Pre-Health Physics Review Johnny B. Holmes, Ph.D..

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Pre-Health Physics Review Johnny B. Holmes, Ph.D.

Transcript of Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Page 1: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Pre-Health Physics Review

Johnny B. Holmes, Ph.D.

Page 2: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

FundamentalsMKS system of units:

M for meters (distance)K for kilograms (mass)S for seconds (time)Also add Q in Coulombs (electric charge)

Other units are combinations of these fundamental units:Speed in m/s; Acceleration in m/s2; Force in Nt=kg*m/s2; Energy in Joules = Nt*m (convert to calories, BTUs);Power in Watts = Joules/secPressure in Pascals = Nt/m2 (convert to psi, atm, mm of Hg, bars); Current in amps = Coul/sec; Voltage in volts = Joule/Coul.

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Common Prefixes

milli = 10-3 (m) Kilo = 103 (k)

micro = 10-6 () Mega = 106 (M)

nano = 10-9 (n) Giga = 109 (G)

pico = 10-12 (p) Tera = 1012 (T)

femto = 10-15 (f)

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VectorsSpace is three-dimensional, so many

quantities in physics have directions as well as magnitudes. For 3-D, need three numbers to specify the quantity (called a vector).

Common forms for vectors are rectangular (x,y,z) in 2-D (x,y)spherical (r,) in 2-D (r,)cylindrical (r,,z)

To add vectors, must add in rectangular!

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Motion

Velocity: v = dx/dt; for constant velocity, have x = xo + vo*t (straight line)v is the slope of the x vs t curve.

Acceleration: a = dv/dt; for constant acc, have y = yo + vo*t + ½*a*t2 (parabola)a is the slope of the v vs t curve.

For 2 & 3 dimensions, work in rectangular components.

For trajectories, ax = 0 and ay = -g = -9.8 m/s2

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Circular Motion

Work and think in rectangular (x,y)Convert to polar (r,) for easier equations:

r = constant changes in timevr = 0 d/dt = constant for uniform CM

v = *r = constant for uniform CM

ar = -2*r directed towards the center due to direction of velocity changing

a = *r where = d/dt is angular acc= 0 for uniform CM

Also, f and 1/f = T (f is frequency, T is period)

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Newton’s Laws of Motion

F = m*a (vector equation so work in rectangular components)

The equation above is Newton’s 2nd law. Newton’s 1st law is a special case of the 2nd :

if F=0, then a=0.Newton’s 3rd law: for every action there is an equal

and oppositely directed action (or, you can’t push yourself). The forces in Newton’s 2nd law are the ones ON the object, not the ones BY the object.

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Forces

• Contact: Fc balances, perpendicular to surface

• Friction: Ff <= *Fc parallel to surface

• Tension: T is same along the string; directed parallel to the string

• Weight: near earth’s surface, W = Fgravity = m*g directed down.

• Gravity far from earth’s surface: Fgravity = G*M*m/r2 with G = 6.67x10-11Nt*m2/kg2

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Example: Circular Orbits

For a satellite to orbit the moon at a height of 100 km, how fast should it go and how long will it take to make one orbit?Radius of Moon is 1,700 km; mass of moon is 7.2 x 1022 kg.

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Satellite Problem – cont.Recognize: 1. Circular orbit, so ar = 2*r, v = *r, =2/T

2. Gravity is the only force: F = G*M*m/r2

3. Use F = m*a

4. Recognize that r = Rmoon + h = 1,700 km + 200 km = 1,900 km.

5. Therefore: G*M*m/r2 = m*2*r6. Recognize that m (mass of satellite)

doesn’t matter, solve for then use v = *r, =2/T to get both v and T.

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Energy and Conservation of Energy

Work = Force through a distance (a scalar)W = F*s*cos(Fs)

Energy is the capacity to do work (in ideal situations)

Kinetic energy = KE = ½*m*v2

Gravitational Potential energy:near the earth, PEgravity = m*g*hin general, PEgravity = -G*M*m/r

Spring Potential energy: PEspring = ½k*(x-xo)2

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Example: Escape Speed

Energiesinitially = Energiesfinally .

KEi + PEi = KEf + PEf

(1/2)mvi2 - Gmearthm/ri = (1/2)mvf

2 - Gmearthm/rf (note that this is equivalent to saying KE = PE, or

½mvf2 – ½mvi

2 = -Gmem/rf - -Gmem/ri)

We see that m is in each term, so we cancel it. (1/2)*(vi)2 - (6.67x10-11 Nt*m2/kg2) *(6.0 x 1024 kg)/(6.4x106m) =

(1/2)*(0 m/s)2 - (6.67x10-11 Nt*m2/kg2) *(6.0 x 1024 kg)/(infinity)

We again have one equation in one unknown (v i).

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Power

We now know what Force and Energy are, but what is Power?

The definition of Power is that it is the rate of change of Energy from one form into another: Power = Energyt .

The units of power are: Joule/sec = Watt.Another common unit is the horsepower, hp.

The conversion factor is: 1 hp = 746 Watts.

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Review of Rotational Equations

Basically we replace F with , m with I, v with , a with andp with L (where L is the angular momentum):

F = ma = IWork = = F ds Work = d

Power = F v Power =

KE = (1/2)mv2 KErotation = (1/2)I2

p = mv L = I F = p/t = L/t .

