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Electromagnetism
Christopher R Prior
ASTeC Intense Beams Group
Rutherford Appleton Laboratory
Fellow and Tutor in Mathematics
Trinity College, Oxford
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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
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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
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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
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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
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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.
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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
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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.
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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
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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
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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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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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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
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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
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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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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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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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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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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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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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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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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
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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
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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
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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
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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.mov7/30/2019 Vale Electromagnetism
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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
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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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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.mov7/30/2019 Vale Electromagnetism
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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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