we have
12 constructive
2 nt m
2 destructivent m
If either both rays are inverted or both are non-inverted we have
2 constructivent m
12 destructive
2 nt m
22
Serway Chapter 38
Diffraction Grating
A diffraction grating is simply a fancy version of two-slit interference with the two
slits replaced by thousands of slits Just as in the two-slit case the bright fringes
occur at angles given by
sin md
but in this case d the distance between neighboring slits is made to be very small
The effect of having many slits instead of two is to make each bright fringe highly
localized with wide dark regions between neighboring maxima
Single Slit Diffraction
When light passes through an opening in an opaque screen an interference pattern is
produced beyond the opening To understand why we may replace the single opening
by many small coherent sources of light These many sources interfere with each
other producing a pattern known as a diffraction pattern (Note that many authors do
not distinguish between interference and diffraction treating them as interchangeable
terms) If the opening is a slit of width a then the diffraction pattern far from the slit
will have a bright central maximum with a succession of minima and weaker maxima
on either side The angle between the incident direction and the minima is given by
sin ma
where m = plusmn1 plusmn2
If the opening is circular with diameter D the angle between the incident direction
and the first minimum is given by
sin 122D
Optical Resolution and Rayleighrsquos Criterion
Two point sources can just be resolved (distinguished from each other) if the peak of
the diffraction image of the first source overlies the first minimum of the diffraction
image of the second source For circular holes of the kind usually encountered in
23
optical devices this condition is approximately satisfied when the angular separation
between the two sources as viewed from the optical instrument is greater than or equal
to the critical angle
min 122D
where λ is the wavelength of the light and where D is the diameter of the aperture in
the instrument
Polarization
We say that an electromagnetic wave is polarized if its electric field vector doesnrsquot
change direction in a random fashion The simplest kind of polarization is linear
polarization in which the electric field vector oscillates back and forth along the
same axis in space Polarized light can be produced from normal unpolarized light by
selective absorption (as in Polaroid sunglasses) by passing light through crystals that
have different indices of refraction for different polarizations (double refraction) by
scattering (the blue sky is polarized) and by reflection (glare) Polarization by
reflection occurs when light reflects from a shiny insulating (non-metallic) surface
The amount of polarization is greatest for reflection at Brewsterrsquos angle
2
1
tan P
n
n
where θP is the incidence angle of light from medium 1 onto medium 2 and where n1
and n2 are the indices of refraction for the media (In our everyday experience n1 = 1
since the light comes in through the air and n2 is the index of refraction of the shiny
insulating material producing the glare eg water glass plastic paint etc)
Malusrsquos Law
The intensity of transmitted polarized light through a perfect polarizer is related to the
incident intensity of polarized light by Malusrsquos law
2cosoI I
where Io is the intensity of the incident light and where θ is the angle between the
electric field vector in the incident wave and the transmission axis the polarizer
24
Serway Chapter 39
Principles of Relativity
All of the weirdness of relativity flows from two simple principles
(1) The laws of physics must be the same in all inertial (non-accelerating) reference
frames
(2) The speed of light in vacuum has the same value in all inertial reference frames
Note that (1) seems reasonable but (2) is very odd It says that if two space ships are
approaching each other at nearly the speed of light and a laser pulse is shot from ship
1 toward ship 2 then when the people on ship 2 measure the speed of the pulse as it
goes by it is moving at 3 times 108 ms the same as if the ships were stationary
Simultaneity
If observer 1 sees two events in her own frame as simultaneous at two different