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Example: Your elbow

Let’s consider as an example of torque how your muscles, bones and joints work.

Consider holding up a ball of weight 5 lb.

How does this work?

First we draw a diagram: biceps

triceps weight

= elbow

rw rb

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Your elbow

From F = 0 we have:

-Fc + Fb - W = 0

And from and measuring from the elbow gives: Fc*rc + Fb*rb - W*rw = 0 .

We have two equations and we have two unknowns (Fc and Fb).

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Your elbow

• We can use the torque equation first, since rc=0 eliminates one of the unknowns, Fc.

Fc*rc + Fb*rb - W*rw = 0 or Fb = W*rw/rb .

• Then we can use the force equation to find Fc : -Fc + Fb - W = 0, or Fc = Fb - W.

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Example: Object rolling down an incline

Conservation of Energy

(KEregular + KErotational + PEgravity)initial =

(KEregular + KErotational + PEgravity)initial + Elost

0 + 0 + mgh = (1/2)mv2 + (1/2)I2 + 0 + 0.

Substituting I=(2/5)mr2 and =v/r gives:

mgh = (1/2)mv2 + (1/2)[(2/5)mr2][v2/r2] , or

mgh = (1/2)mv2 + (1/5)mv2 = (7/10)mv2 .

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Momentum andConservation of Momentum

p = m*v (p and v are vectors!)

Conservation of Momentum

Fxext on 1 + Fx

ext on 2 = (px1 + px2) / t .

If the external forces are small, or if the time of the collision, t, is small, then we have:

(px1 + px2) = 0. This can be re-written as:

(px1 + px2)i = (px1 + px2)f .

This is called Conservation of Momentum. This is a vector law, so a similar equation holds for each component of momentum.

Page 20: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

1-D Collisions

In one dimensional collision cases, we can apply two laws: Conservation of Momentum and Conservation of Energy (here we assume there are no PE’s that change):

(1/2)m1v1i2 + (1/2)m2v2i

2 =

(1/2)m1v1f2 + (1/2)m2v2f

2 + Elost

m1v1i + m2v2i = m1v1f + m2v2f

These are two equations with 7 quantities:

m1, m2, v1i, v2i, v1f, v2f, Elost . Hence if we know five, we can solve for the other two.

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Explosions

Normally, in an explosion the initial object is in one piece and at rest. After the explosion, one piece goes forward. Conservation of Momentum says the other piece must then go backwards. (If we brace ourselves, we can compensate for that backwards push and not fall over.)

0 + 0 = m1v1f + m2v2f or v2f = - m1v1f / m2.

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Pressure

P (pressure, not power or momentum)P = Force/Area (definition)(force is perpendicular to area, not parallel to it)

units of pressure: – Nt/m2

– 1 atmosphere = 1.01 x 105 Nt/m2 = 14.7 lb/in2

– 1 bar = 1.00 x 105 Nt/m2

– 1 Torr = 1 mm of Hg, 760 Torr = 1 atmosphere

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Pressure

effects of gravity:– consider little cube of fluid– consider forces on the fluid in y direction1. weight acts down2. pressure underneath pushes up3. pressure on top pushes down– Fy = -m*g + Pbottom*Abottom - Ptop*Atop = 0 ,– where m = V = *A*h, and A = Abottom = Atop

so: Pbottom*Abottom - Ptop*Atop = m*g = *A*h*g ,

or: Pbottom - Ptop = *g*h .Buoyant Force = P*A = *g*V

W=mg

Pbottom

Ptop

h

Pside

Page 24: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Fluid FlowConservation of Energy:

(1/2)mvi2 + mghi + Won = (1/2)mvf

2 + mghf + Wby

divide each term by Volume, and note m/V=,

also note W = F*s, F=P*A, A*s=V, so Work = P*V:(1/2)vi

2 + ghi + Pi = (1/2)vf2 + ghf + Pf + Plost

examples: • lift on wing of airplane • coffee pot• siphon• oil well

Page 25: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Fluid Flow

Viscous Force: F dv/ds ;

for tubes (cylindrical hoses) with constant velocity (Fapplied = Fresisted, F = P*A )

Pr2rL) dv/dr

Q = P)R4 / (8L) wwhere P = Plost and Q is the volume per time flowing.

Power = Work/t = F*s/t = Pressure*A*s/t = Pressure*V/t = Pressure*Q

Page 26: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Reynolds Number

Have laminar flow (previously assumed layer over layer flow) as long as flow is slow enough; otherwise have turbulent flow

Reynolds number:

R = 2vavgr = 2Q/(r) (dimensionless!)

If R < 2,000, then laminar;

If R > 2,000, then turbulent.

Page 27: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Ideal Gas Law

P*V = N*k*T . We further define R = Na*k, where Na = 6.02 x 1023 = 1 mole. Thus we have: P*V = n*R*T , where

n = N/Na = number of moles in volume, V;

T must be in Kelvin, Not oF or oC !

k = experimental constant = 1.38 x 10-23 J/K ;

R = Na*k = 8.3 Joules/mole*Kelvin .

