We begin our discussion of potentials by considering the two *homogeneous* equations. For example, if we wish to associate a
potential with
such that
is the result of differentiating
the potential, we observe that we can satisfy GLM *by construction*
if we suppose a *vector potential*
such that:
= ×
In that case:
·= ·(×) = 0
as an identity.

Now consider FL. If we substitute in our expression for
:
×+ ×t & = & 0

×(+ t) & = & 0
We can see that if we define:
+ t = -&phis#phi;
then
×(+ t) = -×&phis#phi;= 0
is also an identity. This leads to:
= -&phis#phi;- t

Our next chore is to transform the *inhomogeneous* MEs into
equations of motion for these potentials - motion because MEs (and
indeed the potentials themselves) are now potentially *dynamical*
equations and not just static. We do this by substituting in the
equation for
into GLE, and the equation for
into AL. We
will work (for the moment) in *free space* and hence will use the
*vacuum* values for the permittivity and permeability.

The first (GLE) yields:
(-&phis#phi;- t) & = &
&rho#rho;

&phis#phi;+ ()t & = & -
&rho#rho;

The second (AL) is a bit more work. We start by writing it in terms of
instead of
by multiplying out the
:
& = & + t

() & = & + t(-&phis#phi;-
t)

-+ () & = & - 1c^2
&phis#phi;t - 1c^2t

+ - 1c^2t & = & -
+ ()+ 1c^2&phis#phi;t

+ - 1c^2t & = & -
+ (+ 1c^2&phis#phi;t)