– 1 –

Summary of Intro Physics 1. MECHANICS a. Foundation of kinematics of a point mass The position of a point particle in the x-y plane can be described either in terms of x, y coordinates or in terms of a magnitude and a direction.

�

!

R = xˆ i + yˆ j = R cos!ˆ i + R sin!ˆ j where

�

ˆ i and

�

ˆ j (and

�

ˆ k ) are unit vectors in the x and y (and z) direction.

The particle’s velocity is

�

! v =

d!

R

dt=

dx

dtˆ i +

dy

dtˆ j = vx

ˆ i + vyˆ j

Its acceleration

�

! a =

d! v

dt=

dvx

dtˆ i +

dvy

dtˆ j =

d2

x

dt2

ˆ i +d

2

y

dt2

ˆ j = axˆ i + ay

ˆ j

If the acceleration is constant, the equations of motion are

�

vx

= vox

+ axt

�

vy = voy + ayt

�

x = xo

+ 1

2vx

+ vox

( )t

�

y = yo + 1

2vy + voy( )t

�

x = xo

+ voxt +

1

2axt2

�

y = yo + voyt +1

2ayt

2

�

vx

2

= vox

2

+ 2axx ! x

o( )

�

vy2

= voy2

+ 2ay y ! yo( ) where

�

!

R o = xoˆ i + yo

ˆ j and

�

! v o = vox

ˆ i + voyˆ j = vo cos!ˆ i + vo sin!ˆ j are the particle position and velocity,

respectively, at t = 0. Projectile motion is a special case of two dimensional motion where in the absence of air resistance,

�

ax

= 0 and

�

ay = g . The x-component of velocity is constant. Relative velocity:

�

! v

AC=! v

AB+! v

BC

where

�

! v

ABis the velocity of A with respect to B,

�

! v

BCis the velocity of B with respect to C, and

�

! v

ACis

the velocity of A with respect to C b. Newton's laws, inertial systems An inertial reference frame is one in which Newton’s First Law holds.

The more general version of Newton’s Second Law is

�

!

F ext

=d

dtm! v ( )!

where

�

!

F ext! represents the sum over all forces with sources external to the system.

This reduces to the usual

�

!

F ext

= m! a ! when mass is constant.

Always draw a free-body diagram. Newton’s Laws can be applied either to the system as a whole or to individual parts of the system.

R

!

x = R cos!

y = R sin!

– 2 –

c. Closed and open systems, momentum and energy, work, power Work is done when a force F causes a displacement.

�

dW =

!

F !d! r = F cos"dr

or

�

W =

!

F !d! r

a

b

" = F cos#dra

b

" where θ is the angle between the force and the displacement. If the force is constant

�

W =

!

F !! s = Fscos"

A force is conservative if the work done depends only on the end points a and b and not on the path. The work done by a conservative force is equal to the negative of the change in potential energy.

�

!"U = ! Ub!U

a( ) = W

c=!

F #d! r

a

b

$ The gravitational potential energy near the Earth’s surface is

�

U = mgh where m is the mass and h is the height above the Earth’s surface. The spring potential energy is

�

U =1

2kx

2 where k is the force constant and x is the stretch or compression of the spring. The kinetic energy of a mass m moving with speed v is

�

K =1

2mv

2 The work done by the net force is equal to the change in kinetic energy. The work-energy theorem is

�

Wnet

=

!

F net! d! r = "K# .

The momentum

�

! p of a mass

�

m moving with velocity

�

! v is:

�

! p = m

! v

Power is the rate of doing work or transforming energy

�

P =dW

dt=dU

dt

For a constant force this becomes

�

P =

!

F !! v

d. Conservation of energy, conservation of linear momentum, impulse If

�

Wnc

is the work done by non-conservative forces then

�

Wnc

= !K + !U Or

�

Wnc + K0

+ U0

= Kf + Uf The impulse is equal to the change in momentum and defined

�

!

I = !! p =

!

F dt" . The total momentum of a system of objects is

�

!

P =! p i

i

! = M! v cm

where

�

M is the total mass and

�

! v

cm is the velocity of the center of mass (See Section 2.a)

�

!

F ext! =d!

P

dt=

d

dt

! p i

i

! . If the net external force is zero, the total momentum is conserved.

Types of collisions: If a collision is elastic, both momentum and kinetic energy are conserved. For all other types of collisions kinetic energy is not conserved. If a collision is perfectly inelastic, the objects stick together after the collision.

– 3 –

e. Elastic forces, frictional forces, the law of gravitation, potential energy and work in a gravitational field The elastic modulus is defined to be the stress divided by the strain. For small deformations the modulus is constant, depending only on the material.

Volume elasticity – Bulk Modulus – B:

�

B =!P

!V /V0

where P is the pressure, and V is the volume.

Shape elasticity – Shear Modulus – S:

�

S =F / A

!x / h

See accompanying diagram for the definition of variables.

Length elasticity – Young’s Modulus - Y:

�

Y =F / A

!L /L0

where L is the length and A is the cross-sectional area. Hooke’s Law

�

!

F = !k! x

can be used to represent the relation between the elastic force

�

!

F and the displacement from equilibrium

�

! x . k is a constant called the force constant or spring constant. (See Section 5.a for a

discussion of simple harmonic motion, motion subject to a restoring force.) In the case of an elastic wire Hooke’s Law is related to the Young’s Modulus equation with

�

k =AY

L0

The frictional force is in the direction to oppose the motion. Kinetic friction occurs when the contact surfaces are in relative motion. The magnitude of the force equals

�

fk = µkFN where

�

µk is the coefficient of kinetic friction and

�

FN

is the normal force. Static friction occurs when the contact surfaces are at rest with respect to each other. The magnitude of the force can have any value up to a maximum value given by

�

fs ! µsFN where

�

µs is the coefficient of static friction and

�

FN

is the normal force. The force of gravity between two masses m1 and m2 is given by Newton’s Law of Universal Gravitation

�

!

F 1on 2

= !Gm

1m

2

r12

2ˆ r 12

= !Gm

1m

2

r12

3

! r 12

where G is universal gravitational constant,

�

! r 12

is a vector from 1 to 2 and

�

ˆ r 12

=! r 12

/ r12

is a unit vector pointing in the direction from 1 to 2. The magnitude of the gravitational field at a point outside an astronomical object of mass M is

�

g = GM

r2

where r is the distance from the center of the object to the point.

A

h

F

x

– 4 –

For points inside a spherically symmetric object, Gauss’s Law for gravitation applies and

�

g = GMenc

r 2

where

�

Menc

is the mass enclosed by a sphere of radius r. The gravitational potential energy between two masses m1 and m2 with centers a distance

�

r12

apart is

�

U12

= !Gm1m2

r12

where the potential energy has been chosen to be zero when the masses are infinitely far apart. f. Centripetal acceleration, Kepler's laws. An object in uniform circular motion (motion in a circle of radius r with constant speed v) has an acceleration directed toward the center of the circle. The magnitude of the acceleration is:

�

ac

=v2

r= !

2

r =4"

2

r

T2

where ω = vr is the angular velocity and T is the period. Kepler’s First Law: A planet moves in an elliptical orbit with the sun at one focus. Kepler’s Second Law: A vector drawn form the sun to the planet sweeps out equal areas in equal times. This is equivalent to the law of angular momentum conservation. A planet’s angular momentum

�

!