locations a moving observer 2 will see these two events happening at different times
Relativistic gamma
The factor γ (gamma) appears regularly in the formulas of relativity
2
2
1
1 vc
where v is the relative speed between two inertial frames
Time Dilation
If observer 1 sees two events at the same location in space separated by time Δtp in his
own frame then observer 2 moving at speed v relative to observer 1 will see these
two events separated by a longer time Δt
pt t
Moving clocks run slow
Length Contraction
If observer 1 measures the length of an object along the x-direction at some instant of
time in her frame to be Lp then observer 2 moving in the x-direction at speed v will
measure the length L of the object to be shorter
pLL
25
Moving meter sticks are short
Velocity Addition
If a particle is moving at velocity vac relative to frame c and if frame c is moving at
speed vcb with respect to frame b (with both velocities directed along the same line)
what would be the velocity of the particle in frame b In classical physics the answer is
ab ac cbv v v
but if any of these velocities are an appreciable fraction of the speed of light the
answer changes
21ac cb
abac cb
v Vv
v v c
In using this equation make sure that vac has the same sign in the numerator and in the
denominator and that vbc also has the same sign in both places Note that this equation
is different (and in our opinion easier to understand) than the equations in the text
Momentum and Energy
The momentum and total energy of a particle moving at velocity v (its speed is v v )
are given by
22
2 2
22
1 1 vvc c
m mcm E mc
vp v
Rest Energy
When a particle is at rest (v = 0 and γ = 1) then E = mc2 is called the rest energy It
includes the potential energy of the particle so that if we change the potential energy
of a particle we change its rest mass
Kinetic Energy The kinetic energy is the difference between the total energy and the
rest energy
2 2K mc mc
For v ltlt c this reduces to K = mv22
General Relativity
Einsteinrsquos principle of equivalence in the theory of general relativity is In the vicinity
of a point a gravitational field is equivalent to an accelerated frame of reference in
26
the absence of gravitational effects For example there is no difference whatever in
the physics experienced by an astronaut on earth and one who is in interstellar space
far away from any stars or planets but who has just turned on her spaceshiprsquos engines
and is accelerating at 98 ms2
27
Serway Chapter 40
Photons
We now know that even though light behaves like a wave it is better described as
consisting of small packets of energy called photons The energy of a photon is related
to its frequency by
E hf
where h is Planckrsquos constant
346626 10 J sh
Photons also carry momentum given by
hf hp
c
Since the amount of energy in each photon is fixed the difference between dim light
and intense light (of the same wavelength) is that dim light consists of fewer photons
than intense light
Photoelectric Effect
In the photoelectric effect light shining onto a metal surface gives the electrons in the
metal enough energy to escape and be detected It requires a certain amount of energy
(called the work function typically = 3 ndash 5 eV) just to kick the electron out of the
metal so the light must deliver at least this much energy to an electron to produce the
effect It is observed that red light no matter how intense never produces electrons
But ultraviolet light even if quite dim will eject electrons from the metal Since red
light consists of 2 eV photons and ultraviolet light has photons with energies around
4-6 eV the photon idea explains the behavior of the photoelectric effect The
maximum energy that an ejected electron can have is
maxK hf
where f is the photon frequency The minimum frequency that light can have and cause
electrons to be ejected is called the cutoff energy and is found by setting Kmax = 0
cf h
28
Compton Effect
When high-frequency light interacts with free electrons the scattered light does not
have the same wavelength as the incident light contrary to what classical
electromagnetism would predict But the photon picture in which photons have
momentum and energy allows us to treat scattering as a collision between the photon
and the electron In this collision the electron and the scattered photon both have a
