References: Griffiths, David J. (2007), Introduction to Electrodynamics, 3rd Edition; Pearson Education – Problem 10.8.

In electrostatics and magnetostatics, charge distributions and currents were all constant in time. When they vary, we need to take into account the finite speed of light in calculating potentials and fields. If we want the fields at some point then, if the charge or current changes at some point a distance from , an observer at won’t know about the change until the signal from reaches him, which in vacuum takes a time . To take account of this, the potentials at position and time in a dynamic system are taken to be

where

and

That is, each potential is the sum over all locations where there is charge or current, and each location is sampled at the time in the past which is the time a light signal would have left to arrive at at time . These potentials are called *retarded potentials*, since they depend on the situation at various times in the past to get the fields at the present time.

Griffiths shows in his section 10.2.1 that these potentials (well anyway; the argument for is similar) satisfy the wave equations in the Lorentz gauge

We also need to show that the potentials satisfy the Lorentz gauge condition

Starting from 2 we need to find

Note that the is a derivative with respect to (the observer’s position) and not (the source positions and variable of integration), and that both and depend on both and . We begin by writing

We can also use the derivative with respect to :

We have (you can work this out by using 5 if you don’t believe me):

Therefore

Inserting this into 11 we get

Now we need to work out the two divergences and . To do this, we need to remember that , so it depends on only via but it depends on both explicitly through its first argument and implicitly through . Using the chain rule, we get for the contribution from

where the dot over the is a derivative with respect to , which is the same as a derivative with respect to since and doesn’t depend on time.

The other two coordinates give similar results and we get

For the other divergence, things are a bit trickier since depends explicitly on . Here we used the extended chain rule

Therefore

where we’ve used the specialized notation to indicate the divergence with respect to the *explicit* dependence in . From Maxwell’s fourth equation

where the fields and currents depend on , we can take the explicit divergence to get

The divergence of a curl is always zero, so we get

where we’ve used Maxwell’s first equation

Plugging this into 22 we get

since from 5

Putting 20 and 29 into 16 we get

Finally, from 2 we have

Using the divergence theorem, the second term can be converted to a surface integral at infinity where (presumably) the current is zero, so this term vanishes. Using 1 we then get

which is the Lorentz gauge condition, as required.