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Electromagnetism
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
2
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
3
Basic Equations from Vector Calculus
For a scalar function φx,y,z,t ,
 φ φ φ 
gradient : φ   , , 
 x y z 
Gradient is normal to surfaces
=constant

F  F1 , F2 , F3 ,
 F F F
divergence :   F  1  2  3
x
y
z
  F F F F F F 
curl :   F   3  2 , 1  3 , 2  1 
z z
x x y 
 y
For a vector
4
Basic Vector Calculus


 
 
  ( F  G)  G    F  F    G

    0,     F  0



2
  (  F )  (  F )   F
Stokes’ Theorem
 
 
   F  dS   F  d r
S
C
 
dS  n dS
Oriented
boundary C

n
Divergence or Gauss’
Theorem

 
   F dV   F  dS
V
S
Closed surface S, volume V,
outward pointing normal
5
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 Equations
Relate Electric and Magnetic fields generated by charge
and current distributions.
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
In vacuum

 

D   0 E, B  0 H ,  0 0c 2  1

D  

B  0


B
E 
t


 D
H  j 
t
 
E 
Maxwell’s 1st Equation
0
Equivalent to Gauss’ Flux Theorem:
 
 E 
0

  1
    E dV   E  d S 
V
S
0
Q
  dV  
V
0
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

E
q
40 r
 
q
E

d
S


40
sphere

r
3
dS q

2

r
0
sphere
Area integral gives a measure of the net charge
enclosed; divergence of the electric field gives the density
8
of the sources.

 B  0
Maxwell’s 2nd Equation
Gauss’ law for magnetism:

B  0 
 
 B  dS  0
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.
Gauss’ law for magnetism is then a statement that
There are no magnetic monopoles


B
E  
t
Maxwell’s 3rd Equation
Equivalent to Faraday’s Law of Induction:
 
B 
S   E  dS  S t  dS

 
d
d
  E  dl    B  dS  
dt S
dt
C
(for a fixed circuit C)
The electromotive force round a
circuit
 
   E  dl is proportional to the
rate of change of flux of magnetic
 
field,    B  dS through the circuit.
Faraday’s Law is the basis for electric
generators. It also forms the basis for
inductors and transformers.
N
S


 1 E
  B  0 j  2
c t
Maxwell’s 4th Equation
Originates from Ampère’s (Circuital) Law :


  B  0 j
 
 
 
 B  dl     B  dS  0  j  dS  0 I
C
Ampère
S
S
Satisfied by the field for a steady line current (Biot-Savart Law,
 
1820):
 0 I
B
4

dl  r
r3
For a straight line current
Biot
0 I
B 
2 r
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.
 Apply Ampère to surface 1 (flat disk): line
integral of B = 0I
Surface 2
Surface 1
 Applied to surface 2, line integral is zero
since no current penetrates the deformed
surface.
Current I
dQ
dE
Q , so
 In capacitor, E 
I
 A
ε0 A
Closed loop

0
dt
dt


E
Displacement current density is j  
d
0
t


 

E
  B   0  j  jd    0 j   0 0
t
12
Consistency with Charge
Conservation
Charge conservation:
Total current flowing out of a region
equals the rate of decrease of charge
within the volume.
 
d
j

d
S


 dV


dt


    j dV   
dV
t
 
  j 
0
t
From Maxwell’s equations:
Take divergence of (modified) Ampère’s
equation
 

 1 
    B   0  j  2   E
c t


 0   0  j   0  0  
t   0 
 
 0   j 
t
Charge conservation is implicit in Maxwell’s Equations
13
Maxwell’s
Equations in Vacuum
In vacuum

 

1
D   0 E , B  0 H ,  0 0  2
c
Source-free equations:

B  0

 B
E
0
t
Sourceequations

E 
0

 1 E

B 2
 0 j
c t
Equivalent integral forms
(useful for simple geometries)