different momentum and energy than they did before And since p = hλ for a photon
if the momentum is different the wavelength will also be different
after before 1 cose
h
m c
where the angle θ is the angle between the incoming photon direction and the
direction of the scattered photon
Particles are Waves
Since photons behave like particles sometimes it is not surprising that elementary
particles can behave like waves sometimes The wavelength of a particle with
momentum p is given by
h
p
Wave-particle Duality
Both photons and elementary particles have a dual nature sometimes they behave like
particles and sometimes they behave like waves It is hard for us to comprehend the
nature of such an object by making mental pictures but experiments definitively show
that this is the case Since we have no direct experience with photons and elementary
particles (because their energies are so much smaller than the energies of the everyday
objects we encounter) it is perhaps not surprising that we have a hard time forming a
mental picture of how they behave
Electron Interference
Since an electron is both a particle and a wave just like a photon it should be able to
produce an interference pattern This is observed If an electron beam is shot at two
closely-spaced slits and if the electrons that pass through the slits are detected
downstream it is found that there are some locations where electrons are never detected
29
and others where lots of electrons are detected The pattern is exactly the same as the
one observed for light waves provided that we use the electron wavelength λ = hp in
place of the wavelength of light This pattern is observed even though each electron is
detected as a single dot on the screen Only after many such dots are collected does the
pattern emerge And if we try to understand how this effect could possibly work by
looking closely at each slit to see which one the electron came through the pattern
disappears the act of measurement destroys the interference This means that each
single electron somehow comes through both slits (as a wave would)
Uncertainty Principle
In classical physics we always imagine that the positions and momenta of moving
particles have definite values It might be hard to measure them but surely at each
instant of time a particle should be precisely located at some point in space and have a
similarly precise momentum This turns out not to be true Instead both position and
momentum are required to be uncertain with their uncertainties Δx and Δp satisfying
the Heisenberg uncertainty relation
2x p
So if the particle were known to be precisely at some particular location (so that Δx =
0) we couldnrsquot know anything about its momentum (Δp = infin) and if its momentum
were exactly known we couldnrsquot know its position
There is a similar relation involving the particlersquos energy E and the time interval Δt
over which this energy is measured
2E t
This means that energy is not actually conserved if we are considering very small time
intervals and this brief non-conservation of energy has been observed
30
Serway Chapter 41
Wave Function ψ and Probability
Quantum mechanics does not predict exactly what an electron or a photon will do Instead
it specifies the wave function or probability amplitude ψ of an electron or a photon
This wave function is a complex-valued function of space and time whose squared
magnitude is the probability density P for finding a particle at a particular place in
space at a certain time
2P
where is the complex conjugate of ψ
For example the wave function of an electron with perfectly specified momentum p
would have a wavelength given by
h
p
and its probability amplitude would be proportional to
2i x ipxe e
The corresponding probability density would then be
21ipx ip ipxP e e e
which means that the electron is equally probable to be anywhere along the x axis
This is in accord with the uncertainty principle since we specified the momentum
precisely we canrsquot have any idea about the position of the electron
As another example you have probably seen ldquofuzzy ballrdquo drawings of electron
orbitals in a chemistry book These fuzzy balls are meant to indicate the distribution
of the probability density 2
P in the orbital
