Wave equation in inhomogeneous medium

[palpa]

Homework Statement

a) Assuming the presence of sources (J flux density) and (p charge density) , write out Maxwell’s equations in the time domain in terms of and only for a lossless, but inhomogenous medium in which
ε = ε(r) , μ = μ(r).

b) Derive the vector differential equation (wave equation) satisfied by E(r,t) in a source-free, lossless, inhomogenous medium.

(There are lines on the "r"s indicating that they are position vectors)

Homework Equations

maxwell's equations and the equations that relate D&E and B&H (I am not sure about which forms should be used)

The Attempt at a Solution

I am blowing my mind over this but couldn't see what is being meant by inhomogeneous medium. Obviously I am not asked for the inhomogeneous wave equation (it is not in the curriculum), so I thought this was about anisotropic medium where ε&μ are different for different positions, but when I read about it, I've encountered lots of stuff I haven't even heard about (like tensors).

Please give me a starting point. D=εE , but if ε is not constant, it is not a scalar. If it's not a scalar, how is D=εE true? Or is ε a tensor and since it is a matrix I should treat it like a scalar? Then what is the difference of the answer from constant ε&μ wave equation?

Please help I am desperate.

 

[truth is life]

I am not really the best person to look to for this (I haven't yet done upper-level electromagnetism, just the basic calculus-based stuff), but it sounds like you have epsilon and mu which are controlled by an algebraic function dependent on radius from your origin and you are looking at all of space, yes? So, why not try changing to a different coordinate system (like spherical coordinates), if you haven't done that already?

 

[palpa]

thanks for the response. I am pretty sure that the r there is not radius, it is the vector
r=xi+yj+zk. But while typing that I've notived that using spherical coordinates may simplify it significantly. I will try solving the question again when I have time, meanwhile I would appreciate any other ideas.

 

[jdwood983]

Maxwell's equations in time-domain uses quabla (d'Alembert operator):

$$\square^2\mathbf{E}=\mathrm{some\,fcn\,of\,\mu\,or\,\varepsilon}$$

Use your knowledge of what $$\mathbf{E}$$ in terms of the scalar and vector potentials to finish out Part (a).

 

[gabbagabbahey]

Are you familiar with the derivation of the wave equation in a vacuum? If so, just apply the same general method...