L =! r ! m

! v is constant.

Kepler’s Third Law:

�

T2 is proportional to

�

r3 . For a planet or satellite moving in a circular orbit

about a very large mass M, the relation is

�

T2

=4! 2

GM

"

#

$ %

&

' r3

where T is the period of the motion and r is the radius of the orbit. 2. MECHANICS OF RIGID BODIES a. Statics, center of mass, torque The center of mass of a system of N point particles is

�

!

R cm

=1

MM

i

i=1

N

!!

R i where

�

M = Mi

i=1

N

!

is the total mass of the system and

�

Mi is the mass of the ith point particle located a distance

�

!

R i from

the reference point.

For extended objects

�

!

R cm

=1

M

! r dm! where

�

M = dm!

Torque is defined

�

! ! =

!

R "

!

F . Its magnitude is

�

! =! ! = RF sin"

where θ is the angle between

�

!

R and

�

!

F . The direction of

�

! ! is

given by the right hand rule.

axis

F

!

R

– 5 –

When two equal and opposite forces of magnitude F act on an object, their net force is zero but their net torque is RF where R is the perpendicular distance between them. This pair of forces is called a couple. An object is in equilibrium when

�

!

F = 0! and

�

! ! = 0" .

b. Motion of rigid bodies, translation, rotation, angular velocity, angular acceleration, conservation of angular momentum

If θ describes a rigid body’s angular position, the object’s angular velocity is

�

! =d"

dt

and its angular acceleration is

�

! =d"

dt=d2

#

dt2

.

If the angular acceleration is constant, the rigid body’s equations of motion are

�

! = !o

+"t

�

! = !o

+ 1

2" +"

o( )t

�

! = !o

+"ot +

1

2#t

2

�

!2

= !o

2

+ 2" # $ #o

( ) If an object is rolling without slipping

�

s = !r

�

v = !r

�

a = !r The angular momentum of a point particle about a fixed axis is defined

�

!

L =!

R !! p

where

�

! p is the particles momentum and

�

!

R is its distance from the axis. For a rigid body this becomes

�

!

L = I! !

where I is the moment of inertia. The direction of

�

! ! is along the axis in the direction given by the

right hand rule.

The net torque

�

! ! =

d

!

L

dt"

or for a rigid body (constant I)

�

! ! = I

! " #

For a system of particles.

�

!

L total

=

!

L i

i

!

and

�

! !

ext=

d

!

L total

dt" .

If the net external torque is equal to zero than the total angular momentum of the system is unchanged. Angular momentum is conserved. c. External and internal forces, equation of motion of a rigid body around the fixed axis, moment of inertia, kinetic energy of a rotating body additivity of the moment of inertia The moment of inertia of a single point particle M about an axis a distance R away is

�

I = MR2

The moment of inertia is additive. The moment of inertia of N objects about the same axis is

– 6 –

�

I = Ii

i=1

N

! where

�

Ii is the moment of inertia of the ith object about the axis. For N point particles

�

I = Mi

i=1

N

! Ri

2

where

�

Mi is the mass of the ith particle and

�

Ri is its distance from the axis.

For extended objects

�

I = r2

dm! . where r is the distance of mass element dm from the axis. The moments of inertia of some common objects about an axis through their center of mass is: Hoop of mass M and radius R about an axis perpendicular to the plane of hoop

�

Icm

= MR2

Disk of mass M and radius R about an axis perpendicular to the disk

�

Icm

=1

2MR

2 Sphere of mass M and radius R

�

Icm

=2

5MR

2 Rod of mass M and length L about an axis perpendicular to its length

�

Icm

=1

12ML

2 Parallel Axis Theorem: The moment of inertia about an axis parallel to one through the center of mass and a perpendicular distance h away is

�

I = Icm

+ Mh2 .

The kinetic energy of an object rotating about a fixed axis is

�

K =1

2I!

2 . If the axis is not fixed, the object’s total kinetic energy is the sum of the energy of the center of mass plus the energy about the center of mass

�

K = Kcm

+ Kaboutcm

=1

2Mv

2

+1

2I!

2 The work done by a torque is

�

W = !d""0

" f

#

d. Accelerated reference systems, inertial forces Newton’s laws can only be used in accelerating reference frames if fictitious or inertia forces are included. If the frame has acceleration

�

! a , the inertial force is

�

!

F i

= !m! a .

If the object is accelerating but not rotating, the net torque is not equal to zero about all points. To use

�

! ! = 0" , either take all torques about the center of mass or work in the accelerating frame and

include a torque due to the inertial force.

cm

h

M

– 7 –

3. HYDROMECHANICS Elementary concepts of pressure, buoyancy and the continuity law.

Fluids = liquid or gas Definition: Density ρ = m/V = mass/volume Definition: Pressure p = F/A if constant (or p = dF/dA) 1 atm = 1.013 x 105 Pa = 1.013 bar = 760 torr = 14.7 lb/in2 = 760 mm Hg ≅ 30 in Hg ≅ 10 m H2O Change in pressure with depth dp = -ρg dy or for constant density Δp = –ρgΔy Pascal's principle: A change in pressure applied to an enclosed fluid is transmitted throughout the fluid. Archimedes' Principle: For an object whole or partially immersed in a fluid

Buoyant force = weight of the fluid displaced

Fluids in motion (assume steady state, incompressible, nonviscous, irrotational – if not told and equations provided) Continuity equation: Av = constant Bernoulli's equation: p + 1/2 ρv2 + ρgy = constant Note: pressure depth is a special case. 4. THERMODYNAMICS AND MOLECULAR PHYSICS a. Internal energy, work and heat, first and second laws of thermodynamics Q = W + ΔU First Law of Thermodynamics Q = heat (energy transferred due to a temperature difference) in + (or out –)

– 8 –

W = work done by the gas (expanding a container) expanding + (contracting –) U = internal energy (e. g. KE and PE of molecules) U, p, V, T, S are state variables. Q and W are not. They depend on process. Assumption: Everything done very slowly so that system remains in thermodynamic equilibrium – all points of the system are at the same p, T, etc.

dW =

!

F ! d! x = pAdx = pdV

W = pdV!

W = Area under pV diagram (W>0 if ΔV>0 and W<0 if ΔV<0) Add energy to solids or liquids – Specific heat capacity Q = mcΔT for water c = 1 cal/goC = 4.2 J/goC = 4.2 x 103 J/kgK for ice and steam

c ! 1

2c

w

(T = temperature in Kelvin, TC = temperature in Celsius. Δ T = Δ TC) Molar heat capacities C Q = nCΔT solids and liquids n = number of moles for solids C approaches 25 J/molK For gases define two Qp = nCpΔT at constant pressure

QV = nCVΔT at constant volume Cp = CV + R ! = Cp CV Change of phase Q = mL At normal boiling and freezing points H2O LV = 2.26 x 106 J/kg LF = 0.33 x 106 J/kg

dS =

dQ

T

!S =

dQ

T" S = entropy (state variable)

Reversible process

dS = 0! ΔSTot = ΔSsys + ΔSsur = 0

S increases for an irreversible process (e.g. explosion, melting ice) or in an isolated system. To calculate S for the system, find a reversible process(es) connecting the same states For any process

!STot = !Ssys + !Ssur " 0 Second Law of Thermodynamics

Alternate definition S = k ln Ω k = 1.38 x 10-23 J/K Ω = measure of disorder or Ω = number of different ways the atoms of a system can be arranged without changing the macroscopic properties.