Particle in a Box
A simple example in which we can calculate the wave function is the case of a particle of
mass m constrained to be inside a 1-dimensional box between x = 0 and x = L In this
case the general wave function is a linear superposition of wave functions ψn of the form
sin niE tn
n xx t A e
L
31
where A is a positive constant where n = 1 2 3 and where the energy associated
with each of the quantum states ψn is given by
22
28n
hE n
mL
This wave function is zero at x = 0 and x = L which means that the particle will never
be found at the walls of the box The wave function has maximum values in the
interior and at these places the particle is most likely to be found and it also has
places where it is zero and at these places the particle will also never be found as
expressed by the formula
2sinn x
PL
The particle in the box is interfering with itself producing a probability interference
pattern across the box just like the interference patterns we studied with light and
sound
32
Serway Chapter 42
Atoms
Many people picture an atom as a miniature solar system where electrons orbit around
a massive nucleus at the center This picture is misleading because in atoms the wave
nature of electrons dominates The electrons form 3-dimensional standing waves
(called orbitals) centered at the nucleus And if you ask what is it thatrsquos waving the
answer is ψ the probability amplitude (see Chapter 41)
Standing waves on a string can occur only for certain frequencies (the fundamental
and higher harmonics) Similarly atomic orbitals occur only for certain energies For
the hydrogen atom the energies of the orbitals have the a particularly simple form
2
1136 eVnE
n
where n = 1 2 3 is called the principal quantum number For other atoms the
determination of the orbital energies requires numerical calculation by computers
Atomic Spectra
If an electron is somehow given extra extra energy (we say that it is excited) so that it
occupies a higher orbital it will eventually ldquofallrdquo back down to a lower orbital Each
time an electron falls to a lower orbital it loses the difference in energy between the
two orbitals in the form of a photon Since the orbital energies are discrete so are the
energy differences and so are the wavelengths of the emitted light The entire set of
these discrete wavelengths is called the atomic spectrum and it is unique to each type
of atom For hydrogen the spectrum can be simply written as
H 2 2
1 1 1
f i
Rn n
where RH = 10973732 times 107 m-1 For other atoms the spectrum cannot be expressed as
a formula
Orbital Angular Momentum Electrons in atoms also have quantized values of
angular momentum The orbital quantum number ℓ specifies the value of this
quantized angular momentum through the formula
1L
33
If we want to know the value of the angular momentum along some direction in space
say the z direction the answer is not L but rather
zL m
where mℓ is another quantum number which runs from
1 1m
This quantum number is important when an atom sits in a magnetic field
Spin Angular Momentum It has been found experimentally that electrons and other
charged particles also carry internal angular momentum which we call spin
Electrons have an intrinsic spin angular momentum s along a specified axis that is
extremely quantized it can only take on 2 possible values
1
2zs s
We thus say that electrons have spin s = 12 or that electrons are rdquospin one-half parti-
cles Since s is an angular momentum it obeys the same rule as that for orbital angular
momentum namely that if its value along some axis is s then its total magnitude is
given by
31
2S s s
I know this seems weird but quantum mechanics is weird The only excuse for this
bizarre way of looking at the world is that it predicts what happens in experiments
Exclusion Principle The answer to the question of how many electrons (or any other
spin one-half particle) can be in one particular quantum state was discovered by
Wolfgang Pauli and is called the exclusion principle
ldquoNo two electrons can ever be in the same quantum state therefore no two electrons
in the same atom can have the same set of quantum numbersrdquo
This is the reason that we have atoms with different properties instead of every atom
simple having all of its electrons in the ground state All of the variety we see around
