PHY 712 Electrodynamics
10-10:50 AM MWF Olin 107
Plan for Lecture 14:
Start reading Chapter 6
1. Maxwell’s full equations; effects of
time varying fields and sources
2. Gauge choices and transformations
3. Green’s function for vector and scalar
potentials
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PHY 712 Spring 2014 -- Lecture 14
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PHY 712 Spring 2014 -- Lecture 14
2
Full electrodynamics with time varying fields and sources
Image of statue of
James Clerk-Maxwell
in Edinburgh
"From a long view of the history
of mankind - seen from, say, ten
thousand years from now - there
can be little doubt that the most
significant event of the 19th
century will be judged as
Maxwell's discovery of the laws of
electrodynamics"
Richard P Feynman
http://www.clerkmaxwellfoundation.org/
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PHY 712 Spring 2014 -- Lecture 14
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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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PHY 712 Spring 2014 -- Lecture 14
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Microscopic or vacuum form (P  0; M  0) :
Coulomb' s law :
 E   / 0
1 E
Ampere - Maxwell' s law :   B  2
 0 J
c t
B
Faraday's law :
E 
0
t
No magnetic monopoles :   B  0
1
2
c 
 0 0
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PHY 712 Spring 2014 -- Lecture 14
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Formulation of Maxwell’s equations in terms of vector and
scalar potentials
B  0
 B   A
B
A 

E 
 0  E 
0
t
t 

A
E
 
t
A
or E   
t
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PHY 712 Spring 2014 -- Lecture 14
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Formulation of Maxwell’s equations in terms of vector and
scalar potentials -- continued
 E   / 0 :
   A 
  
  / 0
t
1 E
B  2
 0 J
c t
2

1     A 
    A   2 
 2    0 J
c  t
t 
2
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PHY 712 Spring 2014 -- Lecture 14
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Formulation of Maxwell’s equations in terms of vector and
scalar potentials -- continued
General form for the scalar and vector potential equations:
  
2
   A 
t
  / 0
1       2 A 
    A   2 
 2   0 J
c  t
t 
Coulomb gauge form -- require   A C  0
 2  C   /  0
2

1 A C 1    C 
2
 A C  2
 2
 0 J
2
c t
c
t
Note that J  J l  J t with   J l  0 and   J t  0
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PHY 712 Spring 2014 -- Lecture 14
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Formulation of Maxwell’s equations in terms of vector and
scalar potentials -- continued
Coulomb gauge form - - require   A C  0
  2 C   /  0
2

A C 1   C 
1
2
  AC  2
 2
 0 J
2
c t
c
t
Note that J  J l  J t with   J l  0 and   J t  0
Continuity equation for charge and current density :
  C 


   Jl  0

   J l   0 
t
t
t
  C 
1   C 
 2
  0 0
 0 J l
c
t
t
2

AC
1
2
  AC  2
 0 J t
2
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c t PHY 712 Spring 2014 -- Lecture 14
Formulation of Maxwell’s equations in terms of vector and
scalar potentials -- continued
Analysis of the scalar and vector potential equations :
   A 
2
  
  / 0
t
1      2 A 
    A   2 
 2    0 J
c  t
t 
1  L
Lorentz gauge form - - require   A L  2
0
c t
2
1

L
2
  L  2
  / 0
2
c t
2
1

AL
2
  AL  2
 0 J
2
c t
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PHY 712 Spring 2014 -- Lecture 14
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Formulation of Maxwell’s equations in terms of vector and
scalar potentials -- continued
1  L
Lorentz gauge form - - require   A L  2
0
c t
2
1  L
2
  L  2
  / 0
2
c t
2
1

AL
2
  AL  2
 0 J
2
c t

Alternate potentials : A' L  A L   and  ' L   L 
t
2
1  
2
Yields same physics provided that :    2 2  0
c t
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PHY 712 Spring 2014 -- Lecture 14
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Solution of Maxwell’s equations in the Lorentz gauge
2
L