Page 28: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

HEAT CAPACITYThe amount of energy necessary to heat a

material per temperature change is what we call the heat capacity:

C (heat capacity) = Q/T where Q is the energy to raise temperature of an

amount of material by T.Usually we specify the heat capacity in one of

three ways: per object, per mole (usually for gases), and per mass (usually for liquids and solids).

Page 29: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Heat Capacity of Air

Cmolar-constant P = Cmolar-constant V + R

Air is made up mostly of N2 and O2. These gases act approximately as diatomic ideal gases. Usually, when we heat air it is NOT in a contained volume but expands to keep its pressure constant. This means that most of the time, the heat capacity of air is:

Cmolar - air - constant P = (5/2)R + R = (7/2)R .

Page 30: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Heat Capacity of Materials

By definition, a calorie is the energy necessary to raise the temperature of 1 gram of water up 1oC.

Cwater = 1 cal/gm-oC = 4.186 J/gm-oC

Cethyl alcohol = 2.400 J/gm-oC

Cwood = 1.700 J/gm-oC

Cglass = 0.837 J/gm-oC

Ccopper = 0.387 J/gm-oCSince liquids and solids don’t expand to fill the space like

gases do, we don’t usually distinguish between heat capacities at constant pressure versus constant volume.

Page 31: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Latent Heat

For water, the latent heat of fusion (heat needed to melt ice to water) is 79.7 cal/gm.

For water, the latent heat of vaporization (heat needed to boil water) is 540 cal/gm.

For alcohol, the latent heat of vaporization is less at 204 cal/gm.

Page 32: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Heat TransferThere are four ways of moving heat:

• Evaporation (using latent heat) • Convection (moving heat with a material)

• Conduction (moving heat through a material)

• Radiation (light, usually mainly in the infrared, both emitted and absorbed)

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Heat Transfer: Conduction

Power = Q/t = k*A*T/Lwhere k is a constant that depends on the

material, called the thermal conductivity;

where A is the cross sectional area;

where L is the distance from the hot end to the cold end;

and T is the temperature difference between the hot and cold ends.

A

L

kThi Tlow

L

hot cold

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Blackbody Radiation:Experimental Results

Ptotal = AT4

where = 5.67 x 10-8 W/m2 *K4

peak = b/T where b = 2.9 x 10-3

m*KIntensity(log scale)

wavelengthUV IRblue yellow red

Page 35: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Thermodynamics

The First Law of Thermodynamics is a fancy name for the Law of Conservation of Energy applied to thermal systems. It says:

U = Q - W

where U indicates the change in the internal energy of the system. This internal energy is related to the temperature and heat capacity of the system; Q is the heat energy added to the system; and W is the work done by the system.

Page 36: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Second Law of Thermodynamics

Entropy is a measure of the probability of being in a state. Since things tend to go to their most probable state, we can write the 2nd Law of Thermodynamics as: systems tend to have their entropy increase.

Page 37: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

EfficiencyEfficiency is a measure of how much you get out

versus how much you put in. For heat engines:

Efficiency = = Work done / Heat Added

By the first law, the work done is simply the difference in the heat going into the engine minus the heat coming out of the engine. The total heat added is the heat going into the engine. = (Qhot - Qcold) /Qhot . For the most efficient engine

possible: Carnot = (Thot - Tcold) / Thot

Page 38: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Oscillations with a mass on a spring?

y = A sin(t + o)

The oscillation speed, describes how fast the mass oscillates. But what does this oscillation speed (ω=dphase/dt) depend on?

By putting in our solution for y into Newton’s Second Law (the differential equation), we can get a prediction: = (k/m) .

For stiffer springs and lighter masses, the frequency of the oscillation increases.

Note: the Amplitude does NOT affect the frequency!

Page 39: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Energy: Amplitude and frequency

Since Energy = (1/2) m2A2 , as the frequency goes up (ω), to keep the same energy the amplitude (A) needs to go down. Can you make sense of that relationship?

Since kinetic energy depends on velocity (squared), and since v = dx/dt , a higher frequency means that for the same distance (amplitude) we have a smaller dt. To keep the same v, we need a smaller distance (amplitude) to go with the smaller dt (higher frequency).

Page 40: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Waves (in general)

y = A sin() where is a phase angle

in a moving wave, changes with both– time (goes 2 radians in time T) and– distance (goes 2 radians in distance )

so = (2/)*x +/- (2/T)*t – where 2/T = and– where 2/ = k and so

phase speed: v = distance/time = /T = f = /k

Page 41: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Standing WavesTo create what are called standing waves (we will

play with these in the last lab), we need to create constructive interference from both ends. This leads to the following condition: #(/2) = L ,

which says: we need an integer number of half wavelengths to “fit” on the Length of the string for standing waves.

We can vary the wavelength by either varying the frequency or the speed of the wave: recall that phase speed: v = distance/time = /Tperiod = f . For a wave on a string,

recall that v = f = (Ttension)/) where m/L.

Page 42: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Standing WavesFor stringed instruments (piano, guitar, etc.), the

string vibrates with both ends fixed. However, with wind instruments (trumpet, trombone, etc.), we can have the situation where both ends are free and a different situation where one end is free and one end is fixed.

