Review of E&M, Maxwell’s Equations, and 4-vector potentials
The force equation for a particle of charge Q in an electromagnetic field is



F  Q E   B
c


The fields defined by this equation are found to obey Maxwell’s equations:
(a)  E  4
 b
 E  
(1.1)
(c)  B=0
1 B
c t
d 
 B 
4
1 E
J
c
c t
(1.2)
Adding the time derivative of (a) to the divergence of (d), and recalling that
  V   0 for any vector field V, gives the continuity equation:

 J  0
t
which shows that charge is conserved. Furthermore, since
1 


,  
 c t


(1.3)
(1.4)

is a 4-vector, equation (1.3) shows that J  c , J is also a 4-vector.
In the case of static fields, we know that equations (a) and (c) imply that the fields
can be expressed in terms of potentials  and A :
E  
and
(for static fields only)
(1.5)
B   A
(1.6)
(see Purcell, p. X, for a review of the vector potential A ). When we are dealing with the
more general case of time-varying fields, the equation (1.5) has to be modified since E is
no longer conservative. Substituting B   A into (b) gives
1 A
E   
(1.7)
c t
valid for time-varying fields as well. Consequently one can completely describe an E&M
field by giving the potentials, rather than the fields themselves (4 components per point
rather than 6). But there is redundancy even here, since the potentials producing a given
EM field are not uniquely determined by (1.7) and (1.6). We can transform the potentials
through a “gauge transformation” without changing E and B, by
1 
   
(1.8)
c t
A  A  
(It is left as an exercise, HW8.1, to verify that this leaves the fields unchanged.)
This freedom allows one to choose a gauge that simplifies the equations. In
particular, if we write E and B in terms of the potentials in Maxwell’s source equations
(a) and (d), we get
1
 2 
  A  4
c t
(1.9)
1 2 A
 1 
 4
2
 A  2 2   
  A 
J
c t
 c t
 c
We can always choose a gauge that sets
1 
 A
(the Lorentz gauge)
(1.10)
c t
(see Shankar problem 18.4.3, in Examples). Making this transformation converts
equations (1.9) into
  4
4
2
2
A
c
(1.11)
J
where
2
1 2
 2 2  
c t
(1.12)
We can learn a lot from equations (1.11). First of all, note that
and J   c, J is a 4-vector, so



A  , A

is a 4-vector
2
is a Lorentz scalar,
(1.13)
in the Lorentz gauge.
Secondly, in free spaced (   0, J  0 ) both of these reduce to the wave equation:
1 2
A  r , t   2 A  r , t   0
(1.14)
2
2
c t
Note that this equation is solved, in the case of plane waves moving in the x-direction, by
any function of the form A  x, t   f ( x  0t ) , with 0  c . Such a solution propagates
(undistorted) in the positive x-direction at velocity c.
A sinusoidal plane wave solution has the form
A  r , t   A0 cos k r  t
(1.15)
2
Ar ,t  


The E and B fields carried by the wave are given by
1 A
 
E
    A0 sin k r  t
c t
c


 
B    A   k  A0 sin k r  t
Since   ck , E  B .


(1.16)