– 9 –

b. Model of a perfect gas, pressure and molecular kinetic energy, Avogadro's number, equation of state of a perfect gas, absolute temperature . Ideal gas law (holds low densities) pV = nRT equation of state (relates state variables)

on the molecular level

pV = 1

3mvrms

2= nRT

n = number of moles N = number of molecules NA = Avogadro's number =6.02 x 1023 m = nM = moN M = mass per mole in kg/mol mo = mass per molecule nNA = N nR =Nk R = NAk = 8.31J / mol !K " 2cal / mol !K Average translational KE per molecule

K = 3

2kT

For an ideal monatomic gas

U = 3

2NkT = 3

2nRT CV = 3

2 R Cp = 52 R

Equipartition of energy 1/2 nRT per degree of freedom (independent way to store energy) in U .

If f = number of degrees of freedom, CV =f

2R Cp =

f

2+1

!

" #

$

% & R

For any ideal gas, U depends only on temperature T.

p

T

Triple point 273.16 K

Solid

Liquid

Gas

1 atm Earth

1 atm Mars

Boiling Point Curve

patm

ptot=patm+pwater

pvapor

increases with T

Boiling occurs when pvapor = ptot

Water vaporbubbles are constantly forming and collapsing

– 10 –

c. Work done by an expanding gas limited to isothermal and adiabatic processes Complete the following table Process Special ΔU Q W ΔS pV diagram isothermal ΔT = 0 adiabatic Q = 0 cycle (return to initial state) isobaric Δp = 0 constant volume ΔV = 0

– 11 –

d. The Carnot cycle, thermodynamic efficiency, reversible and irreversible processes, entropy (statistical approach), Boltzmann factor A Carnot cycle is a special reversible cycle consisting of two isotherms and two adiabats.

An isothermal expansion at TH which takes in Qin An adiabatic expansion (Q = 0) which decreases the temperature from TH to TC. An isothermal compression at TC which expels Qout An adiabatic compression which increases the temperature from TC to TH.

For the Carnot cycle of system

!S = 0 =Qin

TH

+Qout

TC

efficiency any cycle e =Wnet

Qin!=1 "

Qout!Qin!

Carnot cycle has max efficiency

emax =1!TC

TH

Not otherwise classified but worth knowing. Triple point of water T3 = 273.16 K

Thermal expansion

!L = L"!T

!V = V#!T if isotropic β = 3 α

!d

d=!L

L

– 12 –

5. OSCILLATIONS AND WAVES a. Harmonic oscillations, equation of harmonic oscillation Simple harmonic motion is motion subject to a restoring force

!

F = !k! x where

! x is the

displacement from equilibrium and k is the force constant or spring constant. Our equation of motion for a particle of mass m is

m! a = !k

! x or in one spatial dimension max = !kx .

Writing this as a differential equation we have md2x

dt2= !kx .

which has solution x = A cos !t + "( )

where A > 0 is the amplitude or maximum displacement, ! = 2"f =2"

T=

k

m, and φ is a phase

angle set by the initial conditions.

The velocity is vx =dx

dt= !"Asin "t +#( ) with maximum magnitude

vmax =!A occurring when the mass passes through the equilibrium position.

The acceleration is ax =d2x

dt2= !"

2Acos "t + #( ) with maximum magnitude

amax =!2A occurring when |x| = A.

A and φ can be determined from the initial conditions: xo = A cos! and vo = !"A sin# . In the absence of damping or driving forces the total mechanical energy of the system is unchanged: E =

1

2kA

2=1

2mvmax

2=1

2kx2+1

2mv

2 The above equations of simple harmonic motion can be generalized to other systems. For rotational systems, such as pendulums or torsional pendulums, the equation of motion for small amplitude oscillations is:

I! = "#$ or I d2!

dt2= "#! and ! =

"

I.

where I is the moment of inertia (rotational inertia) of the pendulum. For a physical (or simple) pendulum ! = mgd where d is the distance from the point of suspension to the center of mass. If, in addition to the restoring force, the system has a damping force proportional to the velocity, the equation of motion is F = !kx ! bv = ma"

or md2x

dt2+ b

dx

dt+ kx = 0

Which has solutions x = Ae!bt /2m cos "t + #( ) with ! =k

m"

b

2m

#

$ %

&

' (

2

.

With a driving forceF = Focos!t the equation becomes m d

2x

dt2+ b

dx

dt+ kx = Fo cos!t

– 13 –

This has solutions x = A cos !t + "( )

with A =Fo

m2!2" !o

2( )2

+ b2!2 and tan! = "

b#

m #o

2 " # 2( ) with !o

2=k

m.

b. Harmonic waves, propagation of waves, transverse and longitudinal waves, linear polarization, the classical Doppler effect, sound waves The equation of a harmonic (sinusoidal) plane wave moving in the positive x direction with transverse displacement in the y direction is

y = Asin kx !"t + #( )

where k = 2!"

is called the wave number, ! = 2"f =2"

T is called the angular frequency, A is the

amplitude or maximum displacement, and ! is a phase angle often set to zero. v = !f or v = !

k.

The equation of a spherical, harmonic wave radiating outward from a point source is

y =A

rsin kr ! "t( )

Transverse waves, e.g., light waves or waves on a string, have particle displacements perpendicular to their direction of propagation. For example for a wave propagation in the +x direction, the displacement

!

D and the amplitude !

A are vectors perpendicular to the x axis.

!

D =!

A sin kx !"t( ) The velocity of a wave depends on properties of the medium.

Waves in a stretched string: v =F

µ where F is the tension in the string and µ is the mass per

unit length of the string.

Speed of sound: v =B

! where B is the bulk modulus and ρ is the density.

For sound in air this is approximately v = 331 m / sT

273 K .

Speed of light in material with index of refraction n: v = c

n

– 14 –

Sound waves can be expressed either as a displacement s = sm cos kx ! "t( ) where sm is the displacement amplitude, or as a pressure variation !p = !pm sin kx " #t( ) where !pm is the maximum variation from normal pressure. Longitudinal waves have displacements that are in the direction of propagation. Sound is a longitudinal wave. Therefore s is in the x direction. These amplitudes are related by !pm = v"#( )sm where v is the velocity and ρ is the density. For any wave, the intensity is proportional to the amplitude squared. For sound intensity I =

1

2!v"

2sm2

Doppler Effect: The observed frequency f' depends on the velocity of the source vs and the velocity of the observer vo. If v is the velocity of sound in air and f is the source frequency, then

! f = fv ± vo

v ! vs

"

#

$ $

%

&

' ' .