us in the world is the result of chemical differences and these differences would not
exist unless electrons obeyed this important principle The entire structure of the
periodic table (see pages 1377-1379 in Serway) is an expression of this principle
34
Serway Chapter 44
Nuclear Properties
The nucleus is composed of protons with charge +e and mass mp = 1007226 u and of
neutrons with zero charge and mass mn = 1008665 u where u is one atomic mass unit
-271 u=1660540 10 kg
The atomic number Z counts the number of protons in a nucleus while the neutron
number N counts the number of neutrons The mass number A is the sum of the two
A N Z
Protons and nuetrons have about the same mass and some times called baryons (heavy
ones) or nucleons (partcles in the nucleus) A is sometimes called the baryon
number The chemical elements are distinguished by Z ie hydrogen has Z = 1 iron
has Z = 26 uranium has Z = 92 etc But for a given Z there might be several nuclei
with different numbers of neutrons These nuclear siblings are called isotopes
These numbers are used to label nuclei according to the pattern
5626 ie FeA
Z X
denotes the isotope of iron with N = 56 ndash 26 = 30 neutrons
The nucleus is roughly spherical with a radius given approximately by
1 3 150 0where 12 10 mr r A r
Nuclear Stability
Because the positively charged protons electrically repel each other with an enormous
force at distances as small as 10-15 m there must be some really strong force that that
overcomes electrical repulsion to hold protons and neutrons together This force is
called with some lack of imagination the strong force It is a very short-range force
(it only acts over a distance of about 2 x 10-15 m) and attracts protons to protons
neutrons to neutrons and neutrons to protons But in spite of this strong nuclear force
the coulomb repulsion of the protons is still present so anything that might keep the
protons from being right next to each other would help keep the nucleus from
35
exploding This role is played by the neutrons and for nuclei with Z le 20 the stable
nuclei roughly have N = Z
For Z gt 20 the coulomb repulsion force is more powerful and more neutrons are
needed to dilute it up to about N = 15Z around Z = 80 For Z gt 83 no amount of
neutrons can help and these nuclei are unstable (radioactive) There is an ldquoisland of
(relative) stabilityrdquo around Z=90-92 (thorium and uranium respectively) These
elements have one or two isotopes with half-lifes of billions of years so there are
substantial amounts of such elements on Earth This fact makes it possible to have
practical fission devices
Radioactive Decay
There are three types of radioactive decay
Alpha decay the nucleus kicks out a helium nucleus (N = 2 Z = 2 A = 4) So if the
original nucleus is called X and the new nucleus is called Y then the decay would look
like this
4 42 2X Y+ HeA A
Z Z
Beta decay the nucleus either kicks out an electron (endash) or its positively-charged
antimatter twin the positron (e+) plus either an electron neutrino v or an electron
anti-neutrino v
1X Y+e electron decayA AZ Z v
1X Y+e positron decayA AZ Z v
A neutrino is a particle with no charge hardly any mass (much less than the electron
mass) and interacts so weakly with matter that most neutrinos upon encountering the
planet earth just pass right through it as if it werenrsquot there
Gamma decay the nucleons in the nucleus X are in an excited energy state X
(perhaps as a result of having undergone alpha or beta decay) and they drop down to a
lower energy state shedding the energy as a high frequency photon
X XA AZ Z
36
This process is exactly analogous to the way that the electrons in atoms emit photons
Decay Rate and Half Life
There is no way to predict exactly when an unstable or excited nucleus will decay but
there is an average rate at which this decay occurs called the decay constant λ The
meaning of this constant is that if there are a large number N of nuclei in a sample
then the number of decays per second that will be observed (called the decay rate R)
is R = λN In mathematical language
dMR N
dt
This simple differential equation has for its solution
0tN t N e
where N0 is the number of nuclei in the sample at time t = 0
The half-life is the time it takes for half of the nuclei in the sample to decay and is
