PHY 712 Electrodynamics
9-9:50 AM MWF Olin 103
Plan for Lecture 15:
Finish reading Chapter 6
1. Some details of Liénard-Wiechert results
2. Energy density and flux associated with
electromagnetic fields
3. Time harmonic fields
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PHY 712 Spring 2017 -- Lecture 15
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Solution of Maxwell’s equations in the Lorentz gauge -- continued
Liénard-Wiechert potentials and fields -Determination of the scalar and vector potentials for a moving
point particle (also see Landau and Lifshitz The Classical
Theory of Fields, Chapter 8.)
Consider the fields produced by the following source: a point
charge q moving on a trajectory Rq(t).
Charge density:  (r, t )  q 3 (r  R q (t ))
Current density: J (r, t )  qR q (t ) (r  R q (t )), where R q (t ) 
3
dR q (t )
dt
Rq(t)
q
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PHY 712 Spring 2017 -- Lecture 15
4
.
Solution of Maxwell’s equations in the Lorentz gauge -- continued
1
 (r, t ) 
4 ò0
 (r ', t ')
  d r 'dt ' | r  r '] |   t ' (t  | r  r ' | / c) 
1
A(r, t ) 
4 ò0 c 2
3
J (r ', t')
  d r 'dt ' | r  r ' |   t ' (t  | r  r ' | / c)  .
3
We performing the integrations over first d3r’ and then dt’
making use of the fact that for any function of t’,

f (tr )
 dt ' f (t ')  t ' (t  | r  R q (t ') | / c)   R q (tr )  (r  R q (tr )) ,
1
c | r  R q (tr ) |
where the ``retarded time'' is defined to be
tr  t 
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| r  R q (tr ) |
c
.
PHY 712 Spring 2017 -- Lecture 15
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Solution of Maxwell’s equations in the Lorentz gauge -- continued
Resulting scalar and vector potentials:
q
1
 (r, t ) 
,
4 ò0 R  v  R
c
q
v
A(r, t ) 
,
2
4 ò0 c R  v  R
c
Notation:
R  r  R q (tr )
v  R q (tr ),
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tr  t 
PHY 712 Spring 2017 -- Lecture 15
| r  R q (tr ) |
c
.
6
Solution of Maxwell’s equations in the Lorentz gauge -- continued
In order to find the electric and magnetic fields, we need to
evaluate
A(r, t )
E(r, t )   (r, t ) 
t
B(r, t )    A(r, t )
The trick of evaluating these derivatives is that the retarded
time tr depends on position r and on itself. We can show the
following results using the shorthand notation:
R
t r  
vR 

c R 

c 

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and
tr
R

.
vR 
t 
R

c 

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Solution of Maxwell’s equations in the Lorentz gauge -- continued
  v2  v 
q
1
v R 
v R
 (r, t ) 
R 1  2    R 
 R 2 ,
3 
4 ò0 
c 
c 
vR    c  c 
R

c 

 vR  v 2 v  R v  R  vR 
A(r, t )
q
1
v  R 


 2  2 R
 2
 .
3 
t
4 ò0 
Rc
c  c 
c 
v R   c  c
R

c 

q
1
E(r, t ) 
3
4 ò0 
v R 
R

c 



vR   v 2  
vR  v   
 R 
 1  2    R   R 
  2   .
c  c  
c  c   






q  R  v  v 2 v  R 
R  v / c  R  E(r, t )
B(r, t ) 
1 2  2  

3 
2
2
4 ò0 c  
c  
cR
v R   c
v R  
 R 

R
 
c
c


 

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Coulomb' s law :
  D   free
D
 J free
t
B
Faraday's law :
E 
0
t
No magnetic monopoles :   B  0
Ampere - Maxwell' s law :   H 
Energy analysis of electromagnetic fields and sources
Rate of work done on source J(r, t) by electromagnetic field :
dW dEmech

  d 3r E  J
dt
dt
Expressing source current in terms of fields it produces :
dW
D 

3
  d r E   H 

dt
t 

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Energy analysis of electromagnetic fields and sources - continued
dW
D 

3
3
  d r E  J   d r E   H 

dt
t 

D
B 

3
   d r    E  H   E 
H

t
t 

Let S  E  H
" Poynting vector"
1
u  E  D  H  B  energy density
2
u

   S  E  J
t
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Energy analysis of electromagnetic fields and sources - continued
dEmech
3
  d r EJ
dt
Electromagnetic energy density :
E field   d r u r, t 
1
u  E  D  H  B 
2
3
Poynting vector :
S  E H
u
From the previous energy analysis :
   S  E  J
t
dEmech dE field
3
2


   d r   Sr, t     d r rˆ  Sr, t 
dt
dt
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Momentum analysis of electromagnetic fields and sources
dPmech
  d 3 r E  J  B 
dt
Follows by analogy with Lorentz force :
F  qE  v  B 
P field   0  d 3 r E  B 
Expression for vacuum fields :
Tij
 dPmech dP field 
3

    d r

dt i
rj
j
 dt
Maxwell stress tensor :


