ElectromagnetismChristopher R Prior

ASTeC Intense Beams Group

Rutherford Appleton LaboratoryFellow and Tutor in Mathematics

Trinity College, Oxford

Contents

Maxwell’s equations and Lorentz Force LawMotion of a charged particle under constant Electromagnetic fieldsRelativistic transformations of fieldsElectromagnetic wavesWaves in a uniform conducting guide

Simple example TE01 modePropagation constant, cut-off frequencyGroup velocity, phase velocityIllustrations

ReadingJ.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

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.

tDjH

tBE

B

D

∂∂

+=∧∇

∂∂

−=∧∇

=⋅∇

=⋅∇

rrr

rr

r

r

0

ρE = electric field

D = electric displacement

H = magnetic field

B = magnetic flux density

ρ= charge density

j = current density

µ0 (permeability of free space) = 4π 10-7

ε0 (permittivity of free space) = 8.854 10-12

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

1,,In vacuum 20000 === cHBED µεµε

rrrr

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

∫∫∫∫∫ ==⋅⇔=⋅∇VS

QdVSdEE000

1ε

ρεε

ρ rrr

02

0

30

4

4

επε

πεq

rdSqSdE

rr

qE

spheresphere

==⋅

=

∫∫∫∫rr

rr

0ερ

=⋅∇ Er

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=⋅∇ Br

∫∫ =⋅⇔=⋅∇ 00 SdBBrrr

Gauss’ law for magnetism is then a statement thatThere 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 tBE∂∂

−=∧∇r

r

dtdSdB

dtdldE

SdtBSdE

C S

SS

Φ−=⋅−=⋅⇔

⋅∂∂

−=⋅∧∇

∫ ∫∫

∫∫∫∫rrr

rr

rr

∫ ⋅= ldErr

ε N S

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

∫ ⋅=Φ ldBrr

Maxwell’s 4th Equation tE

cjB

∂∂

+=∧∇r

rr20

1µ

Originates from Ampère’s (Circuital) Law :

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

ISdjSdBldBC S S∫ ∫∫ ∫∫ =⋅=⋅∧∇=⋅ 00 µµ

rrrrrr

rIB

rrldIB

πµ

πµ

θ 2

40

30

=

∧= ∫

current line straight a For

rrr

jBrr

0µ=∧∇

Ampère

Biot

Need for displacement currentFaraday: vary B-field, generate E-fieldMaxwell: varying E-field should then produce a B-field, but not covered by Ampère’s Law.

Surface 1 Surface 2

Closed loop

Current I

( )tEjjjB d ∂∂

+=+=∧∇r

rrrr0000 εµµµ

Apply Ampère to surface 1 (flat disk): line integral of B = µ0I

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

In capacitor, , so

Displacement current density istEjd ∂∂

=r

r0ε

dtdEA

dtdQI 0ε==

AεQE0

=

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

dVdtdSdj

ρ

ρ

ρ

r

r

rr ( )

tj

tj

Etc

jB

∂∂

+⋅∇=⇒

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+⋅∇=⇒

⋅∇∂∂

+⋅∇=∧∇⋅∇

ρερµεµ

µ

r

r

rr

0

0

1

0000

20

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

Maxwell’s Equations in Vacuo

In vacuum

Source-free equations:

Source equations

Equivalent integral forms (sometimes useful for simple geometries)

200001,,c

HBED === µεµεrrrr

0

0

=∂∂

+∧∇

=⋅∇

tBE

Bv

r

r

jtE

cB

E

rr

v

r

02

0

1 µ

ερ

=∂∂

−∧∇

=⋅∇

∫∫∫∫∫

∫∫∫

∫∫

∫∫ ∫∫∫

⋅+⋅=⋅

Φ−=⋅−=⋅

=⋅

=⋅

SdEdtd

cSdjldB

dtdSdB

dtdldE

SdB

dVSdE

rrrrrr

rrrr

rr

vr

20

0

1

0

1

µ

ρε

Example: Calculate E from B

⎩⎨⎧

><

=0

00

0cos

rrrrtB

Bz

ω

∫∫∫ ⋅−=⋅ dSBdtdldE

rrr

trBE

tBrrErr

ωωωωππ

θ

θ

cos2

cos2

0

02

0

−=⇒

−=<

trBrE

tBrrErr

ωω

ωωππ

θ

θ

cos2

cos2

02

0

02

00

−=⇒

−=>

Also from tBE∂∂

−=∧∇r

r

dtE

cjB

rrr ∂+=∧∇ 20

1µ then gives current density necessary to sustain the fields

r

z

Lorentz force lawSupplement 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:

