Vale Electromagnetism

7/30/2019 Vale Electromagnetism

1/48

Electromagnetism

Christopher R Prior

ASTeC Intense Beams Group

Rutherford Appleton Laboratory

Fellow and Tutor in Mathematics

Trinity College, Oxford


2/48

2

Contents

Review of Maxwells 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


3/48

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: ClassicalElectricity and Magnetism

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


4/48

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

y

x

x,y,z,t

,,:gradient

,functionscalaraFor

Gradient is normal to surfaces

=constant


5/48

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


6/48

What is Electromagnetism?

The study of Maxwells equations, devised in 1863 torepresent the relationships between electric and magneticfields in the presence of electric charges and currents,whether steady or rapidly fluctuating, in a vacuum or inmatter.

The equations represent one of the most elegant andconcise way to describe the fundamentals of electricity andmagnetism. They pull together in a consistent way earlierresults known from the work of Gauss, Faraday, Ampre,Biot, Savart and others.

Remarkably, Maxwells equations are perfectly consistentwith the transformations of special relativity.


7/48

Maxwells Equations

Relate Electric and Magnetic fields generated by

charge and current distributions.

t

DjH

t

BE

B

D

0

1,,In vacuum2

0000 cHBED

E = electric field

D = electric displacement

H = magnetic fieldB = 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


8/48

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

Maxwells 1st Equation

000

1

QdVSdEdVEE

VSV

02

0

3

0

4

4

q

r

dSqSdE

rr

qE

sphe resphe re

0

E

Area integral gives a measure of the net charge

enclosed; divergence of the electric field gives the densityof the sources.


9/48

Gauss law for magnetism:

The net magnetic flux out of any closed

surface is zero. Surround a magneticdipole 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.

Maxwells 2nd Equation0 B

00 SdBB

Gauss law for magnetism is then a statement that

There are no magnetic monopoles


10/48

Equivalent to Faradays 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.

Maxwells 3rd Equationt

BE

dt

dSdB

dt

dldE

Sdt

BSdE

C S

SS

ldE

N S

Faradays Law is the basis for electric

generators. It also forms the basis for

inductors and transformers.

SdB


11/48

Maxwells 4th Equationt

E

cjB

20

1

Originates from Ampres (Circuital) Law :

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

1820):

ISdjSdBldBC S S

00

rIB

r

rldIB

2

4

0

3

0

currentlinestraightaFor

jB 0

Ampre

Biot


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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

Ampres Law.

Surface 1 Surface 2

Closed loop

Current I

Apply Ampre to surface 1 (flat disk): line integral

ofB = 0I

Applied to surface 2, line integral is zero since no

current penetrates the deformed surface.

In capacitor, , so

Displacement current density is

t

Ejd

0

dt

dEA

dt

dQI 0

A

QE

0

tE

jjjBd

0000


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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 Maxwells equations:Take divergence of (modified) Ampres

equation

0

tj

dVt

dVj

dVdt

dSdj

tj

tj

Etc

jB

0

0

1

0

000

20

Charge conservation is implicit in Maxwells Equations


14/48

14

Maxwells

Equations in Vacuum

In vacuum

Source-free equations:

Source equations

Equivalent integral forms (usefulfor 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


15/48

Example: Calculate E from B

0

00

0

sin

rr

rrtBBz

dSBdtdldE

trBE

tBrtBrdt

drErr

cos2

1

cossin2

0

0

2

0

2

0

tr

BrE

tBrtBrdt

drErr

cos2

cossin2

0

2

0

0

2

00

2

00

Also from

t

BE

dt

E

cjB

20

1 then gives current density necessary

to sustain the fields

r

z


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16

Lorentz Force Law

Supplement to Maxwells 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

c

f

c

fv

d

dPF

,1

,

BvEqfvmdt

d 0


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17

Motion of charged particles in constant

magnetic fields

1. Dot product with v:

constantisconstantis0So

1But

0

222

0

vdt

d

dt

dv

dt

dvcv

Bvvm

qv

dt

dv

Bvqvmdtd

BvEqfvmdt

d

00

No acceleration

with a magneticfield

constant,0

0

//

0

vvBdt

d

BvBm

qv

dt

dB

2. Dot product with B:

constantalso

constantandconstant //

v

vv


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Motion in constant magnetic field