related to the decay constant by
1 2
ln 2 0693T
Disintegration Energy
When a nucleus decays it is making a transition to an overall state of lower energy
which means according to Einsteinrsquos famous formula E = mc2 that the sum of the
masses after the decay must be less than the mass before with the lost mass appearing
as kinetic energy among the decay products For example in alpha decay this kinetic
energy called the disintegration energy Q is given by
2X YQ M M M c
37
Serway Chapter 45
Nuclear energy
There are two ways to extract energy from the nucleus fission and fusion For nuclei
with Z greater than 26 breaking the nucleus apart into pieces leads to a lower
2mc energy than the original nucleus so energy can be extracted by fission For
nuclei with Z less than 26 a lower energy is achieved by combining nuclei so energy
can be extracted by fusion
Fission
Since neutrons have no charge they are not repelled from nuclei as protons are for
this reason their behavior is the key to understanding how fission works
Fast neutrons mostly bounce off other nuclei with each collision slowing the neutron
down This elastic energy loss is most effective if the other nuclei have low mass (like
hydrogen) and these materials are called moderators because of their ability to slow
down fast neutrons
The reason that slowing neutrons is important is that slow neutrons are much more
likely to be absorbed by a nucleus which then leads to nuclear reactions of various
kinds For a few very large nuclei like uranium-235 and some plutonium isotopes
absorption of a slow neutron causes the nucleus to split into two large fragments plus
2 or 3 fast neutrons (energy is released in the process too) Thus one slow neutron
can produce energy plus 2 or 3 more neutrons which if moderated from fast to slow
can split 2 or 3 more nuclei leading to a runaway chain reaction If the reaction is
allowed to proceed unchecked it produces a large explosion If a neutron absorbing
material is added to the mix (like the cadmium in reactor control rods) it is possible to
keep the reaction under control and to extract the released energy as heat to drive
steam turbines and produce electricity
Fusion
Fusion involves mashing two nuclei together and since they are both charged and repel
each other this reaction is much harder to make go The nuclei must have enough energy to
overcome the coulomb repulsion which is why this reaction requires a high temperature
(hundreds of millions of degrees K) like that in the sun (which is burning hydrogen to
helium via fusion) or like that in the center of the fission explosion that is used to detonate
38
a hydrogen bomb
This reaction is of interest for power production in spite of this difficult temperature
requirement because of the abundance of fusion fuel on the planet There are about
012 g of deuterium ( 21H ) in every gallon of water on earth and it only costs about 4
cents to extract it The fusion energy available from this minuscule amount of
deuterium would run a 1000-Megawatt power plant for 10 seconds Water is so
abundant on earth that if fusion were to work we would have an essentially
inexhaustible source of energy
So why donrsquot we have fusion power plants Well the fuel is cheap but the match is
incredibly expensive The only way we know to control this difficult high-temperature
reaction is with large and expensive pieces of equipment involving either large
magnetic fields and complex high-power electromagnetic antennas or with gigantic
(football-fieldsized) laser facilities involving more than a hundred of the highest-
energy lasers ever built Power plants based on these current methods for controlling
fusion are unattractive to the fiscally-minded people who run the electric power
industry Hopefully better designs will be discovered as experiments continue
39
c actinium 68 Er erbium 101 Md mendelevium 104 Rf rutherfordium l aluminum 63 Eu europium 80 Hg mercury 62 Sm samarium m americium 100 Fm fermium 42 Mo molybdenum 21 Sc scandium b antimony 9 F fluorine 60 Nd neodymium 106 Sg seaborgium r argon 87 Fr francium 10 Ne neon 34 Se selenium s arsenic 64 Gd gadolinium 93 Np neptunium 14 Si silicon t astatine 31 Ga gallium 28 Ni nickel 47 Ag silver a barium 32 Ge germanium 41 Nb niobium 11 Na sodium k berkelium 79 Au gold 7 N nitrogen 38 Sr strontium e beryllium 72 Hf hafnium 102 No nobelium 16 S sulfur i bismuth 108 Hs hassium 76 Os osmium 73 Ta tantalum h bohrium 2 He helium 8 O oxygen 43 Tc technetium