The upper sign is used for vo (vs ) when the observer (source) moves toward the source (observer). Fermat's Principle of Least time: Light travels between two points along the path that takes the least time. Fermat's principle can be used to derive both the law of reflection

θ1 = θ1' and the law of refraction (Snell's Law)

n1 sin!1 = n2 sin!2 where n is the index of refraction of the material. Total internal reflection occurs when n2 < n1 and !1 > !c where !

c is the critical angle given by:

sin!c=n2

n1

Geometrical Optics

Both spherical mirrors and lenses can be classified by what they do to rays of light that are incident

normal

!1 !1’

!2

n1

n2

– 15 –

parallel to the optic axis (a line perpendicular to the lens or mirror and through its center). Devices that cause the rays to converge to a focal point F are called converging. Devices that cause the rays to diverge so that they appear to come from a focal point F are called diverging.

f = focal length = distance from center of mirror (lens) to the focal point For converging mirrors and lenses, f is positive. For diverging mirrors and lenses, f is negative. Both these devices can be used to form an image of an object. By convention the object is placed to the left, so that the incident rays go from left to right. Defining o = object distance = distance from center of mirror (lens) to the object. i = image distance = distance from center of mirror (lens) to the image. m = magnification = ratio of image size to object size. For both spherical mirrors and thin lenses these quantities are related by the

Thin lens equation

�

1

o+1

i=1

f

and the magnification equation

�

m = !i

o

“Real images” are images that the rays of light pass through. “Virtual images” are images that the rays only appear to come from. Both images shown above are real images. For a single mirror or lens, real images are always inverted and have positive image distance i, while virtual images are always right side up and have negative image distance i.

– 16 –

If more than one lens (or mirror) is present, apply the equations repeatedly. The image of the first lens is the object of the second lens. For two lenses of focal length f1 and f2 which are separated by

a distance L,

�

1

o1

+1

i1

=1

f1

and

�

1

o2

+1

i2

=1

f2

with

�

L = i1

+ o2 and

�

m = m1m2

= !i1

o1

"

# $

%

& ' !

i2

o2

"

# $

%

& '

Note: Either i1 or o2 could be negative. The focal length can be found from the properties of the mirror or lens.

For spherical mirrors of radius of curvature R

�

f =R

2.

For a thin lens constructed of a material with index of refraction nL and surrounded by a material with index of refraction ns, where r1 is the radius of curvature of the first surface the light encounters and r2 is the radius of curvature of the second surface the light encounters,

Lens makers equation

�

1

f=

nLns

!1"

# $

%

& ' 1

r1

!1

r2

"

# $

%

& '

– 17 –

c. Superposition of harmonic waves, coherent waves, interference, beats, standing waves Addition of waves: sin A + B( ) + sin A ! B( ) = 2sin AcosB Beats: If A + B =!1t and A ! B =" 2t then A = 1

2!1 +!2( )t and B = 1

2!1 "!2( )t .

Which yields sin!1t + sin!2t = 2sin!1 +!22

"

# $

%

& ' t

(

) *

+

, - cos

!1 .!22

"

# $

%

& ' t

(

) *

+

, -

The cosine function forms the beat envelop. The beat frequency is f1 ! f2 .

Standing waves on a string: Two waves with the same frequency but traveling in opposite directions. Superposition yields

Asin kx ! "t( ) + Asin kx +"t( ) = 2A sinkx( ) cos"t( ) Points on the string oscillate in simple harmonic motion cos!t with amplitude 2Asin kx , Nodes

(points of zero vibration) occur wherever sin kx = 0 , that is where x = n!k

=n!

2! / "= n

"

2 with

n = 0, 1, 2, 3, .... Nodes are λ/2 apart. Antinodes (points of maximum vibration, sin kx = ±1 ) are also λ/2 apart. Conditions for standing waves:

A system with both ends the same, e.g., a string of length L with two

fixed ends (nodes) or a pipe of length L with two open ends (antinodes): L = n!n

2

The fundamental frequency is fo =v

2L. Other natural frequencies are integer multiples fn = nfo .

A system with the two ends different, e.g., a pipe of length L with

one open end (antinode) and one closed end (node): L = n + 1

2( )!n

2

The fundamental frequency is fo =v

4L. Other natural frequencies are odd integer multiples

f 2n+1( ) = 2n +1( ) fo . Superimposing N waves of the same amplitude A, but potentially different frequencies ωn, the resultant wave is

! t( ) = Acos"nt

n=1

N

# .

– 18 –

Let I = intensity and < I > = time average of I. I = Acos!nt

n=1

N

"#

$

% %

&

'

( (

2

= A2

cos!nt

n=1

N

"#

$

% %

&

'

( (

2

If the waves are coherent (all in phase) !n=!

o, I = A

2cos!ot 1

n=1

N

"#

$

% %

&

'

( (

2

= A2cos!ot( )

2N2

< I >= A2N2< cos! ot( )

2>= A

2N2 1

2( ) If the waves are incoherent (!

n's are not the same)

< I >= A2

cos!nt

n =1

N

"#

$

% %

&

'

( ( cos!kt

k=1

N

"#

$

% %

&

'

( ( = A

2cos!nt cos!kt

k=1

N

"n=1

N

"

Since cos!nt cos! kt =1

2"nk =

1

2

0

#

$ %

& %

if n = k

if n ' k

< I >= A2 1

2!nk

k=1

N

"n=1

N

" = A2 1

2

n=1

N

" = A2N1

2

6. ELECTRIC CHARGE AND ELECTRIC FIELD a. Conservation of charge, Coulomb's law Charge is quantized. All charge that has ever been found is an integer multiple of the fundamental charge.

1e = 1.6 x 10-19 C = magnitude of charge on the electron = magnitude of charge on the proton. Charge is conserved. The net charge on an isolated system is unchanged. Total initial charge = Total final charge Coulomb’s Law gives the force between two point charges q1 and q2. The magnitude of the force is

�

F12

= kq1q2

r12

where

�

k =1

4!"o

= 8.99 #109 Nm

2/C

2 Coulomb’s constant

and r12 = distance between charges. The direction of the force is along the line joining the two charges . It is repulsive if the charges have the same sign and attractive if the charges have opposite sign. In vector form

�

!

F 1 on 2

= kq

1q

2

r12

3

! r 12

= kq

1q

2

r12

2ˆ r 12

where

�

! r 12

is a vector from 1 to 2 and

�

ˆ r 12

=

! r 12

r12

is a unit vector pointing in the direction from 1 to 2.

If multiple charges q1, q2, …, qn are present, the net force on q1 is

�

!

F 1

=

!

F j on 1

j = 2

n

! = kq

1q j

r1 j

2

j =2

n

! ˆ r 1 j = q

1k

qj

r1 j

2

j =2

n

! ˆ r 1j

. .q1 q

2

r12

– 19 –

b. Electric field, potential, Gauss' law The magnitude of the electric field E at point X, a distance r away from a point charge q is

�

E = kq

r2.

Its direction is out from a positive charge and in toward a negative charge. In vector notation

�

!

E = kq

r3

! r = k

q

r2ˆ r

where

�

! r is a vector from the charge to X and

�

ˆ r =

! r

r is a unit vector pointing in the direction of

�

! r .

The force on a infinitesimally small test charge qo at point X is

�

!

F = qo

!

E . If multiple charges q1, q2, …, qn are present, the total field at point X is

�

!

E =

!

E jj =1

n

! = kqj

rj

2

j = 2

n

! ˆ r j where

�

! r j is a vector from qj to X.

For continuous charge distributions

�

!

E = d!

E = k

! r

r3!! dq where

�

! r is a vector from dq to X.