0

0

0

2

0

frequencyangularat

radiuswithmotioncircular

mmm

qBv

qB

vm

Bvm

qv

Bv

m

q

dt

vd

Constant magnetic field givesuniform spiral about B with

constant energy.

rigidityMagnetic

0

q

p

q

vmB


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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

2

2

011

tm

qE

c

v

tm

qE

v

is

11

2

0

2

0

cm

qEt

qE

cmx

v

dt

dx

cmqEtm

qE0

2

0

for2

1

qExEnergy gain is


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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 30

Br

rqE

Evcr

rvqB

23

0 1

4

Evc

BB

2

1

////2

////

,

,

BB

c

EvBB

EEBvEE

Exact:


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21

Potentials

Magnetic vector potential:

Electric scalar potential:

Lorentz Gauge:

ABAB

thatsuch0

01

2

A

tc

AAf(t)

,

t

AE

t

AE

tAE

tAA

ttBE

so,with

0

Use freedom to set


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22

Electromagnetic 4-Vectors

,1

,1

01

42

A

ctcA

tc

Lorentz

Gauge

4-gradient4 4-potential A

Current

4-vector

000where),(),(

jcvcVJ

vj

Continuityequation 0,,14

jtjctcJ

Charge-current

transformations

2,

c

jvvjj xxx


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23

z

y

x

z

y

x

A

A

Ac

c

vc

v

A

A

Ac

1000

0100

00

00

'

'

'

'

Relativistic Transformations

4-potential vector:

Lorentz transformation

Fields:

AcA

,

1

BvEEEEcEv

BBBB

////2////

vtxxyyy

A

x

ABAB x

y

z

',and

2,and

c

vxttzz

t

A

zE

t

AE zz


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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

2

222 ',''so),,0,'('

Origins coincideat t=t=0

Observer P

z

b

charge qx

Frame F v Frame F

z

x

333 '',0',

'

'''

''

r

qbEE

r

qvtEx

r

qE zyxP

tvxtvxx PPP so)(0


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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

BvEEEEc

EvBBBB

////

2////

3

r

rvqB

At non-relativistic energies, 1, restoring the Biot-Savart

law:


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Electromagnetic Energy

Rate of doing work on unit volume of a system is

Substitute forjfrom Maxwells equations and re-arrange into the form

EjEvBjEvfv d

HBDEt

S

HESt

DEt

BHS

t

DEEHHEE

t

DHEj

2

1

where

Poynting vector


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27

HEDEHBt

Ej

2

1

SdHEdVHBDEdtd

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.


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Review of Waves

1D wave equation is with general

solution

Simple plane wave:

2

2

22

2

1tu

vxu

)()(),( xvtgxvtftxu

xktxkt

sin:3Dsin:1D

k

2Wavelength is

Frequency is

2


29/48

Superposition of plane waves. While

shape is relatively undistorted, pulsetravels with the group velocity

Phase and group velocities

kt

xv

xkt

p

0

dkekA kxtki )()(

dk

dvg

Plane wave has constant

phase at peaks

xktsin2 xkt
http://localhost/var/www/apps/conversion/tmp/scratch_8/group%20velocity/GroupVelocity.htmlhttp://localhost/var/www/apps/conversion/tmp/scratch_8/group%20velocity/GroupVelocity.html


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Wave packet structure

Phase velocities of individual plane waves making

up the wave packet are different,

The wave packet will then disperse with time


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Electromagnetic waves Maxwells 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

tEtD

Bt

t

BE

E

EEE

2

2

2

2

2

2

2

2

2

22

:equationwave3D

t

E

z

E

y

E

x

EE


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Nature of Electromagnetic Waves