boron 67 Ho holmium 46 Pd palladium 52 Te tellurium r bromine 1 H hydrogen 15 P phosphorus 65 Tb terbium d cadmium 49 In indium 78 Pt platinum 81 Tl thallium a calcium 53 I iodine 94 Pu plutonium 90 Th thorium f californium 77 Ir iridium 84 Po polonium 69 Tm thulium
carbon 26 Fe iron 19 K potassium 50 Sn tin e cerium 36 Kr krypton 59 Pr praseodymium 22 Ti titanium s cesium 57 La lanthanum 61 Pm promethium 74 W tungsten l chlorine 103 Lr lawrencium 91 Pa protactinium 92 U uranium r chromium 82 Pb lead 88 Ra radium 23 V vanadium o cobalt 3 Li lithium 86 Rn radon 54 Xe xenon u copper 71 Lu lutetium 75 Re rhenium 70 Yb ytterbium m curium 12 Mg magnesium 45 Rh rhodium 39 Y yttrium b dubnium 25 Mn manganese 37 Rb rubidium 30 Zn zinc y dysprosium 109 Mt meitnerium 44 Ru ruthenium 40 Zr zirconium s einsteinium
40
UNITS The SI units are given in parentheses Other commonly used units are given in terms of the SI units Symbols conform with the recommendations of the American National Standards Institute (ANSI) and the American Institute of Physics (AIP) Commonly used multiples of SI units
prefix symbol factor tera T 1012 giga G 109
mega M 106 kilo k 103
centi c 10-2 milli m 10-3
micro μ 10-6 nano n 10-9 pico p 10-12
femto f 10-15
length meter (m) angstrom 1 Ǻ = 10-10 m inch 1 in = 254 x 10-2 m foot 1 ft = 03048 m mile 1 mi = 1609 m
mass kilogram (kg) atomic mass unit 1 u = 1661 x 10-27 kgslug 1 slug = 1459 kg
force newton (N = kg bull ms2) dyne 1 dyn = 10-5 N pound 1 lb = 4448 N
pressure Pascal (Pa = kgm bull s2) atmosphere 1 atm = 1013 x 105 Pa poundssquare inch 1 psi = 6895 Pa cm of mercury 1 cm Hg = 1333 Pa bar 1 bar = 1000 x 105 Pa torr 1 torr = 1333 Pa
time second (s) minute 1 min = 60 s hour 1 h =3600 s
frequency hertz (Hz = s-1) radianssecond 1 rads = 12π Hz
energy joule (J = kg bull m2s2) erg 1 erg = 10-7 J electron volt 1 eV = 1602 x 10-19 J calorie 1 cal = 4187 J kilowatt-hour 1 kWh = 36 x 106 J British thermal unit 1 Btu = 1055 J
power watt (W = kg bull m2s3) horsepower 1 hp = 7457 W
charge coulomb (C = A bull s)
electric potential volt (V = kg bull m2 s3 bull A)
current ampere (A)
resistance ohm (Ω = kg bull m2s3 bull A2)
capacitance farad (F = s4 bull A2kg bull m2)
magnetic field tesla (T = kgs2 bull A) gauss 1 G = 10-4 T
magnetic flux weber (Wb = kg m2s2 bull A) maxwell 1 Mx = 10-8 Wb
magnetic inductance henry (H = kg bull m2s2 bullA2)
temperature kelvin (K) degrees Celsius 0degC = 27315 K
angle radian (rad) degree 1 deg = π180 rad revolution 1 rev = 2π rad
41
Some Physical Constantsa Quantity Symbol Valueb Atomic mass unit μ 1660 538 73 (13) x 10-27 kg
931494 013 (37) MeVc2 Avogadros number NA 6022 141 99 (47) x 1023 particlesmol Avogadrorsquos number 6022 x 1023 mol Bohr magneton
2B e
em 9274 008 99 (37) x 10-24 JT
Bohr magneton μB 927 x 10-24 JT Bohr radius
0
2
2e em e k
a 5291 772 083 (19) x 10-11 m
Boltzmanns constant B A
RNk 1380 650 3 (24) x 10-23 JK
Boltzmannrsquos constant kB 1380 x 10-23 JK Compton wavelength
C eh
m c 2426 310 215 (18) x 10-12 m
Deuteron mass md 3343 583 09 (26) x 10-27 kg 2013 553 212 71 (35) u
electron charge e 1602 x 10-19 C Electron mass me 9109 381 88 (72) x 10-31 kg
5485 799 110 (12) x 10-4 u 0510 998 902 (21) MeVc2
electron mass 911 x 10-31 kg Electron volt eV 1602 176 462 (63) x 10-19 J Elementary charge e 1602 176 462 (63) x 10-19 C Gas constant R 8314 472 (15) JKmol Gravitational constant G 6673 (10) x 10-11 Nm2kg2 Neutron mass mn 1674 927 16 (13) x 10-27 kg
1008 664 915 78 (55) u 939565 330 (38) MeVc2
neutron mass 1675 x 10-27 kg Nuclear magneton
2n p
em 5050 783 17 (20) x 10-27 JT
permeability constant μ0 1257 x 10-6 Hm permittivity constant ε0 8854 x 10-12 Fm Plancks constant h
2h
6626 068 76 (52) x 10-34 Js 1054 571 596 (82) x 10-34 Js
Planckrsquos constant h ħ
6626 x 10-34 Js 1055 x 10-34 Js
Proton mass mp 1672621 58 (13) x 10-27 kg 1007 276 466 88 (13) u 938271 998 (38) MeVc2
proton mass 1673 x 10-27 kg Rydberg constant RH 1097 373 156 854 9 (83) x 107 m-1 Speed of light in vacuum c 2997 92458 x 108 ms (exact) speed of light c 300 x 108 ms a