  1
E  dS 

 
B  dS  0
0
  dV
 
 
d
d
 E  dl   dt  B  dS   dt
 
 
  1 d
 B  dl  0  j dS  c2 dt  E  dS
14
Example: Calculate E from B
 
d 
 E  dl   dt  B  dS
z
r  r0
r
B0 sin  t r  r0
Bz  
r  r0
 0
Also from


B
E  
t

r  r0

d
 r 2 B0 sin  t   r 2 B0 cos  t
dt
1
E   B0 r cos  t
2
2 rE  
d
 r02 B0 sin  t   r02 B0 cos  t
dt
 r02 B0
E  
cos  t
2r
2 rE  


 1 E then gives current density necessary
  B  0 j  2
c dt to sustain the fields
Lorentz Force Law
• Supplement to Maxwell’s equations, gives force on a charged particle
moving in an electromagnetic field:


  
f  q Ev B

• For continuous distributions, have a force density

  
fd   E  j  B
• Relativistic equation of motion
 
v f
dP
F
  
,
d
 c
– 4-vector form:


 1 dE dp 

f  
, 
 c dt dt 

– 3-vector component:


  

d
m0 v   f  q E  v  B
dt

16
Motion of charged particles in constant
magnetic fields

  

d
m0 v   f  q E  v  B
dt





 
d
m0 v   q v  B
dt

1. Dot product with v:
 d 
q   
v   v  
v v  B  0
dt
m0
But
So
 d 
d
 v  v   
dt
dt

d
 0   is constant  v is constant
dt
γv 2  c 2  2  1
No acceleration
with a magnetic
field
2. Dot product with B:
 d

q   
B   v  
Bv  B  0
dt
m0
d  

B  v  0, v//  constant
dt


17
Motion in constant magnetic field

dv
q  

v B
dt m0

v2


Constant magnetic field
gives uniform spiral about B
with constant energy.
q
v B
m0
 circular motion with radius
at angular frequency
ω
v


ρ
qB
m
m0v
qB
m  m0 
m0 v p
B 

q
q
Magnetic rigidity
Motion in constant Electric Field
  
 
d
m0 v   f  q E  v  B
dt

Solution of
is
v 
d 
q 
 v   E
dt
m0
qE
 v 
t   2  1  
m0
c
dx v

dt 



d
m0 v   q E

dt
2
 qE 
   1  
t 
 m0 
2
2


 qEt 
m0c 
  1
 x
1  
qE 
m0c 




2

Energy gain is
1 qE 2
t
2 m0
for qE  m0c
qEx
Constant E-field gives uniform acceleration in straight line
19
Potentials
• Magnetic vector potential:




  B  0   A such that B    A
• Electric scalar potential:






B

A
A 
E  
    A   
    E    0
t
t
t
t 



 A

A
   with E 
  , so E   
t
t

• Lorentz Gauge:
Use freedom to set

    f(t),


A  A  

1 
 A  0
2
c t
20
Electromagnetic 4-Vectors
Lorentz
Gauge

1 
1 
 1 
  A  0  
,     , A   4  Α
2
c t
 c t
 c

4
4-potential A


j  v


J  0V  0 (c, v )  ( c, j ) where   0
4-gradient
Current
4-vector 
 

Continuity
1 

4  J  
,   c , j 
  j  0
equation
t
 c t


Charge-current
transformations

v jx 

jx    jx   v ,        2 
c 

21
Example: Electromagnetic Field of a Single Particle
• Charged particle moving along x-axis of Frame F
Frame F
z
Observer P
v
z’
Frame F’
Origins coincide
at t=t=0
b
x
• P has
charge q
x’
0  xP   ( xP  vt) so xP  vt

x ' P  ( vt' ,0, b), so

 vxp 
2
2 2
x p  r '  b  v t ' , t '    t  2    t
c 

• In F, fields are only electrostatic (B=0), given by

q 
E '  3 x 'P
r'
qvt'
qb
 E ' x   3 , E ' y  0, E ' z  3
r'
r'
Electromagnetic Energy
• Rate of doing work on unit volume of a system is