( )BvEqfrrrr

∧+=

BjEfd

rrrr∧+= ρ

⎟⎠⎞

⎜⎝⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛ ⋅⇒=

dtpd

dtdE

cf

cfv

ddPF

rrrr

,1, γγτ

( ) ( )BvEqfvmdtd rrrrr

∧+==γ0

Motion of charged particles in constant electromagnetic fields

Constant E-field gives uniform acceleration in straight line

Solution of

Energy gain

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

cmqEtmqE

cmqEt

qEcmx

02

0

2

0

20

for21

11

<<≈

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛+=

( ) ( )BvEqfvmdtd rrrrr

∧+==γ0

( ) Emqv

dtd

0

=rγ

qEx=

Bvm

qdtvd rrr

∧=γ0 constantx

constant//

=

=

⊥r

rv

Relativistic Transformations of E and B

According to observer O in frame F, particle has velocity , fields are and 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

( )BvEqfrrrr

∧+=v E

rBr

Eqf ′′=′rr

BvEEqqrrr

∧+=′′= and

0,4 3

0

== BrrqE

rrr

πε

Evcr

rvqBrr

rrr∧−=

∧−= 23

0 14πµ

Evc

BBrrrr

∧−=′ 2

1

( )

////2

////

,

,

BBc

EvBB

EEBvEErr

rrr

rrrrr

=′⎟⎟⎠

⎞⎜⎜⎝

⎛ ∧−=′

=′∧+=′

⊥⊥

⊥⊥

γ

γ

Exact:

Electromagnetic wavesMaxwell’s equations predict the existence of electromagnetic waves, later discovered by Hertz.No charges, no currents:

00 =⋅∇=⋅∇∂∂

−=∧∇∂∂

=∧∇

BDtBE

tDH

rr

rr

rr

( )

( )

2

2

2

2

tE

tD

Bt

tBE

∂∂

−=∂∂

−=

∧∇∂∂

−=

∂∂

∧−∇=∧∇∧∇

rr

r

rr

µεµ

( ) ( )E

EEEr

rrr

2

2

−∇=

∇−⋅∇∇=∧∇∧∇

2

2

2

2

2

2

2

22

:equation wave3D

tE

zE

yE

xEE

∂∂

=∂∂

+∂∂

+∂∂

=∇rrrr

rµε

Nature of electromagnetic wavesA general plane wave with angular frequency ω travelling in the direction of the wave vector has the form

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

kr

)](exp[)](exp[ 00 xktjBBxktjEE rrrrrrrr⋅−=⋅−= ωω

ωjt

kj

↔∂∂

−↔∇r

BEkBE

BkEkBErrr&rr

rrrrrr

ω=∧↔−=∧∇

⋅==⋅↔⋅∇==⋅∇ 00

Waves are transverse to the direction of propagation, and and

are mutually perpendicular

BErs

,kr

xkt rr⋅−ω

Plane electromagnetic wave

Plane Electromagnetic WavesE

cBk

tE

cB

rrrr

r22

1 ω−=∧↔

∂∂

=∧∇

ωω

ω2

thatdeduce

withCombined

kckB

E

BEk

==

=∧

r

r

rrr

ck=⇒ r

ω is in vacuum waveof speed

πων

πλ

2Frequency

k2Wavelength

=

= rThe fact that is an invariant tells us that

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

xkt rr⋅−ω

⎟⎠⎞

⎜⎝⎛=Λ k

c

r,ω

( )vcvckv

+−

=⋅−=′ ωωγωrr

Waves in a conducting mediumFor a medium of conductivity σ,

Modified Maxwell:

Put

Ejrr

σ=

EEEjH &rr&rrrεσε +=+=∧∇

EjEHkjrrrr

εωσ +=∧−

conduction current

displacement current

)](exp[)](exp[ 00 xktjBBxktjEE rrrrrrrr⋅−=⋅−= ωω

Dissipation factor

ωεσ

=D

40

8-

120

7

1057.21.2,103:Teflon

10,108.5:Copper−×=⇒=×=

=⇒=×=

D

D

εεσ

εεσ

Attenuation in a Good Conductor

( ) ( )( )εωσµω

εωσµωµω

µω

+−=⇒

+−−=∧=∧∧⇒

=∧⇒−=∧∇

jkEjHkEkk

HEkBE

2

withCombinerrrrrr

rrr&rr

EjEHkjrrrr

εωσ +=∧−

εωσ >>For a good conductor D >> 1, ( )jkjk −≈⇒−≈ 12

2 µσωµσω

depth-skin theis2where

expexpis form Wave

µσωδ

δδω

=

⎟⎠⎞

⎜⎝⎛ −⎥

⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −

xxtj

copper.mov water.mov

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(jωt)are:

