Inhomogenous electrodynamics wave equation

[Peeter]

I was playing around with some manipulations of maxwell's equations and seeing if I could work out the wave equation for light. I get:

$$(\nabla^2 -{\partial_{ct}}^2) \mathbf{B} = -\mu_0 \nabla \times \mathbf{J}$$

$$(\nabla^2 -{\partial_{ct}}^2) \mathbf{E} = \nabla \rho/\epsilon_0 + \mu_0 \partial_t \mathbf{J}$$

I had plenty of opportunities to mix up signs (and added back in some of the constants at the end) so I was wondering if anybody can confirm for me whether I got this (constants and signs on the RHS) correct.

 

[Andy Resnick]

I get about the same thing. There's a law of continuity for free charge:

$$\frac{\partial \rho}{\partial t}+ \nabla\bullet J = 0$$,

which may be relatable to our extra terms. I have a book at home, "electrodynamics of moving media", I'll see if there's anything fundamentally interesting on this.

 

[Peeter]

I tried a second way after posting this and got the same answer. However, both ways required that I use that condition, or else there'd be more terms.

A side note. Is this conservation really considered a separate law? It seemed to me that it's implied by Maxwell's equations. This can be seen for example by taking gradients of the bivector form of maxwell's equation:

$$\nabla^2 F = \nabla J$$

Since the LHS is a bivector it means that the scalar parts of the RHS is a bivector. Thus:

$$0 = \nabla \cdot J = \sum \partial_{\mu} J^{\mu} = \partial_t \rho + \sum \partial_i J^i$$

 

[Andy Resnick]

Ok, I have some more information:

First off, the (free) charge density and (free) current are zero in dielectrics at rest, so they go away automatically. Landau & Lifgarbagez, in "Electrodynamics of COntinuous Media" (volume 8), has a little bit about conductors, where J is proportional to E in the limit of static fields. But they don't try and derive a wave equation in a conductor, although I'm sure someone has.

When a dielectric moves, one way to account for the movement is to allow the current to appear, because dipoles are crossing boundaries. Penfield and Haus, "Electrodynamics of Moving Media" covers this extensively, but they don't ever try and derive a wave equation. I suspect the reason is that E and D and B and H are not longer simply related. There are many "Maxwell Equations" for moving media, here's an example (the Chu formulation):

$$\nabla\timesE=-\mu_{0}\frac{\partial H}{\partial t}-\frac{\partial}{\partial t}(\mu_{0}M)-\nabla\times(\mu_{0}M\times v)$$

$$\nabla\timesH=\epsilon_{0}\frac{\partial E}{\partial t}+\frac{\partial P}{\partial t}+\nabla\times(P\times v)+ J$$

$$\epsilon_{0}\nabla\bullet E = -\nabla\bullet P + \rho$$

$$\mu_{0}\nabla\bullet H = -\nabla\bullet (\mu_{0}M)$$

From this, I suppose one could try and develop a wave equation for E or B, but it's not clear what the result would look like.