  
 
 

 
 v  f d  v   E  j  B   v  E   j  E
• Substitute for j from Maxwell’s equations and re-arrange into the form


 D  
  
  D
 

  E    E  H  H    E  E 
 j  E     H 
t 
t



  
 B  D
 S  H 
E
where S  E  H
t
t
1     
 S 
ED  BH
Poynting vector
2 t


23



  


 
 1  
 j E 
 BH  E D  E  H
t  2


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.


  
dW
d
1    
  E  D  B  H dV   E  H  dS
dt
dt
2
electric +
magnetic energy
densities of the
fields
Poynting vector
gives flux of e/m
energy across
boundaries
24
Review of Waves
• 1D wave equation is
solution
 2u with
1  2ugeneral
 2 2
2
x
v t
u( x, t )  f (vt  x )  g (vt  x )
• Simple plane wave:
1D : sin  t  k x 
2
Wavelength is   
k
Frequency is


2

 
3D : sin  t  k  x

Phase and group velocities

i  ( k ) t kx 
A
(
k
)
e
dk


Plane wave sin t  k x  has constant
phase t  k x   2 at peaks

t  kx  0
x 
vp 
t

k
Superposition of plane waves. While
shape is relatively undistorted, pulse
travels with the group velocity
vg 
d
dk
Wave packet structure
• Phase velocities of individual plane waves making up
the wave packet are different,
• The wave packet will then disperse with time
27
Electromagnetic waves
•
•
Maxwell’s equations predict the existence of electromagnetic waves, later
discovered by Hertz.

No charges, no currents:




B
    E   
t



B
t


2
2
 D
 E
   2   
t
t 2




D
H 
t

D  0



B
E  
t

B  0





2
    E   E  E

2
  E
3D wave equation :




2
2
2
2

 E  E  E
 E
2
 E  2  2  2   2
x
y
z
t
Nature of Electromagnetic Waves
• A general plane wave with angular frequency  travelling in the direction
of the wave vector k has the form
 
 
 
 
E  E0 exp[ i(t  k  x )] B  B0 exp[ i(t  k  x )]
 
• Phase  t  k= 2
x  number of waves and so is a Lorentz invariant.
• Apply Maxwell’s equations

   ik

 i
t
 
 


E  0  B  k E  0  k B
 



  E  B  k  E   B

 
k and E , B and are mutually perpendicular
Waves are transverse to the direction of propagation,
Plane
Electromagnetic Wave
30
Plane Electromagnetic Waves

 1 E
B  2
c t
 
 
 k B 2E
c
 

k  E  B
Combined with

E  kc2
deduce that
  

B k
Wavelength
2
 
k
Frequency  

2

speed of wave in vacuum is

 c
k
 
Reminder: The fact that  t  k  x is an
invariant tells us that
 
   ,k 
c 
is a Lorentz 4-vector, the 4-Frequency vector.
Deduce frequency transforms as


 
cv

    v k  
cv
Waves in a Conducting Medium
 
 
 
 
E  E0 exp[ i(t  k  x )] B  B0 exp[ i(t  k  x )]
• (Ohm’s Law) For a medium of conductivity ,
• Modified Maxwell:
• Put D 




  E

E
 H  j 
E 
t
t


j E
 


ik  H   E  i E
Dissipation
factor
conduction
current
Copper :   5.8 107 ,    0
 D  1012
Teflon :   3 10-8 ,   2.1 0
 D  2.57 104
displacement
current
Attenuation in a Good Conductor
 


i k  H   E i E

 


B
Combine with   E  
 k  E   H
t
  
 

 k  k  E    k  H    i   E


  
 k  E k  k 2 E    i   E
 

k 2    i    since k  E  0




For a good conductor D >> 1,
   , k  i 
2
x 
1

 x
Wave form is exp i   t   exp   , k  1  i 
    


where  
2
 
is the skin - depth
 k
copper.mov
 
2
1  i 
water.mov
Charge Density in a Conducting Material
• Inside a conductor (Ohm’s law)


j  E
• Continuity equation is


  j  0
t


 

