Potentials

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

Now consider FL. If we substitute in our expression for $\vB$ : ×+ ×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 $\vE$ into GLE, and the equation for $\vB$ 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 $\vB$ instead of $\vH$ by multiplying out the $\mo$ : & = & + 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)