These constants are the values recommended in 1998 by CODATA based on a least-squares adjustment of data from different measurements For a more complete list see P J Mohr and B N Taylor Rev Mod Phys 72351 2000 b The numbers in parentheses for the values above represent the uncertainties of the last two digits
NOTE The ones in red are the ones from Appendix 13 The ones in yellow are the ones that are from the Serway book table but that were already on the Appendix 13 table
42
INDEX
Absolute zero 9
Adiabatic 17
Adiabatic exponent 18
Adiabatic process 17
Alpha decay 46
Amplitude 4
Angle of reflection 25
Angular frequency 3
Angular magnification 30
Angular momentum atomic 43
Angular size 29
Archimedes Principle 1
Atomic spectra 43
Atoms 43
Avogadrorsquos number 10
Beats 8
Bernoullirsquos Equation 2
Beta decay 46
Bifocals 29
Brewsterrsquos angle 34
British Thermal Unit Btu 11
Buoyancy 2
calorie 11
Calorie 11
Camera 28
Carnot cycle 21
Carnot efficiency 22
Celsius scale 9
Ciliary muscle 29
Coefficient of performance 21
Compton effect 39
Constructive interference 7
Continuity equation of 2
Contraction length 35
Convection 14
Curved mirrors 27
Decay constant 46
Decay rate radioactive 46
Decibel scale 5
Degrees of freedom 16
Density 1
Destructive interference 7
Diffraction single slit 33
Diffraction grating 33
Dilation time 35
Disintegration energy 47
Dispersion of light 26
Doppler effect 6
Double slit interference 31
Efficiency 20
Electron interference 39
Emissivity 15
Energy nuclear decay 47
Energy relativistic 36
43
Engineering work 20
Entropy 22
Entropy ideal gas 24
Equipartition of energy 18
Equivalence principle 36
Exclusion principle 44
Expansion thermal 9
Expansion coefficient linear 9
Eye 29
Fahrenheit scale 9
Far point 29
Farsightedness 29
First Law of Thermodynamics 12
Fission 48
Flux volume 2
Free expansion 23
Fringes 31
Fusion 48
Fusion heat of 11
Gamma relativistic 35
Gamma decay 46
General relativity 36
Half-life 46
Heat 11
Heat capacity 11
Heat conduction 13
Heat engine 20
Heat of fusion 11
Heat of vaporization 11
Heat pump 20
Heats of transformation 11
Hydrostatics 1
Ideal Gas Law 10
Images real and virtual 27
Intensity sound 5
Interference 7
Interference two-slit 31
Internal energy 12
Internal energy degrees of freedom 16
Irreversible process 19 23
Isotopes 45
Joule 11
Kelvin scale 9
Kinetic energy relativity 36
Kinetic theory 16
Latent heat 11
Length contraction 35
Linear expansion coefficient 9
Linear polarization 34
Linear superposition 7
Longitudinal wave 3
Loudness 5
Magnification lateral 27
Magnifying glass 30
Malusrsquos law 34
Microscope 30
Momentum relativistic 36
Muscle ciliary 29
44
Musical instruments 8
Musical scale 8
Near point 29
Nearsightedness 29
Nonlinear 7
Nuclear energy 48
Nuclear properties 45
Nuclear stability 45
Octave 8
Optical resolution 33
Orbital quantum number ℓ 43
Orbitals 43
Particle in a Box 41
Particles are waves 39
Pascalrsquos Principle 1
Period 3
Photoelectric effect 38
Photons 38
Pitch 8
Polarization 34
Power sound 5
Presbyopia 29
Pressure 1
Principal quantum number n 43
Principle of equivalence 36
Principle of linear superposition 7
Probability amplitude ψ 41
Processes thermodynamic 13
R-value 14
Radiation thermal 14
Radioactive decay 46
Radioactive decay rate 46
Ray tracing 27
Rayleighrsquos criterion 33
Reading glasses 29
Real image 27
Refraction 25
Refrigerator 20
Relativistic gamma 35
Relativity principles 35
Resolved for light sources 33
Rest energy 36
Reversible process 19 22
Rope wave speed 4
Second Law of Thermodynamics 19
Shock waves 6
Simple magnifier 30
Simultaneity 35
Single slit diffraction 33
Snellrsquos law 25
Sound speed 5
Specific heat 11
Spin Angular momentum 44
Standing waves 7
Stefanrsquos law 14
Strong force 45
Telescope 30
Temperature 9
45
Temperature Scales 9
Thermal conductivity 14
Thermal energy 12
Thermal expansion 9
Thin film interference 31
Thin lenses 28
Time dilation 35
Tone musical 8
Total internal reflection 25
Transverse wave 3
Traveling Waves 3
Two-slit interference 31
Uncertainty principle 40
Vaporization heat of 11
Velocity addition relativity 36
Virtual image 27
Volume flux 2
Wave function 41
Wave Function ψ and Probability 41
Wave speed 3
Wave-particle duality 39
Wavelength 3
Wavenumber 3
Wien Displacement Law 16
Work 11