 E  0 
 
t
t 
• Solution is
  0e
t 
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
Hollow metallic cylinder with perfectly
conducting boundary surfaces
Maxwell’s equations with time dependence exp(it) are:






2
B
 E   E      E
E  
 i  H

t

 i    H

 D


2
H 
 i E
   E

t






 E 
2
 2

         0


  H 




x
y

Assume E ( x, y, z, t )  E ( x, y )e( i t  z )


H ( x, y, z, t )  H ( x, y )e( i t  z )
 is the propagation constant
Can solve for the fields completely
in terms of Ez and Hz




E 
Then  t2  ( 2   2 )     0
H 


Special cases
• Transverse magnetic (TM modes):
– Hz=0 everywhere, Ez=0 on cylindrical boundary
• Transverse electric (TE modes):
– Ez=0 everywhere,
H z on cylindrical boundary
0
n
• Transverse electromagnetic (TEM modes):
– Ez=Hz=0 everywhere
– requires
 2  2  0 or   i 
36
Cut-off frequency, c
2

n
nx i t  z
n



1    , E  A sin
e
, c 
a
a
a 
 c 
 c gives real solution for , so
attenuation only. No wave propagates: cutoff modes.
 c gives purely imaginary solution for ,
and a wave propagates without attenuation.
  ik , k     
2

2
c

1
2
 
   1 
 
2
c
2
For a given frequency  only a finite number



1
2
of modes can propagate.
  c 
n
a 
 n
a


For given frequency, convenient to
choose a s.t. only n=1 mode
occurs.
37
Propagated Electromagnetic Fields


From   E   B , assuming A is real,
t

Ak
 n x 
sin 
 cos t  kz
 Hx  
  a 



i
H
E  
Hy  0


A n
 n x 
H


cos

 sin  t  kz
 z
 a
 a 

x
z
38
Phase and group velocities in the simple wave guide

k     
Wave number:
2
2
c

1
2
  
2
2


, the free  space wavelength
k  
Wavelength:
Phase velocity:
vp 

k

1

,
larger than free - space velocity
Group velocity:
k    
2
2
2
c

d
k
1
 vg 


dk 

smaller than free - space velocity
39
Calculation of Wave Properties
• If a=3 cm, cut-off frequency of lowest order mode is
c
1
3  108
fc 


 5 GHz
2 2a  2  0.03
c 
n
a 
• At 7 GHz, only the n=1 mode propagates and
k     
2

vp 
vg 
2
c

1
2
 2 72  52   109 / 3  108  103 m 1
12
2
 6 cm
k

k
 4.3  108 ms 1  c
k

 2.1  108 ms 1  c
40
Flow of EM energy along the simple guide
Fields (c) are:
n x
cost  kz
a
k
n
n x
Hx  
E y , H y  0, H z  
A cos
sin t  kz

a
a
E x  E z  0, E y  A sin
Total e/m energy
density
Time-averaged energy:
Electric energy
a
2
1
1 2
We    E dx  A a
4 0
8
1 2
W  A a
4
2
2


2
1
1 2  n   k  
Wm    H dx  A a 
  
 
4 0
8
 a     
a
Magnetic energy
 We
since
n 2 2
k  2   2
a
2
41
Poynting Vector
  
Poynting vector is S  E  H  E y H z ,0, E y H x 
Time-averaged:
Integrate over x:
 1
kA2 2 n x
S  0, 0,1
sin
2

a
1 akA2
Sz 
4 
So energy is transported at a rate:
Total e/m energy
density
1 2
W  A a
4
Sz
k

 vg
We  Wm 
Electromagnetic energy is transported down the waveguide
with the group velocity
42