Depending on the type of charge distribution Volume charge dq = !dV ! = charge per unit volume Surface charge dq = !dA ! = charge per unit area Linear charge dq = !dL ! = charge per unit length Gauss’s Law: The net electric flux

�

! =

!

E " ˆ n dA#

equals

�

! =qin

"o

where

�

qin is the net charge enclosed by the Gaussian surface and

�

ˆ n is unit vector normal to the area element dA and pointing in the outward direction. Gauss’s Law can be used to obtain equations for the field due to simple symmetric systems. Just above the surface of a conductor:

!

E =!

"o

ˆ n

Spherically symmetric charge distribution:

!

E = kqin

r2

ˆ r where qin = !dV = 4" !r2dr##

Cylindrically symmetric charge distribution:

!

E =!

in

2"#or

ˆ r

Sheet of uniform charge:

!

E =!

2"o

ˆ n

.q

r

– 20 –

The electric force is conservative. The change in potential energy ΔU as a positive test charge qo

moves from point A to point B is

!U = UB " UA = " qo

!

E # d! s

A

B

$ .

The electric potential difference is defined

!V =!U

qo

= "!

E #d! s

A

B

$

ΔV is path independent. V is a continuous function of position.

Electric potential a distance r from a point charge q (with V = 0 at r = ∞) V = kq

r

Two point charges +q and –q separated by a distance 2a have an electric moment with magnitude p = 2aq and directed from –q to +q. Torque on a dipole in an external field

! ! =! p "!

E Potential energy of a dipole in a field

U = !

! p "!

E c. Capacitors, capacitance, dielectric constant, energy density of electric field Two conductors with equal and opposite charges +Q and –Q are called a capacitor. If the potential difference between them is V, then the capacitance C is defined C =

Q

V

If the conductors are two parallel plates with area A and separation d, C =!oA

d

For two co-axial cylinders with radii a and b, the capacitance per unit length is

�

C

L=

2!"o

ln b / a( )

If the conductors are two concentric spheres with radii a and b C =4!"

oab

b # a( )

Equivalent Capacitance: Capacitors in series: 1

Ceq

=1

Cii

!

Capacitors in parallel Ceq = Ci

i

!

Energy U stored in a capacitor U =1

2QV =

1

2CV

2

=1

2

Q2

C

Energy density u stored in an electric field u =1

2 !oE2

If a material with dielectric constant κ is placed between the plates of a capacitor which initially had capacitance Co, the new capacitance is C = !C

o !

o"#!

o( )

�

! "1

– 21 –

7. CURRENT AND MAGNETIC FIELD a. Current, resistance, internal resistance of source, Ohm's law, Kirchhoff's laws, work and power of direct and alternating currents, Joule's law

Current equals charge per unit time:

�

I =dQ

dt.

Resistance:

�

R =V

I.

Ohm’s Law: For some conductors at constant temperature, R is constant independent of I and V. The resistance of an Ohmic conducting wire with length L, cross sectional area A, and resistivity ρ

is

�

R = !L

A.

The resistivity of most metals depends on temperature

�

! = !o1+ " T # T

o( )[ ]

where α is the temperature coefficient of resistivity. If we can ignore thermal expansion

�

R = Ro1 +! T " T

o( )[ ] .

Real voltage sources have internal resistance. The terminal voltage is

�

Vba

= ! " Ir Kirchhoff’s Junction Law: The total current into a junction equals the total current out (charge conservation)

�

Iin

= Iout!!

Kirchhoff’s Loop Law: The total change in voltage around a closed loop is zero (energy conservation)

�

!V = 0" Power = Rate of doing work

DC Circuits P =dW

dt=dU

dt=d qV( )

dt= VI = I

2

R =V2

R

AC Circuits Pav= I

rms

2R = V

rmsIrmscos!

b. Magnetic field (B) of a current, current in a magnetic field, Lorentz force Lorentz force on a particle with charge q and velocity

�

! v

�

!

F = q!

E + q! v !!

B . A charge moving with velocity

�

! v perpendicular to a uniform magnetic field will undergo uniform

circular motion with frequency

�

2!f = " =v

r=qB

m cycletron frequency.

Magnetic force on a straight, current carrying wire in a uniform magnetic field

�

!

F = I

!

L !

!

B . Magnetic force on a wire segment

�

d

!

F = I d

!

L ( ) !!

B . A current loop of area A with unit normal in direction

�

ˆ n and current I has magnetic dipole moment

�

! µ = IA ˆ n

The torque on a magnetic dipole is

�

! ! =! µ "!

B

Biot Savart Law

d

!

B =µ

oI

4!

d

!

l "! r

r3

=µ

oI

4!

d

!

l " ˆ r

r2

where µo is the permeability of the vacuum. If d!

l and ! r are parallel, d

!

B = 0 .

r

!

b

a

.

.

dl

r

dB

– 22 –

Magnetic field on the axis of a current loop

!

B =µ

oI

2

a2

a2+ x

2( )3/ 2

ˆ i .

At the center of the loop (x = 0)

!

B =µ

oI

2a

ˆ i .

Very far from the loop (x >> a)

!

B !µ

oIa

2

2x3

ˆ i =µ

oIA

2"x3

ˆ i =µ

o

! µ

2"x3

c. Ampere's law Ampere’s Law

!

B ! d!

l = µo" I

in,

where Iin is the current encircled by the Ampere’s Law loop (integration contour). Ampere’s Law can be used to obtain the magnetic field due to simple symmetric systems. A distance r from a long straight wire B =

µoI

2!r

B encircles the wire in the direction given by the right hand rule. Inside a long solenoid B = µ

onI ,

where n is the number of turns per unit length. d. Law of electromagnetic induction, magnetic flux, Lenz's law, self-induction, inductance, permeability, energy

Faraday’s Law

�

! = "Nd#

dt

where the flux through a surface S is

�

! =

!

B "d!

A

S

# and N is the number of turns.

Lenz’s Law: The induced current is in the direction to oppose the change that caused it. The self inductance L can be found from the formula

�

LI = N! .

For a long solenoid

�

L = µon2

Al =µoN2

A

l

where A is the cross-sectional area of the solenoid and l is its length. The energy stored in an inductor is

�

U =1

2LI

2 .

The energy density stored in a magnetic field B is

�

u =B2

2µo

.

If diamagnetic or paramagnetic materials are present replace

�

µo!µ = µ

o1 + "( )

where µ is the permeability and χ is the magnetic susceptibility of the material.

x

a

x + a22

I

– 23 –

e. Alternating current, resistors, inductors and capacitors in AC-circuits, voltage and current (parallel and series) resonances Switched DC circuits:

Capacitor charging, switch closed at time t = 0.

�

V = iR +q

C

with

�

i =dq

dt.

The solutions are

�

q = CV 1! e!t /RC( )

and

�

i =V

Re!t /RC

Capacitive time constant

�

! = RC

Capacitor discharging begins at t = 0 with q = qo

�

0 = iR +q

C

Solutions are

�

q = qoe! t /RC

and

�

i = !qoRC

e!t /RC

Inductor current increasing: Switch is closed at t = 0.

�

V = iR + Ldi

dt

The solution is

�

i =V

R1! e

!Rt /L( )

Inductive time constant

�

! = L / R

Inductor current decreasing: Starting at t = 0 with i = io.

