### 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:

Here is the electric field, the magnetic field, the charge density, the current density, the constant speed of light, and is a constant called the permittivity of space. The charge and current densities are related by the continuity equation

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

 (A.235)

Similarly, for the magnetic field take the curl of (4) and use (2) and (3) to get
 (A.236)

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
since the curl of a gradient is always zero, and
because the two terms drop out against each other.

The fact that and produce the same physical fields is the famous gauge property of the electromagnetic field.

Now you can select so that

That is known as the “Lorenz condition.” A corresponding gauge function is a “Lorenz gauge.”

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),

 (A.237)

From (4), after using (8),
 (A.238)

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.