Electromagnetism Christopher R Prior ASTeC Intense Beams Group Rutherford Appleton Laboratory Fellow and Tutor in Mathematics Trinity College, Oxford.

Electromagnetism

Christopher R Prior

ASTeC Intense Beams Group

Rutherford Appleton Laboratory

Fellow and Tutor in Mathematics

Trinity College, Oxford

2

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

3

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

4

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

Gradient is normal to

surfaces =constant

5

Basic Vector Calculus

FFF

F

GFFGGF

2)()(

0,0

)(

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.

t

DjH

t

BE

B

D

0

1,,In vacuum 20000 cHBED

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

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

Maxwell’s 1st Equation

000

1

Q

dVSdEdVEEVSV

02

0

30

4

4

q

r

dSqSdE

rr

qE

spheresphere

0

E

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

N S

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

SdB

Maxwell’s 4th Equationt

E

cjB

20

1

Originates from Ampère’s (Circuital) Law :

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

ISdjSdBldBC S S 00

r

IB

r

rldIB

2

4

0

30

current line straight a For

jB

0

Ampère

Biot

12

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.

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

13

Consistency with Charge Conservation

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

volume.

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

0

tj

dVt

dVj

dVdt

dSdj

tj

tj

Etc

jB

0

0

1

0000

20

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

14

Maxwell’s Equations in

VacuumIn vacuum

Source-free equations:

Source equations

Equivalent integral forms (useful for simple geometries)

20000

1,,

cHBED

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

16

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

BjEfd

dt

pd

dt

dE

cf

c

fv

d

dPF

,1

,

BvEqfvmdt

d 0

17

Motion of charged particles in constant magnetic fields

1. Dot product with v:

constantisconstantis0So

1But

0

222

0

vdt

ddt

dv

dt

dvcvγ

Bvvm

qv

dt

dv

Bvqvmdt

dBvEqfvm

dt

d 00

No acceleration with a magnetic field

constant,0

0

//

0

vvBdt

d

BvBm

qv

dt

dB

2. Dot product with B:

constantalso

constantandconstant //

v

vv

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

19

Motion in constant Electric Field

Solution of Em

qv

dt

d

0

Constant 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

cmqEtm

qE0

2

0

for2

1

qExEnergy gain is

• According to observer O in frame F, particle has velocity v, fields

are E and B and Lorentz force is

• In Frame F, particle is at rest and force is

• Assume measurements give same charge and force, so

• Point charge q at rest in F:

• See a current in F, giving a field

• Suggests

Relativistic Transformations of E and B

BvEqf

Eqf

BvEEqq

and

0,4 3

0

Br

rqE

Evcr

rvqB

23

0 1

4

Evc

BB

2

1

Rough

idea

////2

////

,

,

BBc

EvBB

EEBvEE

Exact:

21

Potentials• Magnetic vector potential:

• Electric scalar potential:

• Lorentz Gauge:

ABAB

thatsuch0

01

2

Atc

AAf(t)

,

t

AE

t

AE

t

AE

t

AA

tt

BE

so,with

0

Use freedom to set

22

Electromagnetic 4-Vectors

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

23

z

y

x

z

y

x

A

A

Ac

cv

cv

A

A

Ac

1000

0100

00

00

'

'

'

'

Relativistic Transformations• 4-potential vector:

• Lorentz transformation

• Fields:

A

cA

,

1

BvEEEEc

EvBBBB

////2////

vtxxyyy

A

x

ABAB xy

z

',and

2

,andc

vxttzz

t

A

zE

t

AE z

z

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,'('

Origins coincide at t=t=0

Observer Pz

b

charge qx

Frame F v Frame F’z’

x’

333 '',0',

'

'''

''

r

qbEE

r

qvtEx

r

qE zyxP

tvxtvxx PPP so)(0

Transform to laboratory frame F:

23

2222

23

2222

'

0

'

tvb

bqEE

E

tvb

vtqEE

zz

y

xx

0

'2

zx

zzy

BB

Ec

Ec

vB

33 '',0',

'

''

r

qbEE

r

qvtE zyx

BvEEEE

c

EvBBBB

////

2////

3r

rvqB

At non-relativistic energies, ≈ 1, restoring the Biot-Savart law:

26

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

EjEvBjEvfv d

HBDEt

S

HESt

DE

t

BHS

t

DEEHHEE

t

DHEj

2

1

where

Poynting vector

27

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

30

Wave packet structure

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

• The wave packet will then disperse with time

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

t

E

t

D

Bt

t

BE

E

EEE

2

2

2

2

2

2

2

2

2

22

:equation wave3D

t

E

z

E

y

E

x

EE

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

33

Plane Electromagnetic

Wave

Plane Electromagnetic Waves

Ec

Bkt

E

cB

22

1

2

thatdeduce

withCombined

kc

kB

E

BEk

ck

is in vacuum waveof speed

2Frequency

k

2Wavelength

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

t

EE

t

EjH

EiEHki

conduction current

displacement current

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

D

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

z

y

x

Hollow metallic cylinder with perfectly conducting boundary surfaces

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

022

2

2

H

E

E

Hi

EEE

Eit

DH

Hit

BE

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

39

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

0

n

H z

i or022

40

A simple model: “Parallel Plate Waveguide”

Transport between two infinite conducting plates (TE01 mode):

22222

22t

)(

,

satisfies )( where)()0,1,0(

KEKdx

EdE

xEexEE zti

KxAE

cos

sini.e.

To satisfy boundary conditions, E=0 on x=0 and x=a, so

integer ,,sin na

nKKKxAE n

Propagation constant is

n

cc

n

K

a

nK

where1

2

22

z

x

y

x=0 x=a

41

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.

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

42

Propagated Electromagnetic Fields

From

kzta

xn

a

nAH

H

kzta

xnAkH

Ei

H

At

BE

z

y

x

sincos

0

cossin

real,isassuming,

z

x

43

Phase and group velocities in the simple wave guide

21

22ckWave

number:wavelengthspacefreethe,

22

kWavelength:

velocityspace-freethanlarger

,1

kvpPhase

velocity:

velocityspace-freethansmaller

1222

k

dk

dvk gc

Group velocity:

44

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

a

nc

45

Waveguide animations

• TE1 mode above cut-off ppwg_1-1.mov• TE1 mode, smaller ppwg_1-2.mov• TE1 mode at cut-off ppwg_1-3.mov• TE1 mode below cut-off ppwg_1-4.mov• TE1 mode, variable ppwg_1_vf.mov• TE2 mode above cut-off ppwg_2-1.mov• TE2 mode, smaller ppwg_2-2.mov• TE2 mode at cut-off ppwg_2-3.mov• TE2 mode below cut-off ppwg_2-4.mov

46

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

Flow of EM energy along the simple guide

kzta

xnA

a

nHHE

kH

kzta

xnAEEE

zyyx

yzx

sincos,0,

cossin,0

Fields (c) are:

Time-averaged energy:Total e/m energy density

aAW 2

4

1

47

Poynting VectorPoynting 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

48