�

0 = iR + Ldi

dt

The solution is

�

i = ioe!Rt /L

8. ELECTROMAGNETIC WAVES a. Oscillatory circuit, frequency of oscillations, generation by feedback and resonance

When a charged capacitor is connected to an inductor, the total energy is

U =UC +UL =Q2

2C+1

2LI2 .

Applying Kirchhoff's loop rule: Q

C+ L

dI

dt=Q

C+ L

d2Q

dt2= 0

This is similar to the equation of motion for simple harmonic motion. The solution is

Q =Qmax cos !t + "( ) with ! =1

LC.

The circuit oscillates.

R

C

V

R

L

– 24 –

Adding resistance R to the circuit and applying Kirchhoff's loop rule: Q

C+ RI + L

dI

dt=Q

C+ R

dQ

dt+ L

d2Q

dt 2= 0

The solutions here are analogous to those for damped harmonic motion

Q =Qmaxe!Rt /2L

cos"t with ! =1

LC"

R

2L

#

$ %

&

' (

2

If an emf is added to the circuit: Q

C+ R

dQ

dt+ L

d2Q

dt2=!max cos"t ,

the situation is analogous to forced harmonic oscillations with resonance occurring when the driving frequency ω is equal to the natural frequency ωo

!o =1

LC

b. Wave optics, diffraction from one and two slits, diffraction grating, resolving power of a grating, Bragg reflection (See Modern Notes for Bragg Scattering)

Single slit diffraction for a long narrow slit of width a:

The intensity of the pattern is I! = Io

sin""

#

$ %

&

' (

2

where Io= intensity at ! = 0 and ! =

"a

#sin$ .

Minimum intensity I! = 0 occurs for m! = asin" for m = ±1,±2,±3,.... but not m = 0 . If the angle θ is small, sin! " tan! = y / L Two slit interference for two infinitesimally narrow slits with spacing d (sources are assumed to be coherent):

– 25 –

The intensity of the pattern is I! = Iocos

2" where I

o= intensity at ! = 0 and ! =

"d

#sin$ .

Maximum intensity I! = Io

occurs for m! = d sin" for m = 0,±1,±2, ±3,.... . m is called the order of interference.

Minimum intensity I! = 0 occurs for m + 1

2( )! = dsin" . Two slit interference from slits of width a and separation d:

The intensity of the pattern is I! = Io

sin""

#

$ %

&

' (

2

cos2 ) where I

o= intensity at ! = 0 , ! =

"a

#sin$ ,

and ! ="d

#sin$ .

Missing orders: If a md order interference maximum occurs at the same θ as a ma diffraction

minimum, the order md is missing. md! = d sin" and ma! = asin" . md

d=ma

a

Thin film interference: Light reflected from a surface of higher index of refraction undergoes a phase shift of π radians. If the light is approximately normally incident

2t +0

1

2!n

" # $

% $

& ' $

( $ =

m!n

m + 1

2( )!n

" # $

% $

& ' $

( $

with !n=!

n= wavelength in the film , n = the index of refraction of the film material,

m = 0,1,2,3,..., and t = film thickness. Determine the total number of reflection phase shifts. If it is 0 or 2, select the top choice in the left { }. If it is 1, select the bottom choice. If you want maximum reflection (or minimum transmission) select the top choice in the right { }. For minimum reflection (or maximum transmission) select the bottom choice. For any two initially coherent, in phase, same amplitude waves that travel different paths and recombine, maximum constructive interference occurs when the path difference is an integer number of wavelengths and complete destructive interference occurs when the path difference is an odd half integer number of wavelengths. Diffraction gratings: Bright lines (fringes) occur at m! = d sin" for m = 0,±1,±2, ±3,.... where d is the spacing between adjacent grating lines. The ability of a grating to separate two wavelengths that differ by Δλ is called the resolving power

�

R =!

"!= Nm

where λ is the average wavelength, N is the number grating lines, and m is the order of interference. c. Dispersion and diffraction spectra, line spectra of gases Visible light has wavelengths from 400 nm to 700 nm (violet to red). Characteristic wavelengths for other parts of the electromagnetic spectrum are 0.1 nm (size of atom) for X-rays, 10 fm (size of nucleus) for gamma rays, and 20 cm (size of oven) for microwaves.

– 26 –

All the above effects of interference and diffraction depend on wavelength. Different wavelengths produce lines at different angles. See above.

Dispersion of a grating with slit spacing d in mth order d!d"

#

$ %

&

' (

max

=m

d cos!

d. Electromagnetic waves as transverse waves, polarization by reflection, polarizers Reflected light is partially polarized parallel to the surface of reflection. The amount of polarization depends on angle of reflection. The reflected light is completely polarized at Brewster's angle !P

(see diagram)given by

tan!P=n2

n1

.

If unpolarized light of intensity I

o

is incident on a polarizer, the transmitted intensity I1 is

I1 =1

2Io

If polarized light of intensity I1 is incident on a polarizer, the transmitted intensity I2 is I2 = I1 cos

2! (Malus' Law) (Malus'

where θ is the angle between the axes of the two polarizers. e. Resolving power of imaging systems The Rayleigh Criterion: When the central maximum of one image falls on the first minimum of the other, the images are said to be just barely resolved. Since the angles are small, sin!

R" !

R and the

required minimum angular separation is !

R" # / a for a slit of width a

or !R" 1.22# / D for a circular aperture of diameter D

f. Black body, Stefan-Boltzmann's law Stefan-Boltzmann's law: I = e!T

4 σ = 5.67 x 10–8 W/(m2K4) Stefan-Boltzmann constant I = total intensity (energy flux = power per unit area) e = emissivity Wien's Law !maxT = 2.898x10

"3 m#K

Planck's constant h = 6.626 x 10–34Js

! =h

2!

Planck assumed radiators quantized. Einstein assumed radiation quantized. Photons γ E = h! = !" p = hν/c = h/λ

– 27 –

9. QUANTUM PHYSICS a. Photoelectric effect, energy and impulse of the photon hν = φ + Κ

φ = work function = minimum energy to free an electron = hνth

Measure K by finding the stopping potential, voltage needed to stop the current

K = e Vstop

Compton Effect Photons scattered from free or nearly free electrons λ' – λ = (h/mec)(1 – cos θ) Compton wavelength of the electron h/mec = 0.00243 nm b. De Broglie wavelength, Heisenberg's uncertainty principle

De Broglie wavelength λ = h/p (valid for any velocity) k =

2!

"=

2!

hp =

p

!

Heisenberg's uncertainty principle

!x!px " ! / 2 and !E!t " ! / 2 10. RELATIVITY a. Principle of relativity, addition of velocities, relativistic Doppler effect Einstein based his Theory of Special Relativity on two postulates:

All the laws of physics are the same in all inertial reference frames. The speed of light (in vacuum) is the same in all inertial reference frames.

Consider the two inertial reference frames S and S' where frame S' is moving with velocity V in the x direction with respect to frame S. An event can be described by (x, y, z, t) in S and (x', y', z', t') in S'. These are related by the Lorentz transformations

! y = y ! z = z

! x = " x # $ct( ) c ! t = " ct # $x( )

where c is the speed of light in the vacuum,

! =V

c, and ! =

1

1" #2=

1

1" V / c( )2.