1. If both ends are free, we get the same resonance condition as for both ends fixed: #(/2) = L.

2. If one end is free and the other end is fixed, we get a different condition: #odd(/4) = L, where #odd is an odd number (1, 3, 5, etc.).

Page 43: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Sound Intensity

I(dB) = 10*log10(I/Io) where Io = 10-12 W/m2

The weakest sound intensity we can hear is what we define as Io. In decibels this becomes:

I(dB) = 10*log10(10-12 W/m2 / 10-12 W/m2) = 0 dB .

The loudest sound without damaging the ear is 1 W/m2, so in decibels this becomes:

I(dB) = 10*log10(1 W/m2 / 10-12 W/m2) = 120 dB .

Page 44: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Electric Force

To account for repulsive and attractive forces, we find that like charges repel, and unlike charges attract.

We also find that the force decreases with distance between the charges just like gravity, so we have Coulomb’s Law:

Felectricity = k q1 q2 / r122 where k, like G in

gravity, describes the strength of the force in terms of the units used.

Page 45: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Electric Force

Charge is a fundamental quantity, like length, mass and time. The unit of charge in the MKS system is called the Coulomb.

When charges are in Coulombs, forces in Newtons, and distances in meters, the Coulomb constant, k, has the value:

k = 9.0 x 109 Nt*m2 / Coul2 . (Compare this to G which is 6.67 x 10-11 Nt*m2 / kg2 !)

Page 46: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Fundamental ChargesWhen we break matter up, we find there are just a few

fundamental particles: electron, proton and neutron. (The proton and neutron are now thought to be made up of more elementary particles called quarks, while the electron remains elementary.)

electron: qe = -1.6 x 10-19 Coul; me = 9.1 x 10-31 kg

proton: qp = +1.6 x 10-19 Coul; mp = 1.67 x 10-27 kg

neutron: qn = 0; mn = 1.67 x 10-27 kg(note: despite what appears above, the mass of neutron and proton are

NOT exactly the same; the neutron is slightly heavier; however, the charge of the proton and electron ARE exactly the same - except for sign)

Page 47: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Electric Field for a point charge

If I have just one point charge setting up the field, and a second point charge comes into the field, I know (from Coulomb’s Law) that

Fon 2 = k q1 q2 / r122 and

Fon 2 = q2 * Eat 2 which gives:

E at 2 due to 1 = k q1 / r122 for a point charge.

Page 48: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Electric Potential EnergySince Coulomb’s Law has the same form as

Newton’s Law of Gravity, we will get a very similar formula for electric potential energy:

PEel = k q1 q2 / r12

Recall for gravity, PEgr = - G m1 m2 / r12 .

Note that the PEelectric does NOT have a minus sign. This is because two like charges repel instead of attract as in gravity.

Page 49: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Voltage

Just like we did with forces on particles to get fields in space,

(Eat 2 due to 1 = Fon 2/ q2)

we can define an electric voltage in space (a scalar):

Vat 2 due to 1 = PEof 2 / q2 .

We often use this definition this way:

PEof 2 = q2 * Vat 2 .

Page 50: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Voltage and Field

V = - E s , or Ex = -V / x .

Note also the minus sign means that electric field goes from high voltage towards low voltage. Note also that this means that

positive charges will tend to “fall” from high voltage to low voltage (like masses tend to fall from

high places to low places) , but that

negative charges will tend to “rise” from low voltage to high voltage (like bubbles tend to rise) !

Page 51: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Review

F1on2 = k q1 q2 / r122 PE12 = k q1 q2 / r12

Fon 2 = q2 Eat 2 PEof 2 = q2 Vat 2

Eat 2 = k q1 / r122 Vat 2 = k q1 / r12

use in use in

F = ma KEi + PEi = KEf +PEf +Elost

VECTOR scalar

Ex = -V / x

Page 52: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Electric CircuitsIn electricity, the concept of voltage will be like

pressure. Water flows from high pressure to low pressure (this is consistent with our previous analogy that Voltage is like height since P = gh for fluids) ; electricity flows from high voltage to low voltage.

But what flows in electricity? Charges!

How do we measure this flow? By Current:

current = I = q / t

UNITS: Amp(ere) = Coulomb / second

Page 53: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Resistance

By experiment we find that if we increase the voltage, we increase the current: V is proportional to I. The constant of proportionality we call the resistance, R:

V = I*R Ohm’s Law

UNITS: R = V/I so Ohm = Volt / Amp.

The symbol for resistance is (capital omega).

Page 54: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

ResistanceThe resistance depends on material and

geometry (shape). For a wire, we have:

R = L / A

where is called the resistivity (in Ohm-m) and measures how hard it is for current to flow through the material, L is the length of the wire, and A is the cross-sectional area of the wire. The second lab experiment deals with Ohm’s Law and the above equation.

Page 55: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Electrical Power

The electrical potential energy of a charge is:

PE = q*V .

Power is the change in energy with respect to time:Power = PE / t .

Putting these two concepts together we have:

Power = (qV) / t = V(q) / t = I*V.

Page 56: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Capacitance

We define capacitance as the amount of charge stored per volt: C = Qstored / V.

