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) & = &
&phis#phi;+ ()t & = & - &rho#rho;
The second (AL) is a bit more work. We start by writing it in terms 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)