Given a gyrotropic material with

$$\vec D = \epsilon \vec E + \lambda / c \vec H \\ \vec B = v / c \vec E + \mu \vec H$$

where $\epsilon, \lambda, v, c, \mu$ are constants (no relation is given between them), determine the wave equation that all fields must obey using phasors (only in frequency domain).

Taking the curl of $\nabla \times \vec E$ and $\nabla \times \vec H$ respectively and by using substitutions I managed to extract the following relations: \begin{align} -\nabla^2 \hat E &= [\mu \epsilon - (v/c)^2] \omega^2 \hat E + \mu / c [\lambda - v] \omega^2 \hat H \\ \nabla^2 \hat H &= \epsilon / c [\lambda - v] \omega^2 \hat E + [(\lambda/c)^2 - \mu \epsilon] \omega^2 \hat H \end{align}

which seem to be separable if $\mu \epsilon = 1 / c^2$ and $\lambda = v$ and therefore they reduce to $$\hat \square \hat E = \hat \square \hat H = \vec 0$$

where $\hat \square := \nabla^2 + \frac {1 - v^2} {c^2} \omega^2$. Am I missing any solutions using this practice?


The initial system is overdetermined: you could simply write $$\vec D = \alpha \vec E + \beta \vec H \\ \vec B = \gamma \vec E + \delta \vec H,$$ and it describes exactly the same class of fields, but with only four constants instead of the five in your description. As such, you're perfectly free to define $\epsilon=\alpha$, $\mu=\delta$, $c=1/\sqrt{\mu\epsilon}=1/\sqrt{\alpha\delta}$, $\lambda=\beta c=\beta/\sqrt{\alpha\delta}$ and $v=\gamma c=\gamma/\sqrt{\alpha\delta}$, or in other words, you can impose the condition $\mu\epsilon=1/c^2$ for free.

However, it's much less clear to me that you can impose a specific relation between the two mixed coefficients $\lambda$ and $v$, and in general if the material is not reciprocal (cf. this Wikipedia page) then you won't be able to relate the two. Either way, your set-piece explicitly disavows the ability to impose constraints on the five constants you're given, so setting $\lambda=v$ is at odds with the problem as set.

Thus, the best you can do is to take the wave equations as you've derived them, \begin{align} -\nabla^2 \hat E &= [\mu \epsilon - (v/c)^2] \omega^2 \hat E + \mu / c [\lambda - v] \omega^2 \hat H \\ \nabla^2 \hat H &= \epsilon / c [\lambda - v] \omega^2 \hat E + [(\lambda/c)^2 - \mu \epsilon] \omega^2 \hat H, \end{align} and then assume that both fields are in Helmholtz eigenstates of the laplacian, giving you \begin{align} k^2 \hat E &= [\mu \epsilon - (v/c)^2] \omega^2 \hat E + \mu / c [\lambda - v] \omega^2 \hat H \\ -k^2 \hat H &= \epsilon / c [\lambda - v] \omega^2 \hat E + [(\lambda/c)^2 - \mu \epsilon] \omega^2 \hat H \end{align} or in other words $$ \begin{pmatrix} k^2-\left(\mu\epsilon-\frac{v^2}{c^2}\right)\omega^2 & \frac{\mu}{c}(\lambda -v)\omega^2 \\ \frac{\epsilon}{c}(\lambda -v)\omega^2 & k^2-\left(\mu\epsilon-\frac{\lambda^2}{c^2}\right)\omega^2 \end{pmatrix} \begin{pmatrix} \vec E\\\vec H \end{pmatrix} =0, $$ which then requires that the determinant vanish, i.e. $$ \left(k^2-\left(\mu\epsilon-\frac{v^2}{c^2}\right)\omega^2 \right) \left(k^2-\left(\mu\epsilon-\frac{\lambda^2}{c^2}\right)\omega^2 \right) - \frac{\mu}{c}(\lambda -v)\omega^2 \times\frac{\epsilon}{c}(\lambda -v)\omega^2 =0 $$ which fails to simplify in ways that make suspect you've made a sign error somewhere, so I'll leave it to you to double-check your workings and pull out the dispersion relation from there.

  • $\begingroup$ Thank you very much for your detailed question. I extracted the dispersion relation in terms of the refraction index: $\eta^2 = 1 / (1 - v^2)$ $\endgroup$ – bolzano Jun 30 '17 at 13:28
  • $\begingroup$ That's unlikely to be general enough to capture the full set of possible solutions, but whatever works for you. $\endgroup$ – Emilio Pisanty Jun 30 '17 at 13:29
  • $\begingroup$ Yes that seems true. This problem was from an exam in Electromagnetic Fields and it was asked to determine the wave equations assuming only time dependent phasors (the addition of the term $\exp(j \vec k \cdot \vec r)$ was asked next) but the obvious requirement was the above. $\endgroup$ – bolzano Jun 30 '17 at 13:32

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