UNITS: Farad = Coulomb / Volt

Just as the capacity of a water tower depends on the size and shape, so the capacitance of a capacitor depends on its size and shape. Just as a big water tower can contain more water per foot (or per unit pressure), so a big capacitor can store more charge per volt.

Page 57: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

CapacitanceWhile we normally define the capacity of a water

tank by the TOTAL AMOUNT of water it can hold, we define the capacitance of an electric capacitor as the AMOUNT OF CHARGE PER VOLT instead.

There is a TOTAL AMOUNT of charge a capacitor can hold, and this corresponds to a MAXIMUM VOLTAGE that can be placed across the capacitor. Each capacitor DOES HAVE A MAXIMUM VOLTAGE.

Page 58: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Review:Capacitors: C = Q/V

PE = ½CV2; C// = KA/[4kd] Series: 1/Ceff = 1/C1 + 1/C2

Parallel: Ceff = C1 + C2

series gives smallest Ceff , parallel gives largest Ceff .

Resistors: V = IR Power = IV; R = L/A Series: Reff = R1 + R2

Parallel: 1/Reff = 1/R1 + 1/R2

series gives largest Reff , parallel gives smallest Reff .

Page 59: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Magnetic Force Lawmagnitude: Fmagnetic = q v B sin(vB)

direction: right hand rule:

thumb = hand fingers

Point your right hand in the direction of v, curl you fingers in the direction of B, and the force will be in the direction of your thumb; if the charge is negative, the force direction is opposite that of your thumb (or use you left hand).

Page 60: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Magnetic Force and Motion

Since the magnetic field is perpendicular to the velocity, and if the magnetic force is the only force acting on a moving charge, the force will cause the charge to go in a circle:

F = ma, Fmag = q v B, and a = 2r = v2/r

gives: q v B = mv2/r, or r = mv/qB .This is the basis of the mass spectrometer and the cyclotron.

Page 61: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Torque on rectangular current loop

Recall that torque is: = r F sin(rF). For magnetic force, F = qvB becomes F = ILB. In the figure below we can see that r = w/2. Thus the Fleft gives a torque of (w/2)ILB, and the Fright also gives a torque of (w/2)ILB.

r

F F

N B S L

w

Page 62: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Lenz’s Law

V =(N B A cos(BA) ] / t

The above formula is for determining the amount of voltage generated. But what is the direction of that voltage (what direction will it try to drive a current)?

The answer is Lenz’s Law: the direction of the induced voltage will tend to induce a current to oppose the change in magnetic field through the area.

Page 63: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

RMS Voltage and Current

In order to work with AC circuits just as we did with DC circuits, we create a voltage and current called rms (root mean square).

Vrms = Vo (1/2)1/2 and Irms = Io (1/2)1/2

so that we have

Pavg = Irms Vrms and Vrms = Irms R .

Note that the power formula and Ohm’s Law are the same for DC and for AC-rms, but NOT for instantaneous AC.

Page 64: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Review of Circuit Elements

Resistor: VR = R I where I = q/t

Capacitor: VC = (1/C)q (from C = q/V)

Inductor: VL = -L I/t

We can make an analogy with mechanics:

q is like x; V is like F;

t is like t; L is like m;

I = q/t is like v = x/t; C is like 1/k (spring);

I/t is like a = v/t; R is like air resistance.

Page 65: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

AC Circuits

A resistor obviously limits the current in a circuit. But, as we just saw, a capacitor and an inductor also limit the current in an AC circuit. However, the reactances do not just add together. Using the fundamental relations and the calculus, we come up with the concept of impedance, Z: V = IZ where Z takes into account all three reactances: XR=R, XL=L and XC= 1/C:

Z = [R2 + (L - 1/C)2]1/2.

Power, however, is still: Pavg = I2R (not P=I2Z).

Page 66: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Property 1: Speed of Light

particle (photon): no prediction

wave (E&M): Maxwell’s Eqs.in vacuum:

v = [1 / {o o}]1/2 where

o = 1/{4k} = 1 / {4 * 9x109 Nt-m2/Coul2}

o = 4 * 1x10-7 T-s /Coulv = [4*9x109 / 4*1x10-7 ]1/2 = 3 x 108 m/s = c

Page 67: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Property 2: Color

Experiment:– invisible as well as visible– total spectrum order:

• radio• microwave• IR• visible• UV• x-ray and gamma ray

Page 68: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Property 2: Color

Experiment:– visible order:

• red• orange• yellow (yellow)• green• blue• violet

Page 69: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Property 3: ReflectionA white paper is rough on a microscopic level, and

so a beam of light is reflected in all directions:

A mirror is smooth on a microscopic level, and so a beam of light is all reflected in just one direction.

rough paper smooth mirror

Blue is incoming, red is outgoing

Page 70: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Property 4: Refraction

Snell’s Law: n1 sin(1) = n2 sin(2)• NOTE: If n1 > n2 (v1 < v2), THEN 1 < 2 .

• NOTE: All 2 values (in the faster medium) between 0 & 90 degrees work fine.

• NOTE: Not all values of 1 (in the slower medium) work!