( ) ( )

44444444444 344444444444 21

r

r

r

r

rrr

rr

r

rr

r

022

2

2

=+∇

−=

∧∇=

∧∇∧∇−⋅∇∇=∇

⇒

=∂∂

=∧∇

−=∂∂

−=∧∇

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

HE

E

Hj

EEE

EjtDH

HjtBE

µεω

εµω

µωεω

µω

Assume )(

)(

),(),,,(

),(),,,(ztj

ztj

eyxHtzyxH

eyxEtzyxEγω

γω

−

−

=

=rr

rr

Then [ ] 0)( 222 =⎭⎬⎫

⎩⎨⎧

++∇HE

t r

r

γεµω

γ is the propagation constant Can solve for the fields completely in terms of Ez and Hz

Special casesTransverse 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 everywhererequires

0=∂∂

nH z

εµωγεµωγ j±==+ or022

A simple model with Ez=0

z

x

y

Transport between two infinite parallel conducting plates:

x=0 x=a

22222

22t

)(

,

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

γεµω

γω

+=−==∇

= −

KEKdx

EdE

xEexEE ztjr

KxAE⎭⎬⎫

⎩⎨⎧

=cossin

i.e.

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

integer ,,sin na

nKKKxAE nπ

===

Propagation constant is

εµω

ωωπ

εµωγ

nc

c

n

Ka

n

K

=⎟⎟⎠

⎞⎜⎜⎝

⎛−=

−=

where12

22

Cut-off frequency, ωc

ω<ωc gives real solution for γ, so attenuation only. No wave propagates: cut-off modes.ω>ωc gives purely imaginary solution forγ, and a wave propagates without attenuation.

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

εµπωπ

ωωπγ γω

ane

axnAE

an

cztj

c

==⎟⎟⎠

⎞⎜⎜⎝

⎛−= − ,sin,1

2

εµπω

εµπωω an

an

c <⇒=>For given frequency,

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

( )2

1

2

22

122 1, ⎟⎟⎠

⎞⎜⎜⎝

⎛−=−==ωωεµωωωεµγ c

ckjk

Propagated electromagnetic fieldsFrom

( )

( )⎪⎪

⎩

⎪⎪

⎨

⎧

−⎟⎠⎞

⎜⎝⎛−=

=

−⎟⎠⎞

⎜⎝⎛−=

⇒∧∇=

∂∂

−=∧∇

kzta

xna

nAH

H

kzta

xnAkH

EjH

tBE

z

y

x

ωππµω

ωπµω

µωsincos

0

cossin

,

rr

rr

(assuming A is real)

z

x

Phase and group velocities

Plane wave exp j(ωt-kx) has constant phase ωt-kx at peaks

ktxv

xkt

pω

ω

=∆∆

=⇔

=∆−∆ 0Superposition of plane waves. While shape is relatively undistorted, pulse travels with the group velocity

[ ]∫∞

∞−

− dkekA kxtkj )()( ω

dkdvgω

=

Wave packet structure

Phase velocities of individual plane waves making up the wave packet are different, The wave packet will then disperse with time

Phase and group velocities in the simple wave guide

Wave number is

so wavelength in guide the free-space wavelength

Phase velocity is larger than free-space velocity

Group velocity is less than infinite space value

( ) εµωωωεµ <−= 2122

ck

( )εµωεµ

ωωωεµ 1222 <==⇒−=k

dkdvk gc

,22εµωππλ >=

k

,1εµ

ω>=

kvp

Calculation of wave propertiesIf a=3 cm, cut-off frequency of lowest order mode is

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

GHz52

12

≅==εµπ

ωa

f cc

( )

18

18

12122

ms101.2

ms103.4

cm62m103

−

−

−

×==

×≈=

≈=

≈−=

ωεµ

ω

πλ

ωωεµ

kv

kv

k

k

g

p

c

Waveguide animationsTE1 mode above cut-off ppwg_1-1.movTE1 mode, smaller ω ppwg_1-2.movTE1 mode at cut-off ppwg_1-3.movTE1 mode below cut-off ppwg_1-4.movTE1 mode, variable ω ppwg_1_vf.movTE2 mode above cut-off ppwg_2-1.movTE2 mode, smaller ppwg_2-2.movTE2 mode at cut-off ppwg_2-3.movTE2 mode below cut-off ppwg_2-4.mov