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Recall the following morphs of Maxwell's equations, this time with
the sources and expressed in terms of potentials by means of the
homogeneous equations. Gauss's Law for magnetism is:

(11.3) 
This is an identity if we define
:

(11.4) 
Similarly, Faraday's Law is
and is satisfied as an identity by a scalar potential such that:
Now we look at the inhomogeneous equations in terms of the potentials.
Ampere's Law:



(11.10) 



(11.11) 



(11.12) 



(11.13) 



(11.14) 
Similarly Gauss's Law for the electric field becomes:
In the the Lorentz gauge,

(11.18) 
the potentials satisfy the following inhomogeneous wave equations:
where and
are the charge density and current density
distributions, respectively. For the time being we will stick with
the Lorentz gauge, although the Coulomb gauge:

(11.21) 
is more convenient for certain problems. It is probably worth reminding
y'all that the Lorentz gauge condition itself is really just one out of
a whole family of choices.
Recall that (or more properly, observe that in its role in these wave
equations)

(11.22) 
where is the speed of light in the medium. For the time being,
let's just simplify life a bit and agree to work in a vacuum:

(11.23) 
so that:
If/when we look at wave sources embedded in a dielectric medium, we can
always change back as the general formalism will not be any different.
Next: Green's Functions for the
Up: Maxwell's Equations, Yet Again
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Robert G. Brown
20130104