!

"#

#’

-

+

K

h!

– 28 –

The space and time intervals between two events A and B are Δx = xA - xB, Δy = yA - yB, Δz = zA - zB, and Δct = ctA - ctB. The quantity

!x( )2 + !y( )2

+ !z( )2 " !ct( )2 = ! # x ( )2

+ ! # y ( )2

+ ! # z ( )2" !c # t ( )

2 is an invariant. It is the same in all inertial reference frames.

Events for which !x( )2 + !y( )2+ !z( )2 " !ct( )2 > 0 are called space like. If one is at the origin, the

other is in the "ELSEWHERE." These events can occur at the same time in some inertial reference frame. Events for which !x( )2 + !y( )

2+ !z( )2 " !ct( )2 < 0 are called time like. If one is at the origin, the

other is in the "PAST" or the "FUTURE." These events can occur at the same point in some inertial reference frame. If !x( )2 + !y( )

2+ !z( )2 " !ct( )2 = 0 and one event is at the origin, the other is on the light cone. The

events can be connected by a light signal. Time dilation !t = "!t

o

Δto = time measured in a frame where both events occur at the same point (Δx' = 0) and called proper time interval. E.g., If you were on a space ship, your clock, which is at rest, measures Δto . Any other clock measures Δt for the same interval.

Length contraction L =Lo

!= L

o1 " #

2

Lo = length measured in a frame where the object is at rest and called proper length Addition of velocities. If the components of an object's velocity are vx ,vy ,vz in S and ! v x , ! v y , ! v z in ! S where S' has velocity V in the x direction with respect to S, then

light conect

x

world line

of a particle

FUTURE

PAST

ELSEWHEREELSEWHERE

ct ct’

x=ctx’= ct’

x’

x

light signalA

B

Events A and B occur at the same time in S’ but different times in S

– 29 –

! v x =vx " V

1"vxV

c2

! v y =vy 1" V / c( )2

1"vxV

c2

! v z =vz 1" V / c( )2

1"vxV

c2

Relativistic Doppler Effect For a source at rest in S' which emits EM radiation of frequency νo measured in S'.

If moving away ! = !o

1 " #

1 + #

If moving toward ! = !o

1 + "

1 # "

b. Relativistic equation of motion, momentum, energy, relation between energy and mass, conservation of energy and momentum The relativistic momentum and energy of a particle of intrinsic mass m moving with velocity

! v is

! p = m

! v ! E = mc

2!

where ! =1

1" #2=

1

1" v / c( )2 and ! =

v

c.

The kinetic energy of the particle is K = E !mc2= mc

2" !1( )

The relativistic equation of motion has the same form as the Newtonian equation when expressed in terms of momentum. E.g., the equation of motion for a relativistic particle with mass m, velocity

! v , and charge q in an electric and magnetic field is

d

dt

m! v

1! v / c( )2= q!

E + q! v "!

B

Any quantity that has the same transformations from S to S' as

x, y, z,ct( ) =

! x , ct( ) does is called a

four vector. c! p ,E( ) is another four vector. Assuming all velocities are in the x direction,

c ! p = " cp # $E( ) ! E = " E # $cp( ) The quantity

c! p ( )2! E

2= c! " p ( )2! " E

2

is an invariant and has physical meaning. Usually it is written E2! c

2 ! p 2= mc

2( )2

For light with m = 0, E = cp. Energy and momentum are conserved in all inertial reference frames.

– 30 –

11. MATTER a. Simple applications of the Bragg equation X-rays or particle waves scattering off crystals ! " 0.1nm Path difference = 2d sin θ Maximum if nλ = 2d sin θ For a given λ , measure θ 's, deduce d. Many plane spacings possible, b. Energy levels of atoms and molecules (qualitatively), emission, absorption, spectrum of hydrogenlike atoms Hydrogenlike atoms (single electron atoms with nuclear charge Ze)

En = !

me(keZe2 )2

2!2n

2= !13.6eV

Z2

n2

h! = "E

Bohr radius for hydrogen (Z = 1)

ao =!

2

mekee2

other

rn =n

2ao

Z

Z ! Zeff for others

e.g. X-rays Accelerated electrons strike a metal target. The threshold wavelength depends only on the kinetic energy Ek of the accelerated electron

!th =hc

Ek

and is target independent. The characteristic wavelengths depend on the target.

Zeff = Z !1 for

transitions to the K shell (innermost) Pauli Exclusion principle No two electrons in the same atom can have the same set of quantum numbers. n, l, s, ml, ms or n, l, s, j, mj with l = 0, 1, 2, ... , n – 1 n values of l

ml = –l, –l+1, ..., l–1, l etc (2l+1) values of ml and

! j =!

l +! s j = |l-s|, |l-s+1|, ... , l+s-1, l+s

Each nl shell holds 2(2l+1) electrons

I

!!th

! !

!d

– 31 –

Molecules: Vibrational levels En = (n + 1/2)hν n = 0, 1, 2, 3, ... simple harmonic oscillator levels

Rotational levels

Erot

=

! L 2

2I=

L(L +1)"2

2I L = 0, 1, 2, 3, ... = angular momentum

c. Energy levels of nuclei (qualitatively), alpha-, beta- and gamma-decays, absorption of radiation, half-life and exponential decay, components of nuclei, mass defect, nuclear reactions Z = number of protons = charge on nucleus (determines element) = atomic number N = number of neutrons A = Z + N = number of nucleons = mass number Nuclei with the same number of protons and different numbers of neutrons are called isotopes. Using X and Y to represent the chemical symbol of the parent and daughter nucleus, respectively, α decay (!" 2

4He )

ZA

X!Z"2

A" 4Y +2

4He

β decay (!"

#"10e# electron ) Z

AX!Z + 1

AY +"1

0e +# e (!

e" electron neutrino )

(!+"+1

0e" positron ) Z

AX!Z–1

AY ++1

0e +"e

electron capture –1

0e+Z

AX!Z–1

AY + "e usually accompanied by X-rays

γ decay (! " high energy photon ) Z

AX

*!Z

AX + "

Decay: Let N(t) = the number at time t and No = the number at time t = 0 dN = - λNdt λ = decay constant N (t ) = Noe

!"t Half life t1/2 = time for 1/2 of the sample to decay t1/2 = ln(2)/λ At t = n t1/2 N(t) = (1/2)n No Absorption (attenuation)

I = Ioe!µx

µ = linear attenuation coefficient (depends on energy, type of radiation) Energy levels see Figure 10.10 and 10.9. Shell model used to predict energy levels The strong nuclear force is short range with a strong dependence on total angular momentum J. Nuclei with magic numbers (2, 8, 20, 28, etc.) particularly stable, comparable to atomic inert gases. Figure 10.8 shows stable isotopes. For low A, N ! Z . For large A, more neutrons. Coulomb repulsion increases energy of proton energy levels. mass defect = ZmH +Nmn – m where m = mass of the atom Binding energy B = (mass defect)c2. Usually represented as B/A (see Figure 10.2)

– 32 –

!t+"

A sin(!t+" )A

Two Methods to Add Waves A. Phasors To add wavefunctions: The function ! = Asin("t + #) can be represented as the projection on the y axis of phasor of length A rotating with angular velocity ω. To add three waves:

! = Asin("t) + Asin("t + #) + Asin("t + 2# ) ! = AR sin("t + #) Where AR and ! can either be determined by geometry or by means of a careful scale drawing. AR is a maximum when φ = 0, 2π, 4π, ....

