Electromagnetism

Contents

• Review of Maxwell’s equations and Lorentz Force Law• Motion of a charged particle under constant Electromagnetic fields• Relativistic transformations of fields• Electromagnetic energy conservation• Electromagnetic waves

– Waves in vacuo– Waves in conducting medium

• Waves in a uniform conducting guide– Simple example TE01 mode

– Propagation constant, cut-off frequency– Group velocity, phase velocity– Illustrations

2

Reading

• J.D. Jackson: Classical Electrodynamics

• H.D. Young and R.A. Freedman: University Physics (with Modern Physics)

• P.C. Clemmow: Electromagnetic Theory

• Feynmann Lectures on Physics

• W.K.H. Panofsky and M.N. Phillips: Classical Electricity and Magnetism

• G.L. Pollack and D.R. Stump: Electromagnetism

3

Basic Equations from Vector Calculus

y

F

x

F

x

F

z

F

z

F

y

FF

z

F

y

F

x

FF

FFFF

123123

321

321

,,:curl

:divergence

,,,vectoraFor

z

φ

y

φ

x

φφ

x,y,z,tφ

,,:gradient

,functionscalaraFor

4

Gradient is normal to

surfaces =constant

Basic Vector Calculus

FFF

F

GFFGGF

2)()(

0,0

)(

5

S C

rdFSdF

dSnSd

Oriented boundary C

n

Stokes’ Theorem

Divergence or Gauss’ Theorem

SV

SdFdVF

Closed surface S, volume V, outward pointing normal

What is Electromagnetism?• The study of Maxwell’s equations, devised in 1863 to represent

the relationships between electric and magnetic fields in the presence of electric charges and currents, whether steady or rapidly fluctuating, in a vacuum or in matter.

• The equations represent one of the most elegant and concise way to describe the fundamentals of electricity and magnetism. They pull together in a consistent way earlier results known from the work of Gauss, Faraday, Ampère, Biot, Savart and others.

• Remarkably, Maxwell’s equations are perfectly consistent with the transformations of special relativity.

Maxwell’s EquationsRelate Electric and Magnetic fields generated by charge and current distributions.

1,,In vacuum 20000 cHBED

t

DjH

t

BE

B

D

0

E = electric field

D = electric displacement

H = magnetic field

B = magnetic flux density

= charge density

j = current density

0 (permeability of free space) = 4 10-7

0 (permittivity of free space) = 8.854 10-12

c (speed of light) = 2.99792458 108 m/s

Maxwell’s 1st Equation

000

1

Q

dVSdEdVEEVSV

02

0

30

4

4

q

r

dSqSdE

rr

qE

spheresphere

0

E

8

Equivalent to Gauss’ Flux Theorem:

The flux of electric field out of a closed region is proportional to the total electric charge Q enclosed within the surface.

A point charge q generates an electric field

Area integral gives a measure of the net charge enclosed; divergence of the electric field gives the density of the sources.

Gauss’ law for magnetism:

The net magnetic flux out of any closed surface is zero. Surround a magnetic dipole with a closed surface. The magnetic flux directed inward towards the south pole will equal the flux outward from the north pole.

If there were a magnetic monopole source, this would give a non-zero integral.

Maxwell’s 2nd Equation 0 B

00 SdBB

Gauss’ law for magnetism is then a statement that There are no magnetic monopolesThere are no magnetic monopoles

Equivalent to Faraday’s Law of Induction:

(for a fixed circuit C)

The electromotive force round a

circuit          is proportional to the

rate of change of flux of magnetic field,

             through the circuit.

Maxwell’s 3rd Equation t

BE

dt

dSdB

dt

dldE

Sdt

BSdE

C S

SS

ldE

SdB

N S

Faraday’s Law is the basis for electric generators. It also forms the basis for inductors and transformers.

Maxwell’s 4th Equation

jB

0

t

E

cjB

20

1

ISdjSdBldBC S S 00

Originates from Ampère’s (Circuital) Law :

Satisfied by the field for a steady line current (Biot-Savart Law, 1820):

r

IB

r

rldIB

2

4

0

30

current line straight a For

Ampère

Biot

Need for Displacement Current

• Faraday: vary B-field, generate E-field• Maxwell: varying E-field should then produce a B-field, but not covered by Ampère’s

Law.

12

Surface 1 Surface 2

Closed loop

Current I

Apply Ampère to surface 1 (flat disk): line integral of B = 0I

Applied to surface 2, line integral is zero since no current penetrates the deformed surface.

In capacitor, , so

Displacement current density ist

Ejd

0

dt

dEA

dt

dQI 0

Aε

QE

0

t

EjjjB d

0000

Consistency with Charge Conservation

0

tj

dVt

dVj

dVdt

dSdj

Charge conservation: Total current flowing out of a region equals the rate of decrease of charge

within the volume.

