Time variation
Combining electrostatics and magnetostatics:
(1) .E = r/eo
where r = rf + rb
(2) .B = 0
“no magnetic monopoles”
(3)  x E = 0
“conservative”
(4)  x B = moj
where j = jf + jM
Under time-variation:
(1) and (2) are unchanged,
(3) becomes Faraday’s Law
(4) acquires an extra term, plus 3rd component of j
Faraday’s Law of Induction
emf x induced in a circuit equals the rate of change of magnetic
flux through the circuit

x 
   B.d S x   E.d 
t
B

 E.d   t  B.dS
B
   E.dS   t .dS Stokes' Theorem
B
  E  
t
dS
dℓ
 E.d  0 in general, only  0 for electrosta tic fields for which E  
C
x  0 so E no longer representi ble simply by E  
Displacement current
  B mo j  j 
 .j 
1
1
mo
 B
Ampere’s Law
.  B  0
Problem!
mo
r
Continuity equation
 .j  
 0 for non - steady currents
t
Steady current implies constant charge density so Ampere’s
law consistent with the Continuity equation for steady currents
Ampere’s law inconsistent with the continuity equation
(conservation of charge) when charge density time dependent
Extending Ampere’s Law
add term to LHS such that
taking Div makes LHS also
identically equal to zero:
.E 
r
 e o .E  r
eo
j  ?  
1
mo
 B
. j  .?   0
or . j  . j  0

e o.E  r  . e o E   .j
t
t
 t 
E  1
 
j  e o
 B

The extra term is in the bracket
 t  m o

extended Ampere’s Law
E
  B m o j  e o mo
t
Illustration of displacement current
I
 .jD dv  
V
jD  e o
V
r
dv   jD .d S
t
S
E
t
instantane ous charge density   

E
eo
C
S
E  1 Q I
eo



t
t A t A
I ˆ
S jD .dS  S A n.dS  I
R
Q
A
Discharging capacitor
eo E
-
+
eo E/t
-
+
Displacement current magnitude
Suppose E varies harmonically in time
E  Eo sin(t  kx)
E
 E
t
j  E
 Cu  6.10
7
Ωm1
NB σ here is conductivi ty
E
 e oE
t
e o 8.854.10 12
19



1.47.10

7

6.10
eo
 ~ 10.19 rads-1 for eoE/t to be comparable to E
Types of current j
E
  B m o j  e o m o
t
j  jf  jM  jP Total current
P
jP 
t
jM   x M
k
M = sin(ay) k
j
i
jM = curl M = a cos(ay) i
• Polarisation current density from oscillation of charges in
electric dipoles
• Magnetisation current density variation in magnitude of
magnetic dipoles in space/time
1st form of Maxwell’s Equations
r
(1).E 
eo
( r  r f  rb )
(2).B  0
B
0
t
E
(4)  B  e o m o
 mo j
t
(3)  E 
( j  jf  jM  jP )
all field terms on LHS and all source terms on RHS
The sources (r and j) are multiple (free, bound, mag, pol)
special status of free source suggests 2nd Form
Extending Ampere’s Law to H
E
  B m o j  e o m o
t

1
mo
  B  jf  jM  jP  e o
E
t
P
E
 jf    M   e o
t
t
B


D
    M   jf  e oE  P     H  jf 
t
t
 mo

D/t is displacement current postulated by Maxwell (1862)
to exist in the gap of a capacitor being charged
In vacuum D = eoE and displacement current exists throughout
space
2nd form of Maxwell’s Equations
Applies only to well behaved LIH media
D  e r e oE  eE and B  mr moH  mH
Focus on sources means equations (2) and (3) unchanged!
Recall Gauss’ Law for D
rf
.D  r f  .eE   r f  .E 
e
In this version of (1), r  rf and eo  e
Recall H version of (4)
D
D
  H 
 jf
t
t
D
E
m  H  m
 mjf    B  em
 mjf
t
t
  H  jf 
In this version of (4), j  jf , also mo  m and eo  e
2nd and 3rd forms
rf
(1).E 
e
.D  r f
(2).B  0
.B  0
B
(3)  E 
0
t
E
(4)  B  em
 mjf
t
B
 E 
0
t
D
 H 
 jf
t
LHS: 2nd form, free sources only, other sources hidden in
permittivity and permeability constants
RHS: 3rd form (Minkowsky) free sources only, mixed fields, no
constants
Electromagnetic Wave Equation
First
form
r
(2).B  0
eo
B
E
(3)  E 
 0 (4)  B  e o m o
 mo j
t
t
(1).E 