Example: If n1 = 1.33, n2 = 1, and 1 = 75o, then

2 = inv sin [n1 sin(1) / n2] = inv sin [1.28] = ERROR - this is called total internal reflection

Page 71: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Refraction and Thin Lenses

We break the THIN LENS equation:

(nglass – nair) 1 1 1 1

nair R1 R2 s s’

Into the LENS MAKERS equation

and the LENS USERS equation:

(nglass – nair) 1 1 1 1 1 1

nair R1 R2 f f s s’

where f is a distance called the focal length.

* { =+ +

* {

}

+ } = & = +

Page 72: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Refraction and the Lens-users Eq.

f f

For s>f (lens used with camera or projector)

– Note that a real image is formed.

– Note that the image is up-side-down.

ray 1

ray 2

ray 3

object

image

Page 73: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Refraction and the Lens-users Eq.

f f

For s<f (lens used as a magnifying glass)

Notice that: s’ is on the “wrong” side, which

means that s’ < 0 , and that |s’| > |s| so f > 0.

s

ray 1

ray 2

ray 3

s’

h’

Example: 1 / 5 cm = 1 / 4 cm + 1 / -20 cm

Page 74: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Refraction and the Lens-users Eq.

Notes on using a lens as a magnifying glass:

• hold lens very near your eye

• want IMAGE at best viewing distance

which has the nominal value of 25 cm

so that s’ = -25 cm.

Page 75: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Microscope

M1 = -s1’/s1 M2 = -s2’/s2

Mtotal = M1 * M2 = (s2’*s1’) / (s2*s1)

Object 1

objective lens

Image 1Object 2

eyepiece

Image 2

s1 s1’ s2

s2’

L = s1’ + s2

1/s1 + 1/s1’ = 1/f1

1/s2 + 1/s2’ = 1/f2

NOTE: s2’ = -25 cmso Mtotal < 0 !

Page 76: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Property 5: ShadowsDouble Slit Experiment

n = d sin(n) d tan(n) = d (xn / L)

d

SCREEN

L

xbright

bright

dim

dim

bright

Page 77: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Diffraction: single slit

REVIEW:• For double (and multiple) slits:

n = d sin(n) for MAXIMUM

(for ALL n)

• For single slit:

n = w sin(n) for MINIMUM

(for all n EXCEPT 0)

-2 -1 0 1 2

-2 -1 0 1 2

Page 78: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Diffraction: circular opening

If instead of a single SLIT, we have a CIRCULAR opening, the changein geometry makes:

the single slit pattern into a series of rings; and

the formula to be: 1.22 n = D sin(n) .

Page 79: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Rayleigh Criterion: a picture

In this case: limit = sin-1(1.22 /D)

= tan-1(x’/s’) = tan-1(x/s) .

lensD

s’

x’

s

x

Page 80: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Limits on Resolution:

• Imperfections in the eye (correctable with glasses)

• Rayleigh Criterion due to wavelength of visible light

• Graininess of retinal cells (Note that in low light where only the rods are activated, we cannot resolve very well because the rod cells are not packed as closely as the cone cells are. Also in low light we only see in black and white – not in color.)

Page 81: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Cone cells & Color Recognition

400 450 500 550 600 650 700 (in nm)

If only the “blue” cone is activated, the color is violet.If both the “blue” and “green” cones are activated, and the “blue” gives

a stronger signal, the color is blue.If both the “blue” and “green” cones are activated, and the “green”

gives a stronger signal, the color is green.

Cone cell sensitivity to different wavelengths

Page 82: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Polarization: Wave Theory

Three polarizers in series:

Sailboat analogy:

North wind

sail

force onsail boat goes along

direction of keel

Page 83: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Polarization: Wave Theory

#2 Polarization by reflection– Brewster Angle: when refracted + reflected = 90o

– Sunglasses and reflected glare

surface

incident rayvertical

horizontal

refracted ray

reflected rayno problem with horizontal

almost no vertical since vertical is essentially longitudinal now

vertical can be transmitted

Page 84: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Interference: Thin Films

• Recall that the light is in the FILM, so the wavelength is not that in AIR: f = a/nf.

air

film

water

reflected red interferes withrefracted/reflected/refracted blue.

t

Page 85: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Interference: Thin Films

• reflection: no difference if nf < nw;

180 degree difference if nf > nw.

• distance: no difference if t = a/2nf

180 degree difference if t = a/4nf

• Total phase difference is sum of the above two effects.

Page 86: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Interference: Thin Films

• Total phase difference is sum of the two effects of distance and reflection

• For minimum reflection, need total to be 180 degrees.– anti-reflective coating on lens

• For maximum reflection, need total to be 0 degrees.– colors on oil slick

Page 87: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Photons and Colors

• Electron volts are useful size units of energy

1 eV = 1.6 x 10-19 Coul * 1V = 1.6 x 10-19 J.• radio photon: hf = 6.63 x 10-34 J*s * 1 x 106 /s =

6.63 x 10-28 J = 4 x 10-9 eV = 4 neV• red photon: f = c/3 x 108 m/s / 7 x 10-7 m = 4.3 x 1014 Hz, red photon energy = 1.78 eV

• blue: = 400 nm; photon energy = 3.11 eV .• X-ray photon with = 1 nm; photon energy =

1,243 eV = 1.24 keV

Page 88: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Photoelectric Effect

Light hits a metal plate, and electrons are ejected. These electrons are collected in the circuit and form a current.