13

From Maxwell’s equations: Take divergence of (modified) Ampère’s equation

tj

tj

Etc

jB

0

0

1

0000

20

Charge conservation is implicit in Maxwell’s EquationsCharge conservation is implicit in Maxwell’s Equations

Maxwell’s Equations in Vacuum

20000

1,,

cHBED

In vacuum

Source-free equations:

Source equations

14

Equivalent integral forms (useful for simple geometries)

0

0

t

BE

B

jt

E

cB

E

02

0

1

SdEdt

d

cSdjldB

dt

dSdB

dt

dldE

SdB

dVSdE

20

0

1

0

1

Example: Calculate E from B

0

00

0

sin

rr

rrtBBz

dSBdt

dldE

trBE

tBrtBrdt

drErr

cos2

1

cossin2

0

02

02

0

tr

BrE

tBrtBrdt

drErr

cos2

cossin2

02

0

02

002

00

Also from t

BE

dt

E

cjB

20

1 then gives current density necessary to sustain the fields

r

z

Lorentz Force Law

• Supplement to Maxwell’s equations, gives force on a charged particle moving in an electromagnetic field:

• For continuous distributions, have a force density

• Relativistic equation of motion

– 4-vector form:

– 3-vector component:

BvEqf

dt

pd

dt

dE

cf

c

fv

d

dPF

,1

,

16

BjEfd

BvEqfvmdt

d 0

Motion of charged particles in constant magnetic fields

Bvqvmdt

dBvEqfvm

dt

d 00

17

1. Dot product with v:

constantisconstantis0So

1But

0

222

0

vdt

ddt

dv

dt

dvcvγ

Bvvm

qv

dt

dv

No acceleration with a magnetic field

constant,0

0

//

0

vvBdt

d

BvBm

qv

dt

dB

2. Dot product with B:

Motion in constant magnetic field

0

0

0

2

0

frequencyangularat

radiuswithmotioncircular

mmm

qBvω

qB

vmρ

Bvm

qv

Bvm

q

dt

vd

Constant magnetic field gives uniform spiral about B with constant energy.

rigidityMagnetic

0

q

p

q

vmB

Motion in constant Electric Field

Solution of Em

qv

dt

d

0

cmqEtm

qE0

2

0

for2

1

19Constant E-field gives uniform acceleration in straight line

Eqvmdt

dBvEqfvm

dt

d 00

2

0

22

0

11

t

m

qE

c

vt

m

qEv is

11

2

0

20

cm

qEt

qE

cmx

v

dt

dx

qExEnergy gain is

Potentials• Magnetic vector potential:

• Electric scalar potential:

• Lorentz Gauge:

ABAB

thatsuch0

AAf(t)

,

20

01

2

Atc

t

AE

t

AE

t

AE

t

AA

tt

BE

so,with

0

Use freedom to set

Electromagnetic 4-Vectors

21

Α,1

,1

01

42

Actc

Atc

Lorentz

Gauge

4-gradient

4 4-potential A

Current 4-vector

000 where),(),(

jcvcVJ

vj

Continuity equation

0,,1

4

jt

jctc

J

Charge-current transformations

2,

c

jvvjj x

xx

Example: Electromagnetic Field of a Single Particle

• Charged particle moving along x-axis of Frame F

• P has

• In F, fields are only electrostatic (B=0), given by

tc

vxtttvbrxbvtx p

pP

2222 ','' so ),,0,'('

333 '',0',

'

'''

''

r

qbEE

r

qvtEx

r

qE zyxP

Origins coincide at t=t=0

Observer Pz

b

charge qx

Frame F v Frame F’z’

x’

tvxtvxx PPP so)(0

Electromagnetic Energy

• Rate of doing work on unit volume of a system is

• Substitute for j from Maxwell’s equations and re-arrange into the form

HBDEt

S

HESt

DE

t

BHS

t

DEEHHEE

t

DHEj

2

1

where

23

EjEvBjEvfv d

Poynting vector

24

HEDEHBt

Ej

2

1

SdHEdVHBDEdt

d

dt

dW

2

1

electric + magnetic energy densities of the

fields

Poynting vector gives flux of e/m energy across

boundaries

Integrated over a volume, have energy conservation law: rate of doing work on system equals rate of increase of stored electromagnetic energy+ rate of energy flow across boundary.

Review of Waves• 1D wave equation is with general

solution

• Simple plane wave:

2

2

22

2 1

t

u

vx

u

)()(),( xvtgxvtftxu

xktxkt sin:3Dsin:1D

k 2

Wavelength is

Frequency is2

Superposition of plane waves. While shape is relatively undistorted, pulse travels with the group velocity

Phase and group velocities

kt

xv

xkt

p

0

dkekA kxtki )()(

dk

dvg

Plane wave has constant phase at peaks

xkt sin2 xkt

Wave packet structure

• Phase velocities of individual plane waves making up the wave packet are different,

• The wave packet will then disperse with time

27

Electromagnetic waves• Maxwell’s equations predict the existence of electromagnetic waves, later

discovered by Hertz.• No charges, no currents:

