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:
= 0
This is an identity if we define
:
() = 0

Similarly, Faraday's Law is
+ t & = & 0

+ t & = & 0

(+ t) & = & 0
and is satisfied as an identity by a scalar potential such that:
+ t & = & -&phis#phi;

& = & -&phis#phi;- t

Now we look at the inhomogeneous equations in terms of the potentials.
Ampere's Law:
& = & &mu#mu;(+
&epsi#epsilon;t)

() & = & &mu#mu;(+
&epsi#epsilon;t)

() - &nabla#nabla;^2& = & &mu#mu;+
&mu#mu;&epsi#epsilon;t

() - &nabla#nabla;^2& = & &mu#mu;-
&mu#mu;&epsi#epsilon;&phis#phi;t - &mu#mu;&epsi#epsilon;
t

&nabla#nabla;^2- &mu#mu;&epsi#epsilon;t & = & -&mu#mu;
+ (+ &mu#mu;&epsi#epsilon;&phis#phi;t)

Similarly Gauss's Law for the electric field becomes:
& = & &rho#rho;&epsi#epsilon;

(-&phis#phi;- t) & = &
&rho#rho;&epsi#epsilon;

&nabla#nabla;^2 &phis#phi;+ t & = &
-&rho#rho;&epsi#epsilon;

In the the Lorenz gauge,

(13.3) |

the potentials satisfy the following

(13.4) | |||

(13.5) |

where and are the charge density and current density distributions, respectively. For the time being we will stick with the Lorenz gauge, although the Coulomb gauge: = 0 is more convenient for certain problems. It is probably worth reminding y'all that the Lorenz 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)
&mu#mu;&epsi#epsilon;= 1v^2
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:
&mu#mu;_0&epsi#epsilon;_0 = 1c^2
so that:
&nabla#nabla;^2 &Phi#Phi;- 1c^2^2&Phi#Phi;t^2 & = & -
&rho#rho;&epsi#epsilon;_0

&nabla#nabla;^2 **A** - 1c^2 ^2**A**t^2 & = &
-&mu#mu;_0

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.