(3)     E    B  0
t
 
E 
(4)     E   m o j  e o m o
0
t 
t 
 
E 
2
identity  .E    E   m o j  e o mo
0
t 
t 
 2E
r
j
(1)   E  e o m o 2    mo
t
eo
t
2
Electromagnetic Waves in Vacuum
E  Re E exp it kz  
E is a constant vector

o
 exp
2
it kz 
o
2
 2 exp it kz   k 2exp it kz 
z
2
it kz 
2
it kz 
exp



exp
= ck = 2pc/l
t 2
2
k

E
 2E  e o m o 2  0  k 2E   2e o m oE  0
Dispersion relation
t

1
speed of light
2
2
 k   e o mo  v p  
c
in vacuum
k
e o mo
If Eo || z then .E  0
If Eo  z then .E  0
Eo  z for form chosen implies transverse wave
Relationship between E and B
k || k
E  Re E exp it kz  ˆi
o
B
 xE ik Eo exp it kz  ˆj
t
k
it kz  ˆ E o
B  Eo exp
j
exp it kz  ˆj

c
B || j
E || i
Plane waves in a nutshell
exp i(kz)  exp i(k.r )
k.r  k.(r  r|| )  k.r  0
k.r  k r
k r  2p  r  l
i.e. k 
2p
l
k r
Consecutive wave fronts
r
r||
l
EM Waves in insulating LIH medium
 2E r f
jf
 E  em 2 
m
 0 (2nd form )
t
e
t
r f  jf  0
E  E exp it kz 
2

Slope=±c/n
o
2

E
 2E  em 2  0
t
 k 2E   2emE  0
= ck/n = 2pc/nl
k
Dispersion relation
k 2   2em
vp 

k

1
em

c
e r mr
c
Refractive index n 
 e r mr
vp
Less than speed of light in vacuum
e,m complex in general, real (as has been assumed) if hn<<Eg
Bound Charges
mx  mx  mo2 x  eEocos(t)




  2   2 cos(t)

- eEo  o
 sin(t)

x

m  o2   2 2   2 o2   2 2   2 


vB


+p/2
1
xB
xB
0.5
1
2
3
4
5
6
t
-0.5
-1

0
-p/2
o
vB
Bound charge displacement xB
Or velocity vB versus time
-p
Phase relative to driving field vs frequency
Free Charges
For a free charge, spring constant and o tend to zero
mx  mx  eEocos(t)
- eEo sin(t)
x(  0) 
m

- eEo cos(t)
x(  0) 
m

Ne 2Eo cos(t) Ne 2
j  rx 

Eocos(t)  E
m

m
Ne 2
Ne 2

c.f.  
Drude conductivi ty
m
m
Dielectric susceptibility
p
V
p  e oE  -ex
P  e o  EE 
ex
E  eoV
Polarisati on
Dipole moment
Dielectric susceptibi lity




2
 2

e2  o   cos(t)
 sin(t)

E 

2
2
e omV  o2   2   2 o2   2   2 




EM waves in conducting LIH medium
2

E r f
j
 2E  em 2 
m f 0
(2nd form)
t
e
t
r f  0 and jf  E
Ohmic conduction
jf
E
m
 m
t
t
2

E
E
2
 E  em 2  m
0
t
t
E  E exp it kz 
o
k 2   2em  im
k is complex
EM waves in conducting LIH medium
E
E
 E  emi 
 m
0
t
t
E
 2E  m
0
t
e  
2
1
(1  i) m
2
E  Eo exp it kz   Eo exp z  exp it  z 
k 2  im k   i m 
  
m
2
skin depth
2
m
 1cm in Cu at 50Hz
EM wave is attenuated within ~ skin depth in conducting media
NB Insulating materials become ‘conducting’ when radiation frequency tuned above Eg
Energy in Electromagnetic Waves
Energy density in matter for static fields
dU 1
 D.E  B.H
dv 2
In vacuum D  e oE B  m oH e r  m r  1
E  Eo exp i(t kz) H  Ho exp i(t kz)
dU 1
 e oE.E  m oH.H
dv 2
1
 e oEo .E o  m oHo .Ho cos2 (t  kz)
2
1
dU 1
2
cos (t  kz) 
 e oEo .E o  m oHo .Ho 
2
dv 4
Average obtained over one cycle of light wave
Energy in Electromagnetic Waves
Average energy over one cycle of light wave
dU 1
 e oEo .E o  moHo .Ho 
dv 4
Distance travelled by light over one cycle = 2pc/= c
Average energy in volume ab c
c
b
a
Energy in Electromagnetic Waves
1
e oEo .E o  moHo .Ho abc
4
B
E
1
Ho  o c 
Bo  o
mo
c
e o mo
U
 1 Eo
U
1
1 Ho 
 EoHo 
eo 
mo 
abc 2
2 Eo 
 2 Ho
Eo Boc

mo 
Ho Bo
mo
mo

eo
m oe o
 1 mo
 1
e
U
1
1
o
 EoHo 
eo 
m o   EoHo
abc 2
2 m o  2c
 2 eo
U
1
 EoHo Energy crossing unit area (ab) per periodic time ( )
ab 2
Poynting Vector
N = E x H is the Poynting vector
Equal to the instantaneous energy flow associated
with an EM wave
In vacuum N || wave vector k
Example If the E amplitude of a plane wave is 0.1 Vm-1
Energy crossing unit area per second is
1
1 2 mo
EoHo  Eo
 1.3.10 5 Wm 2
2
2
eo