Maxwell’s equations in medium without charges read

begin{align}

0&=nablacdot D,,\

0&=nablacdot B,,\

0&=nablatimes E + dot B,,\

0&=nablatimes H – dot D,.

end{align}

Here, $D = varepsilon E$ and $mu H = B$, with $varepsilon$ and $mu$ the relative permittivity and permeability of the medium.

One may derive wave equations for $E$ and $B$ of the form

begin{align}

0&=square E,,\

0&=square B,,

end{align}

if $varepsilon$ and $mu$ are constant. However, when they vary in space, derivatives of $mu$ and $varepsilon$ spoil this nice relationship. Even so, if the medium is slowly changing as a function of space, we can usually neglect derivatives of $mu$ and $varepsilon$, leading to the interpretation of $1/sqrt{muvarepsilon}$ as the effective speed of light. My question is whether there are inhomogeneous media for which this interpretation is exact, i.e. for which changing $mu$ and $varepsilon$ may be exactly interpreted as changing $c$ in the wave equation.

I think you can have a position-dependent speed of light $$ c_{rm local}= frac 1{sqrt{muepsilon}} $$ without much else happening as long as you keep the medium's wave impedence $$ Z_{rm local}= sqrt{frac{mu}{epsilon}} $$ constant. This is suggested by the Riemann-Siberstein rewriting of Maxwell's equations in which the ${boldsymbol Psi}^{pm} ={bf E}pm iZ_{rm local}{bf H}$ parts of the E&M field decouple when $Z$ is constant. The "curl" Maxwell become $$ ifrac{partial}{partial t} Psi_i^{pm}= c_{rm local}epsilon_{ijk} frac{partial}{partial x_j} Psi^{pm}_k, $$ and the $nabla cdot B=nablacdot D=0$ equations follow from these when the frequency is non-zero. Of course it is rather difficult to keep $Z$ constant as many transparant materials have large $epsilon$ but not many dielectrics have large $mu$'s

Correct answer by mike stone on January 1, 2021

The relation $c_{medium}=1/epsilonmu$ is not exact if the medium is not homogeneous. The question is how exact do you want it to be?

You apparently knew that this would be the answer.

Note that at atomic scale no medium is homogenous.

Answered by my2cts on January 1, 2021

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