1
2
  /  0
 L  2
2
c t
2
AL

1
2
  0 J
 AL  2
2
c t
Consider the general form of the 3 - dimensional wave equation :
2


1
2
   2 2  4f
c t
 r, t   wave field
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f r, t   source
PHY 712 Spring 2014 -- Lecture 14
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Solution of Maxwell’s equations in the Lorentz gauge -- continued
2
1

 r, t 
2
  r, t   2
 4f r, t 
2
c
t
Green' s function :
 2 1 2 
   2 2 G r, t ; r ' , t '  4 3 r  r ' t  t '
c t 

Formal solution for field  r, t  :
 r, t    f 0 r, t    d 3 r ' dt 'G r, t ; r ' , t ' f r ' , t '
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PHY 712 Spring 2014 -- Lecture 14
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Solution of Maxwell’s equations in the Lorentz gauge -- continued
Determination of the form for the Green' s function :
 2 1 2 
   2 2 G r, t ; r ' , t '  4 3 r  r ' t  t '
c t 

For the case of isotropic boundary values at infinity :
  1
1

G r, t ; r ' , t ' 
  t ' t  r  r '  
r  r'   c

Formal solution for field  r, t  :
 r, t    f 0 r, t  
  1
1

 d r ' dt ' r  r'   t ' t  c r  r'   f r' , t '
3
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PHY 712 Spring 2014 -- Lecture 14
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Solution of Maxwell’s equations in the Lorentz gauge -- continued
Analysis of the Green' s function :
 2 1 2 
   2 2 G r, t ; r ' , t '  4 3 r  r ' t  t '
c t 

Fourier analysis in the time domain - - note that
1
 t  t ' 
2

i  t  t ' 
d

e


Define :
1
G r, t ; r ' , t ' 
2

 d e
i  t  t ' 
~
G r, r ' ,  

 2 2  ~
    2 G r, r ' ,    4 3 r  r '
c 

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PHY 712 Spring 2014 -- Lecture 14
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Solution of Maxwell’s equations in the Lorentz gauge -- continued
Analysis of the Green' s function (continued) :
 2 2  ~
   2 G r, r ' ,    4 3 r  r '
c 

For the case of isotropic boundary values at infinity :
~
~
G r, r ' ,    G r  r ' ,  
~
Further assuming that G r  r ' ,   is isotropic in r  r '  R :
 1 d2
2  ~
3




r  r '
R

G
r
,
r
'
,



4

2
2 
c 
 R dR
1  i R / c
~
Solution : G r, r ' ,    e
R
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Solution of Maxwell’s equations in the Lorentz gauge -- continued
Analysis of the Green' s function (continued) :
1
~
 i r  r ' / c
G r, r ' ,   
e
r  r'
1
G r, t ; r ' , t ' 
2
1

2

i  t  t ' 
~
G r, r ' ,  
i  t  t ' 
1
 i r  r ' / c
e
r  r'
 d e


 d e

1  1


r  r '  2


d e


1
1

 t  t ' r  r ' / c  
 t 't  r  r ' / c 
r  r'
r  r'
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
i t t '  r r ' / c 
PHY 712 Spring 2014 -- Lecture 14
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Solution of Maxwell’s equations in the Lorentz gauge -- continued
1
G r, t ; r ' , t ' 
 t 't  r  r ' / c 
r  r'
Solution for field  r, t  :
 r, t    f 0 r, t  
  1
1

 d r ' dt ' r  r'   t ' t  c r  r'   f r' , t '
3
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Solution of Maxwell’s equations in the Lorentz gauge -- continued
Liènard-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 )  q R 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 2014 -- Lecture 14
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.
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 rd]rdt' ' | r  r ' |   t ' (t  | r  r ' | / c)  .
3
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 2014 -- Lecture 14
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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 2014 -- Lecture 14
| r  R q (tr ) |
c
.
21
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 

02/17/2014
and
tr
R

.
vR 
t 
R

c 

PHY 712 Spring 2014 -- Lecture 14
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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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PHY 712 Spring 2014 -- Lecture 14
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