A.36 Maxwell’s wave equations

This note derives the wave equations satisfied by electromagnetic fields. The derivation will use standard formulae of vector analysis, as found in, for example, [40, 20.35-45].

The starting point is Maxwell’s equations for the electromagnetic field in vacuum:

To get a wave equation for the electric field, take the curl,
, of (3) and apply the standard vector identity
(D.1), (1) and (4) to get

These are uncoupled inhomogeneous wave equations for the components of
and , for given charge and current densities.
According to the theory of partial differential equations, these
equations imply that effects propagate no faster than the speed of
light. You can also see the same thing pretty clearly from the fact
that the homogeneous wave equation has solutions like

which are waves that travel with speed in the -direction.

The wave equations for the potentials and are next. First note from (2) that the divergence of is zero. Then vector calculus says that it can be written as the curl of some vector. Call that vector .

Next define

Plug this into (3) to show that the curl of is zero. Then vector calculus says that it can be written as minus the gradient of a scalar. Call this scalar . Plug that into the expression above to get

Next, note that if you define modified versions
and of and by setting

where is any arbitrary function of , , , and , then still

The fact that and produce the
same physical fields is the famous gauge property

of
the electromagnetic field.

Now you can select so that

To find the gauge function that produces this condition, plug
the definitions for and in terms of
and into the left hand side of the Lorentz condition.
That produces, after a change of sign,

That is a second order inhomogeneous wave equation for .

Now plug the expressions (6) and (7) for and in
terms of and into the Maxwell’s equations.
Equations (2) and (3) are satisfied automatically. From (2),
after using (8),

You can still select the two initial conditions for . The smart thing to do is select them so that and its time derivative are zero at time zero. In that case, if there is no charge density, will stay zero for all time. That is because its wave equation is then homogeneous. The Lorenz condition will then ensure that is zero too.

Instead of the Lorenz condition, you could select to make
zero. That is called the “Coulomb gauge” or “transverse gauge” or “transverse gauge.” It requires that satisfies the
Poisson equation

Then the governing equations become

Note that now satisfies a purely spatial Poisson equation.