A general plane wave with angular frequency travelling in the

direction of the wave vectork has the form

Phase = 2 number of waves and so is a Lorentz

invariant. Apply Maxwells 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


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Plane

Electromagnetic Wave


34/48

Plane Electromagnetic Waves

Ec

BktE

cB

221

2

thatdeduce

withCombined

k c

kB

E

BEk

ck

isin vacuumwaveofspeed

2Frequency

k

2Wavelength

Reminder: The fact that is aninvariant tells us that

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

xkt

kc

,

vc

vckv


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Waves in a Conducting Medium

(Ohms Law) For a medium of conductivity ,

Modified Maxwell:

Put

Ej

t

EE

t

EjH

EiEHki

conductioncurrent

displacementcurrent

)](exp[)](exp[00

xktiBBxktiEE

D

4

0

8-

12

0

7

1057.21.2,103:Teflon

10,108.5:Copper

D

D

Dissipation

factor


36/48

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-skintheis2

where

11

,expexpisformWave

ik

xxti copper.mov water.mov
http://localhost/var/www/apps/conversion/tmp/scratch_8/copper.movhttp://localhost/var/www/apps/conversion/tmp/scratch_8/water.movhttp://localhost/var/www/apps/conversion/tmp/scratch_8/water.movhttp://localhost/var/www/apps/conversion/tmp/scratch_8/copper.mov


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Charge Density in a Conducting Material

Inside a conductor (Ohms law)

Continuity equation is

Solution is

Ej

tE

t

j

t

0

0

t

e

0So 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

M ll E ti i U if


38/48

Maxwells Equations in a Uniform

Perfectly Conducting Guide

z

y

x

Hollow metallic cylinder with perfectlyconducting boundary surfaces

Maxwells equations with time dependence exp(it) are:

022

2

2

H

E

EHi

EEE

Eit

DH

Hit

BE

Assume)(

)(

),(),,,(),(),,,(

zti

zti

eyxHtzyxHeyxEtzyxE

Then 0)( 222

H

Et

is the propagation constant

Can solve for the fields completely

in terms ofEz and Hz


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Special cases

Transverse magnetic (TM modes):H

z=0 everywhere,E

z=0 on cylindrical boundary

Transverse electric (TE modes):E

z=0 everywhere, on cylindrical boundary

Transverse electromagnetic (TEM modes):

Ez=Hz=0 everywhere

requires

0

n

Hz

i or022

A i l d l P ll l Pl t W id


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A simple model: Parallel Plate Waveguide

Transport between two infinite conducting plates (TE01 mode):

2222

2

22

t

)(

,

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

KEKdx

EdE

xEexEEzti

KxAE

cos

sini.e.

To satisfy boundary conditions,E=0 onx=0andx=a, so

integer,,sin na

nKKKxAE n

Propagation constant is

nc

c

n

K

a

nK

where1

2

22

z

x

y


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Cut-off frequency, c

cgives real solution for, soattenuation only. No wave propagates: cut-

off modes. cgives purely imaginary solution for,

and a wave propagates without attenuation.

For a given frequencyonly a finite numberof modes can propagate.

ane

axnAE

an

czti

c

,sin,1

2

an

a

nc For given frequency, convenient to

choose a s.t. only n=1 mode

occurs.

2

1

2

2

21

221,

cckik

P t d El t ti Fi ld


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Propagated Electromagnetic Fields

From

kzta

xn

a

nA

H

H

kzta

xnAkH

Ei

H

A

t

BE

z

y

x

sincos

0

cossin

real,isassuming,

z

x

Ph d l iti i th i l


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Phase and group velocities in the simple

wave guide

2122 ckWave number:

wavelengthspacefreethe,22

kWavelength:

velocityspace-freethanlarger

,1

kvpPhase velocity:

velocityspace-freethansmaller

1222

k

dk

dvk gc

Group velocity:


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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

af cc

ck

v

ck

v

k

k

g

p

c

18

18

189212221

22

ms101.2

ms103.4

cm62

m103103/10572

a

nc


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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