00

BD

t

BE

t

DH

2

2

2

2

2

2

2

22

:equation wave3D

t

E

z

E

y

E

x

EE

2

2

2

2

t

E

t

D

Bt

t

BE

E

EEE

2

2

Nature of Electromagnetic Waves• A general plane wave with angular frequency travelling in the direction

of the wave vector k has the form

• Phase = 2 number of waves and so is a Lorentz invariant.• Apply Maxwell’s equations

)](exp[)](exp[ 00 xktiBBxktiEE

it

ki

BEkBE

BkEkBE

00

Waves are transverse to the direction of propagation,

and and are mutually perpendicularBE

,k

xkt

Plane Electromagnetic Wave

30

Plane Electromagnetic Waves

Ec

Bkt

E

cB

22

1

2Frequency

k

2Wavelength

2

thatdeduce

withCombined

kc

kB

E

BEk

ck

is in vacuum waveof speed

Reminder: The fact that is an invariant tells us that

is a Lorentz 4-vector, the 4-Frequency vector. Deduce frequency transforms as

xkt

k

c

,

vc

vckv

Waves in a Conducting Medium

• (Ohm’s Law) For a medium of conductivity ,

• Modified Maxwell:                                                    

• Put

Ej

D

t

EE

t

EjH

EiEHki

conduction current

displacement current

)](exp[)](exp[ 00 xktiBBxktiEE

40

8-

120

7

1057.21.2,103:Teflon

10,108.5:Copper

D

D

Dissipation factor

Attenuation in a Good Conductor

0since

withCombine

2

2

Ekik

EiEkkEk

EiHkEkk

HEkt

BE

EiEHki

For a good conductor D >> 1, ikik 12

, 2

depth-skin theis2

where

11

,expexpis form Wave

ik

xxti copper.mov water.mov

Charge Density in a Conducting Material

• Inside a conductor (Ohm’s law)

• Continuity equation is

• Solution is

Ej

tE

t

jt

0

0

te 0

So charge density decays exponentially with time. For a very good conductor, charges flow instantly to the surface to form a surface charge density and (for time varying fields) a surface current. Inside a perfect conductor () E=H=0

Maxwell’s Equations in a Uniform Perfectly Conducting Guide

022

2

2

H

E

E

Hi

EEE

Eit

DH

Hit

BE

z

y

x

Hollow metallic cylinder with perfectly conducting boundary surfaces

Maxwell’s equations with time dependence exp(it) are:

Assume )(

)(

),(),,,(

),(),,,(zti

zti

eyxHtzyxH

eyxEtzyxE

Then 0)( 222

H

Et

is the propagation constant

Can solve for the fields completely in terms of Ez and Hz

Special cases• Transverse magnetic (TM modes):

– Hz=0 everywhere, Ez=0 on cylindrical boundary

• Transverse electric (TE modes):– Ez=0 everywhere, on cylindrical boundary

• Transverse electromagnetic (TEM modes):– Ez=Hz=0 everywhere– requires

36

0

n

H z

i or022

Cut-off frequency, c

c gives real solution for , so attenuation only. No wave propagates: cut-off modes.

c gives purely imaginary solution for , and a wave propagates without attenuation.

For a given frequency only a finite number of modes can propagate.

37

a

ne

a

xnAE

a

nc

zti

c

,sin,1

2

a

na

nc For given frequency, convenient to

choose a s.t. only n=1 mode occurs.

2

1

2

22

122 1,

c

ckik

Propagated Electromagnetic Fields

From

kzta

xn

a

nAH

H

kzta

xnAkH

Ei

H

At

BE

z

y

x

sincos

0

cossin

real,isassuming,

38z

x

Phase and group velocities in the simple wave guide

39

21

22ckWave

number:wavelengthspacefreethe,

22

kWavelength:

velocityspace-freethanlarger

,1

kvpPhase

velocity:

velocityspace-freethansmaller

1222

k

dk

dvk gc

Group velocity:

Calculation of Wave Properties

• If a=3 cm, cut-off frequency of lowest order mode is

• At 7 GHz, only the n=1 mode propagates and

GHz503.02

103

2

1

2

8

a

f cc

ck

v

ck

v

k

k

g

p

c

18

18

189212221

22

ms101.2

ms103.4

cm62

m103103/10572

40

a

nc

Flow of EM energy along the simple guide

kzta

xnA

a

nHHE

kH

kzta

xnAEEE

zyyx

yzx

sincos,0,

cossin,0

22

222

22

2

0

2

2

0

2

since

8

1

4

1energy Magnetic

8

1

4

1energy Electric

a

nkW

k

a

naAdxHW

aAdxEW

e

a

m

a

e

41

Fields (c) are:

Time-averaged energy:Total e/m energy density

aAW 2

4

1

Poynting Vector

42

Poynting vector is xyzy HEHEHES ,0,

Time-averaged: a

xnkAS

22

sin1,0,02

1

Integrate over x:

2

4

1 akASz

So energy is transported at a rate:g

me

z vk

WW

S

Electromagnetic energy is transported down the waveguide with the group velocity

Total e/m energy density

aAW 2

4

1