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