Flow of EM energy along the simple
http://localhost/var/www/apps/conversion/tmp/scratch_8/ppwg_1-1.movhttp://localhost/var/www/apps/conversion/tmp/scratch_8/ppwg_1-2.movhttp://localhost/var/www/apps/conversion/tmp/scratch_8/ppwg_1-3.movhttp://localhost/var/www/apps/conversion/tmp/scratch_8/ppwg_1-4.movhttp://localhost/var/www/apps/conversion/tmp/scratch_8/ppwg_1_vf.movhttp://localhost/var/www/apps/conversion/tmp/scratch_8/ppwg_2-1.movhttp://localhost/var/www/apps/conversion/tmp/scratch_8/ppwg_2-2.movhttp://localhost/var/www/apps/conversion/tmp/scratch_8/ppwg_2-3.movhttp://localhost/var/www/apps/conversion/tmp/scratch_8/ppwg_2-4.movhttp://localhost/var/www/apps/conversion/tmp/scratch_8/ppwg_2-4.movhttp://localhost/var/www/apps/conversion/tmp/scratch_8/ppwg_2-4.movhttp://localhost/var/www/apps/conversion/tmp/scratch_8/ppwg_2-4.movhttp://localhost/var/www/apps/conversion/tmp/scratch_8/ppwg_2-3.movhttp://localhost/var/www/apps/conversion/tmp/scratch_8/ppwg_2-3.movhttp://localhost/var/www/apps/conversion/tmp/scratch_8/ppwg_2-3.movhttp://localhost/var/www/apps/conversion/tmp/scratch_8/ppwg_2-2.movhttp://localhost/var/www/apps/conversion/tmp/scratch_8/ppwg_2-2.movhttp://localhost/var/www/apps/conversion/tmp/scratch_8/ppwg_2-2.movhttp://localhost/var/www/apps/conversion/tmp/scratch_8/ppwg_2-1.movhttp://localhost/var/www/apps/conversion/tmp/scratch_8/ppwg_2-1.movhttp://localhost/var/www/apps/conversion/tmp/scratch_8/ppwg_2-1.movhttp://localhost/var/www/apps/conversion/tmp/scratch_8/ppwg_1_vf.movhttp://localhost/var/www/apps/conversion/tmp/scratch_8/ppwg_1-4.movhttp://localhost/var/www/apps/conversion/tmp/scratch_8/ppwg_1-4.movhttp://localhost/var/www/apps/conversion/tmp/scratch_8/ppwg_1-4.movhttp://localhost/var/www/apps/conversion/tmp/scratch_8/ppwg_1-3.movhttp://localhost/var/www/apps/conversion/tmp/scratch_8/ppwg_1-3.movhttp://localhost/var/www/apps/conversion/tmp/scratch_8/ppwg_1-3.movhttp://localhost/var/www/apps/conversion/tmp/scratch_8/ppwg_1-2.movhttp://localhost/var/www/apps/conversion/tmp/scratch_8/ppwg_1-2.movhttp://localhost/var/www/apps/conversion/tmp/scratch_8/ppwg_1-2.movhttp://localhost/var/www/apps/conversion/tmp/scratch_8/ppwg_1-1.movhttp://localhost/var/www/apps/conversion/tmp/scratch_8/ppwg_1-1.movhttp://localhost/var/www/apps/conversion/tmp/scratch_8/ppwg_1-1.mov


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2

2

222

22

2

0

2

2

0

2

since

8

1

4

1energyMagnetic

8

1

4

1energyElectric

a

nkW

k

a

naAdxHW

aAdxEW

e

a

m

a

e

Flow of EM energy along the simple

guide

kzt

a

xnA

a

nHHE

kH

kzta

xnAEEE

zyyx

yzx

sincos,0,

cossin,0

Fields (c) are:

Time-averaged energy:Total e/m energy

density

aAW2

4

1


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Poynting Vector

Poynting vector is xyzy HEHEHES ,0,

Time-averaged: a

xnkAS

22

sin1,0,02

1

Integrate overx:

2

4

1 ak ASz

So energy is transported at a rate:g

me

z

v

k

WW

S

Electromagnetic energy is transported down the waveguide

with the group velocity

Total e/m energy

density

aAW2

4

1


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