A

light

+ -

V

ejected electron

Page 89: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Photoelectric Effect

Put into a nice equation:

• hf = W + e*Vstop

– where f is the frequency of the light– W is the “WORK FUNCTION”, or the amount

of energy needed to get the electron out of the metal

– Vstop is the stopping potential

• When Vstop = 0, f = fcutoff , and hfcutoff = W.

Page 90: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Rutherford Scattering

The results of the scattering were consistent with the alphas scattering off a tiny positive massive nucleus rather than the diffuse positive pudding.

The results indicated that the positive charge and heavy mass were located in a nucleus on the order of 10-14 m

(Recall the atom size is on the order of 10-10 m).

Page 91: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

The Bohr TheoryBohr Theory: angular momentum, radius and Energy are all

quantized (with quantum number, n)

r = n22/(meke2) = (5.3 x 10-11 m) * n2

(for n=1, this is just the right size radius for the atom) and

E = [-mek2e4/22]*(1/n2) = -13.6 eV / n2

(where 1 eV = 1.6 x 10-19 Joules).

This says the electron energy is QUANTIZED.

Page 92: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

The Bohr Theory - an example

E = hf = [-13.6 eV]*[(1/nf2) - (1/ni

2)]

Example:

In the case of ni = 3, and nf = 2,

E = (-13.6 eV)*(1/4 - 1/9) = 1.89 eV

E = hf = hc/ , so in this case,

emitted = hc/E =

(6.63x10-34 J-sec)*(3x108 m/s)/(1.89 x 1.6x10-19 J)

= 658 nm (red light).

Page 93: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

The Bohr Theory

• Note that we have quantized energy states for the orbiting electron.

• Note that for all nfinal = 1, we only get UV photons.

• Note that for all nfinal > 2, we only get IR photons.

Page 94: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

DeBroglie Hypothesis

Problem with Bohr Theory: WHY L = n ?

• have integers with standing waves:

n(/2) = Length

• consider circular path for standing wave:

n = 2r , and so from Bohr theory:

L = mvr = n = nh/2get 2r = nh/mv = nwhich means = h/mv = h/p .

Page 95: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Quantum Theory

What we are now dealing with is the Quantum Theory:

• atoms are quantized (you can have 2 or 3, but not 2.5 atoms)

• light is quantized (you can have 2 or 3 photons, but not 2.5)

• in addition, we have quantum numbers

(L = n , where n is an integer)

Page 96: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Heisenberg Uncertainty Principle

A formal statement of this (from Fourier analysis) is: x * k

(where k = 2/, and indicates the uncertainty in the value)

But from the DeBroglie Hypothesis, = h/p, this uncertainty relation becomes:

x * (2/) = x * (2p/h) = 1/2 , or

x * p = /2.

Page 97: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Heisenberg Uncertainty Principle

A similar relation from Fourier analysis for time and frequency: t * = 1/2 leads to another part of the Uncertainty Principle (using E = hf = ): t * E > /2 .

There is a third part: * L > /2 (where L is the angular momentum value).

All of this is a direct result of the wave/particle duality of light and matter.

Page 98: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Nuclear Physics

Stability: see sheet detailing stable isotopes

Radiations:

1) , are all emitted;

2) protons and neutrons are NOT emitted, except in the case of mass numbers 5 and 9;

3) alphas are emitted only for mass numbers greater than 209, except in the case of mass number 8.

Page 99: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Alpha () decay

example: 92U238 90Th234 + 24 +

(it is not obvious whether there is a gamma emitted; this must be looked up in each case) Mass is reduced!

NOTE: 1. subscripts must be conserved (conservation of charge) 92 = 90 + 2

2. superscripts must be conserved

(conservation of mass) 238 = 234 + 4

Page 100: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

eta minus-) decayexample: 6C14

7N14 + -10 + 00

(a neutron turned into a proton by emitting an electron; however, one particle [the neutron] turned into two [the proton and the electron].

Charge and mass numbers are conserved, but since all three (neutron, proton, and electron) are fermions [spin 1/2 particles], angular momentum, particle number, and energy are not! Need the

anti-neutrino [0] to balance everything!

Page 101: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Positron (+) decayexample: 6C11

5B11 + +10 + 00

a proton turns into a neutron by emitting a positron; however, one particle [the proton] turned into two [the

neutron and the positron]. Charge and mass numbers are conserved, but

since all three are fermions [spin 1/2 particles], angular momentum, particle number, and energy are not! Need the

neutrino [0] to balance everything!

Page 102: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Nuclear Physics

General Rules:

1) emitted to reduce mass, only emitted if mass number is above 209

2) emitted to change neutron into proton, happens when there are too many neutrons

3) emitted (or electron captured) to change proton into neutron, happens when there are too few neutrons

4) emitted to conserve energy in reaction, may accompany or .

