Consider a medium in which the electric displacement $\mathbf{D}(t, \mathbf{x})$ and magnetising field $\mathbf{H}(t, \mathbf{x})$ are linearly related to the electric and magnetic fields respectively with corresponding polarisation constants $\varepsilon$ and $\mu$;

$$\mathbf{D}=\varepsilon \mathbf{E}, \quad \mathbf{B}=\mu \mathbf{H} .$$

Write down Maxwell's equations for $\mathbf{E}, \mathbf{B}, \mathbf{D}$ and $\mathbf{H}$ in the absence of free charges and currents.

Consider EM waves of the form,

$$\begin{aligned} &\mathbf{E}(t, \mathbf{x})=\mathbf{E}_{0} \sin (\mathbf{k} \cdot \mathbf{x}-\omega t) \\ &\mathbf{B}(t, \mathbf{x})=\mathbf{B}_{0} \sin (\mathbf{k} \cdot \mathbf{x}-\omega t) \end{aligned}$$

Find conditions on the electric and magnetic polarisation vectors $\mathbf{E}_{0}$ and $\mathbf{B}_{0}$, wave-vector $\mathbf{k}$ and angular frequency $\omega$ such that these fields satisfy Maxwell's equations for the medium described above. At what speed do the waves propagate?

Consider two media, filling the regions $x<0$ and $x>0$ in three dimensional space, and having two different values $\varepsilon_{-}$and $\varepsilon_{+}$of the electric polarisation constant. Suppose an electromagnetic wave is incident from the region $x<0$ resulting in a transmitted wave in the region $x>0$ and also a reflected wave for $x<0$. The angles of incidence, reflection and transmission are denoted $\theta_{I}, \theta_{R}$ and $\theta_{T}$ respectively. By constructing a corresponding solution of Maxwell's equations, derive the law of reflection $\theta_{I}=\theta_{R}$ and Snell's law of refraction, $n_{-} \sin \theta_{I}=n_{+} \sin \theta_{T}$ where $n_{\pm}=c \sqrt{\varepsilon_{\pm} \mu}$ are the indices of refraction of the two media.

Consider the special case in which the electric polarisation vectors $\mathbf{E}_{I}, \mathbf{E}_{R}$ and $\mathbf{E}_{T}$ of the incident, reflected and transmitted waves are all normal to the plane of incidence (i.e. the plane containing the corresponding wave-vectors). By imposing appropriate boundary conditions for $\mathbf{E}$ and $\mathbf{H}$ at $x=0$, show that,

$$\frac{\left|\mathbf{E}_{R}\right|}{\left|\mathbf{E}_{T}\right|}=\frac{1}{2}\left(1-\frac{\tan \theta_{R}}{\tan \theta_{T}}\right)$$