AR vanishes when ! =

2"

3,4"

3,8"

3,...

To add voltages: LRC series circuit has alternating current I t( ) = I

msin!t

and with applied emf ! t( ) = !msin "t + #( ) .

The voltage across the resistor is vR = RI t( ) = RIm sin!t = VR sin!t . The voltage across the capacitor is vC =

Q

C=1

CI t( )dt! =

1

CIm sin"tdt! = #

Im

"Ccos"t =

Im

"Csin "t # $

2( ) = VC sin "t # $2( )

The voltage across the inductor is

vL = LdI t( )

dt= L

d

dtIm sin!t = LIm! cos!t =!LIm sin !t + "

2( ) = VL sin !t + "

2( )

Defining impedances XC =1

!C X

L=!L X

R= R

!tA

"

"A

A

#

AR

A A A

AR

2!/3

A A

A

– 33 –

the voltage amplitudes are VC = XCIm VL = XLIm VR = XRIm . Applying Kirchhoff's loop rule we have:

! = vR + vC + vL

!m sin "t + #( ) = XRIm sin"t + XCIm sin "t $%2( ) + XLIm sin "t + %

2( ) The diagram to the right shows the phasors at time t = 0. From the diagram

!m = Im XR2 + XL " XC( )

2= Im R

2+ #L "

1

#C$

% &

'

( )

2

and

tan! =

"L #1

"C

R

The total impedance in the circuit Z is defined Z =!m

Im

Z = R2

+ !L "1

!C#

$ %

&

' (

2

At the resonance frequency

! =!o =1

LC, tan! = 0 and !

m= RI

m.

Power in AC Circuits P

av= I

rms

2R = !

rmsIrmscos" ,

where Irms

=Im

2 and !

rms=!

m

2

B. Complex Numbers (not required according to the syllabus but usable) The imaginary quantity i is defined i = !1 . A complex number z can be represented two ways z = x + iy = Re

i! . x and y are respectively called the real and imaginary parts of z x = Re z y = Im z Since x = Rcos! y = Rcos! , e

i!= cos! + i sin! and cos! = Re e

i!( ) sin! = Im ei!( ) .

cos! = 1

2ei!+ e

" i!( ) sin! = 1

2iei!" e

"i!( ) Therefore to add cosines or sines we can add complex exponentials and then take the real or the imaginary part as a final step.

!m

"

XRIm

XCIm

XLIm

!

Ry

x

Real axis

Imaginary axis

– 34 –

The "*" in z* denotes taking the complex conjugate, i! "i z2= zz

*= x + iy( ) x ! iy( ) = x2 + y2 = Re

i"( ) Re!i"( ) = R2

To add waves: N slit interference The path difference between rays through adjacent slits is !L = d sin" . This gives rise to a phase shift between adjacent rays of ! =

2"

#$L =

2"

#d sin%

The resultant wave at angle θ is

! = Asin "t( ) + Asin "t + #( ) + A sin "t + 2#( ) + ... + Asin "t + N $1( )#( )

! = Asin "t + n#( )n=0

N$1

%

Using complex numbers ! = Im Aei "t+ n#( )

n=0

N$1

% = Im&

! = Aei "t+n#( )

n= 0

N $1

% = Aei"t

ein#

n= 0

N$1

% = Aei"t

ei#( )

n

=n= 0

N$1

% Aei"t 1 $ e

iN#

1 $ ei#

! = Aei"t e

iN# /2

ei# /2

$

%

& &

'

(

) ) eiN# /2 * e*iN# /2

ei# /2 * e*i# /2

$

%

& &

'

(

) ) = Aei"tei N*1( )# / 2 sin N# / 2( )

sin # / 2( )

$

% & &

'

( ) )

! = Im" = Im Aei #t+ N$1( )% /2( ) sin N% / 2( )

sin % / 2( )

&

' ( (

)

* + +

,

-

.

.

/

0

1 1

= Asin #t + 1

2N $1( )%( )

sin N% / 2( )

sin % / 2( )

&

' ( (

)

* + +

Itot = ! 2= A

2sin "t + 1

2N #1( )$( )

2 sin N$ / 2( )

sin $ / 2( )

%

& ' '

(

) * *

2

= 1

2A2 sin N$ / 2( )

sin $ / 2( )

%

& ' '

(

) * *

2

Setting Io =1

2A2 and ! =

"

2=#d

$sin% ,

Itot = Io

sin N!( )

sin !( )

"

# $ $

%

& ' '

2

Maximum intensity of Itot = N

2Io occurs at sin ! = 0

or equivalently ! = m" ="d

#sin$ or m# = d sin$ . m = 0,1,2,3,...

This is just the grating equation. Minimum (zero) intensity occurs atsinN! = 0 but sin! " 0 . or equivalently

N! =N"d

#sin$ = m" or

m#

N= d sin$ with m = 1,2,...,N %1,N +1,N + 2,... .

d

!L

"

– 35 –

AC Circuits: Represent all voltages and currents as complex exponentials. In our prior example of the series LRC circuit I t( ) = Ime

i!t vR = RI t( ) = RIme

i!t

vL = LdI t( )

dt= LIm

d

dtei! t

= i!LImei!t

= i!LI t( )

vc =1

CI t( )! dt =

1

CIme

i"t! dt =1

i"CIme

i"t=I t( )

i"C

Defining complex impedances Z =

v

I t( ) for each circuit element.

ZR= R ZL = i!L ZC =

1

i!C=

"i

!C

One advantage of this formalism is relative simple rules for impedances in series and parallel. Impedances in series Zs = Zj

j

!

Impedances in parallel 1

Zp=

1

Zjj

!

For our prior example of the series LRC circuit the total impedance of the circuit is

Ztot = ZR + ZC + ZL = R !i

"C+ i"L = Ze

i#

which has magnitude Z = Ztot!Ztot = Re Ztot( )

2+ ImZtot( )

2= R

2+ "L #

1

"C$

% &

'

( )

2

and phase tan! =ImZtot

ReZtot

=

"L #1

"C

R.

Both as previously obtained. An LRC parallel circuit Here the total impedance is

1

Ztot

=1

ZR

+1

ZL

+1

ZC

=1

R+1

i!L+ i!C =

1

R+ i !C "

1

!L#

$ %

&

' (

Ztot =1

1

R+ i !C " 1

!L#

$ %

&

' (

#

$

% % % %

&

'

( ( ( (

=

1

R" i !C "

1

!L#

$ %

&

' (

1

R

#

$ %

&

' (

2

+ !C "1

!L#

$ %

&

' (

2

#

$

% % % % %

&

'

( ( ( ( (

= Zei)

C LR

– 36 –

Z = Ztot!Ztot =

1

1

R" i #C "

1

#L$

% &

'

( )

$

% &

'

( ) 1

R+ i #C "

1

#L$

% &

'

( )

$

% &

'

( )

=1

1

R

$

% &

'

( )

2

+ #C " 1

#L$

% &

'

( )

2

and phase tan! =ImZtot

ReZtot

= R1

"L#"C

$

% &

'

( )