Page 103: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Mass Defect & Binding Energy

Similarly, the missing mass was converted into energy (E=mc2) and emitted when the carbon-12 atom was made from the six protons and six neutrons:

m = 6*mproton + 6*mneutron - mC-12 =

6(1.00782 amu) + 6(1.008665 amu) - 12.00000 amu

= .099 amu; BE = m*c2 =

(0.099 amu)*(1.66x10-27kg/amu)*(3x108m/s)2

= 1.478x10-11J*(1 eV/1.6x10-19J) = 92.37 MeV

Page 104: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Activity

Review: N(t) = No e-t

A = N = Aoe-t

T(half life) = ln(2) / .

If the half life is large, is small. This means that if the radioactive isotope will last a long time, its activity will be small; if the half life is small, the activity will be large but only for a short time!

Page 105: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Radioactivity around usan example

For 1 gram of carbon in a living plant, Ao = 15.0/min . Also, carbon-14 has a half life of 5,730 years.

If a 1 gram carbon sample from a dead plant has an activity of 9.0/min, then using:

A = Aoe-t , we have 9.0/min = 15.0/min * e-(ln2/5730yrs)t , or

-(ln2/5730 yrs)*t = ln(9/15) , ort = 5730 years * ln(15/9) / ln(2) = 4,200 years.

Page 106: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Radioactivity around us

Another radioactive isotope found in the earth is

92U238 . Since it is well above the 209 mass limit, it gives rise to a whole series of radioactive isotopes with mass numbers 238, 234, 230, 226, 222, 218, 214, 210. The 226 isotope is 88Ra226, which is the isotope that Marie Curie isolated from uranium ore. The 222 isotope is 86Rn222 which is a noble gas.

Other radioactive isotopes found in nature are

90Th232 , 92U235, and 19K40 . Both 90Th232 and 92U235 have decay chains that lead down to 82Pb (lead).

Page 107: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

X-raysexample

ionization = 13.6 eV * (Z-1)2 where the -1 comes from the other inner shell electron. If the electrons have this ionization energy, then they can knock out this inner electron, and we can see the characteristic spectrum for this target material.

For iron with Z=26, the ionization energy is:

13.6 eV * (26-1)2 = 1e * 8,500 volts.

Page 108: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

X and ray penetration

I = Io e-x where depends on the material the x-ray is going through and the energy of the x-ray.

In a similar way to half lives, we can define a half-value-layer, hvl, where hvl = ln(2)/ .

Page 109: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Measuring Radioactivity• How do we measure radioactivity?The Bq (dis/sec) and Curie (1 Ci = 3.7 x 1010 Bq) measure

how many decays happen per time. However, different radioactive materials emit different particles with different energies.

• What is the source of the health effects of radiation?Radiation () ionizes atoms. Ionized atoms

are important to biological function, and so radiation may interfere with biological functions.

• Can we devise a way to measure the health effects of radiation?

Page 110: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Measuring Health Effects

Can we devise a way to measure the health effects of radiation?

A unit that directly measures ionization is the Roentgen (R) = (1/3) x 10-9 Coul created per cc of air at STP. This uses air, since it is relatively easy to collect the charges due to ionization. It is harder to do in biological material, so this method is best used as a measure of EXPOSURE dose.

Page 111: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Measuring Health Effects

Can we devise a way to measure the health effects of radiation?

In addition to measuring ionization ability in air, we can also measure the energy that is absorbed by a biological material: Rad = .01 J/kg MKS: Gray (Gy) = 1 J/kg = 100 rads. This is called an ABSORBED dose.

Generally, one Roentgen of exposure will give one rad of absorption.

Page 112: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Measuring Health Effects

This difference in penetrating ability (and localization of ionization) leads us to create an RBE (radiation biological equivalent) factor and a new unit: the rem. The more localized the ionization, the higher the RBE.

# of rems = RBE * # of rads . This is called an EFFECTIVE dose.

RBE for gammas = 1; RBE for betas = 1 to 2; RBE for alphas = 10 to 20.

Page 113: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Levels of Radiation and Measurable Health Effects

200 millirems/year: background

Here are some more benchmarks based on our experience with acute (short time) doses:

20,000 millirems: measurable transient blood changes;

150,000 millirems: acute radiation sickness;

200,000 millirems: death in some people;

350,000 millirems: death in 50% of people.

Page 114: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Chain Reactions

In some cases, a very heavy nucleus, instead of undergoing alpha decay, will spontaneously split in two. Example:

92U238 50Sn129 + 42Mo106 + 30n1 + energy

The amount of energy coming from this reaction is on the order of 200 MeV, which is about 200 million times more than a chemical reaction.

This fissioning of uranium does not always result in these two resultant atoms - there is a whole range of resulting atoms. But it always gives a few neutrons.

In some cases a neutron can stimulate a heavy nucleus to split in two. This, if properly set up, can cause a chain reaction. This chain reaction is the basis for both the nuclear bomb and the nuclear power station.

Page 115: Pre-Health Physics Review Johnny B. Holmes, Ph.D..

Fusion

1H1 + 1H1 1D2 + +10 + + energy

1H1 + 1D2 1T3 + +10 + + energy

1H1 + 1T3 2He4 + energy

so we have four hydrogens becoming one helium, with about 24 MeV of energy and two neutrino’s produced plus two positrons which will combine with the extra two electrons from the 4 H’s to give another 2 MeV’s of energy.