1


2
2
Tij   0  Ei E j  c Bi B j   ij E  E  c B  B 
2 15
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

12
Comment on treatment of time-harmonic fields
Fourier transformation in time domain :
1
E(r,t) 
2

~
 i t
d

E
(
r
,

)
e



~
E(r,)   dt E(r,t) e it

~
~*
Note that E(r,t) is real  E(r,)  E (r,  )
These relations and the notion of the superposition principle,
lead to the common treatment :

 
1 ~
~
~*
 i t
 i t
i t
E(r,t)   E(r,)e
 E(r,)e  E (r,)e
2
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13
Comment on treatment of time-harmonic fields -- continued
Equations for time harmonic fields :

 
1 ~
~
~*
 i t
 i t
E(r,t)   E(r,)e
 E(r,)e  E (r,)eit
2
Equations
Coulomb' s law :
in time domain
  D   free
D
Ampere - Maxwell' s law :   H 
 J free
t
B
Faraday's law :
E 
0
t
No magnetic monopoles :   B  0
in frequency domain
~
  D  ~ free
~
~ ~
  H  iD  J free
~
~
  E  i B  0
~
B  0
Note -- in all of these, the real part is taken at the end of the
calculation.
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
14
Comment on treatment of time-harmonic fields -- continued
Equations for time harmonic fields :

 
1 ~
~
~*
 i t
 i t
E(r,t)   E(r,)e
 E(r,)e  E (r,)eit
2

Poynting vector : Sr,t   E(r,t)  H(r,t)
1
Sr,t  
4
1

4


~
~*
~
~*
 i t
i t
 i t
E(r,)e  E (r,)e  H(r,)e  H (r,)eit


~
~*
~*
~
E(r,)  H (r,)  E (r,)  H(r,)

1 ~
~
~*
~*
 2 i t
 E(r,)  H(r,)e
 E (r,)  H (r,)e 2it
4
~*
1 ~

Sr,t  t avg   E(r,)  H (r,) 
2


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

PHY 712 Spring 2017 -- Lecture 15
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
Summary and review
Coulomb' s law :
  D   free
D
Ampere - Maxwell' s law :   H 
 J free
t
B
Faraday's law :
E 
0
t
No magnetic monopoles :   B  0
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For linear isotropic media - - D  E; B  H
and no sources :
Coulomb' s law :
E  0
E
Ampere - Maxwell' s law :   B  
0
t
B
Faraday's law :
E 
0
t
No magnetic monopoles :   B  0
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Analysis of Maxwell’s equations without sources -- continued:
Coulomb' s law :
E  0
Ampere - Maxwell' s law :   B  
E
0
t
B
0
t
No magnetic monopoles :   B  0
E 
   E 

2
     B  


B



t 
t

Faraday's law :
E 
 2B
  B   2  0
t
2
B 
   B 

2
   E 


E


t 
t

2

E
  2 E   2  0
t
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Analysis of Maxwell’s equations without sources -- continued:
Both E and B fields are solutions to a wave equation:
1
 B 2
v
1
2
 E 2
v
2
B
0
2
t
2
E
0
2
t
2
 0 0 c
where v  c
 2

n
2
2
2
Plane wave solutions to wave equation :

Br, t    B 0 e
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ik r it


Er, t    E 0 e
PHY 712 Spring 2017 -- Lecture 15
ik r it
19

Analysis of Maxwell’s equations without sources -- continued:
Plane wave solutions to wave equation :

Br, t    B 0 e
ik r it
    n 
k   

v  c 
2
2

2

Er, t    E 0 e
ik r it


where n 
 0 0
Note: , , n, k can all be complex; for the moment we will
assume that they are all real (no dissipation).
Note that E 0 and B 0 are not independent;
B
from Faraday's law :   E 
0
t
k  E 0 nkˆ  E 0
 B0 


c
also note : kˆ  E 0  0 and kˆ  B 0  0
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Analysis of Maxwell’s equations without sources -- continued:
Summary of plane electromagnetic waves :
 nkˆ  E 0 ik r it 

Br, t   
e

c


    n 
k   

v  c 

Er, t    E 0 e ik r it


where n 
and kˆ  E 0  0
 0 0
Poynting vector for plane electromagnetic waves :
2
2
2
S
avg
*


ˆ


n
k

E
1 
1
ik r it
0 ik r it  

  E0e
 
e
 

2
 c
 

2
1 
2
ˆ

k
E 0 kˆ
2 c
2 
n E0
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Analysis of Maxwell’s equations without sources -- continued:
Transverse Electric and Magnetic (TEM) waves
Summary of plane electromag netic waves :
 nkˆ  E0 ik r it 

Br, t   
e

c


2
    n 
k   

v
c
  

2
2
where n 

Er, t    E0 eik r it


and kˆ  E0  0
 0 0
Energy density for plane electromagnetic waves :




1
ik r it
ik r it *
u avg   E 0 e
 E0e

4
*

ˆ
ˆ
1  1 nk  E 0 ik r it  nk  E 0 ik r it  


e
 
e
 
4  
c
c

 

1
2
  E0
2
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More general result (for homework)
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