(a) A dielectric medium exhibits a linear response if the electric displacement $\mathbf{D}(\mathbf{x}, t)$ and magnetizing field $\mathbf{H}(\mathbf{x}, t)$ are related to the electric and magnetic fields, $\mathbf{E}(\mathbf{x}, t)$ and $\mathbf{B}(\mathbf{x}, t)$, as

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

where $\epsilon$ and $\mu$ are constants characterising the electric and magnetic polarisability of the material respectively. Write down the Maxwell equations obeyed by the fields $\mathbf{D}, \mathbf{H}, \mathbf{B}$ and $\mathbf{E}$ in this medium in the absence of free charges or currents.

(b) Two such media with constants $\epsilon_{-}$and $\epsilon_{+}$(but the same $\mu$ ) fill the regions $x<0$ and $x>0$ respectively in three-dimensions with Cartesian coordinates $(x, y, z)$.

(i) Starting from Maxwell's equations, derive the appropriate boundary conditions at $x=0$ for a time-independent electric field $\mathbf{E}(\mathbf{x})$.

(ii) Consider a candidate solution of Maxwell's equations describing the reflection and transmission of an incident electromagnetic wave of wave vector $\mathbf{k}_{I}$ and angular frequency $\omega_{I}$ off the interface at $x=0$. The electric field is given as,

$$\mathbf{E}(\mathbf{x}, t)=\left\{\begin{array}{cc} \sum_{X=I, R} \operatorname{Im}\left[\mathbf{E}_{X} \exp \left(i \mathbf{k}_{X} \cdot \mathbf{x}-i \omega_{X} t\right)\right], & x<0 \\ \operatorname{Im}\left[\mathbf{E}_{T} \exp \left(i \mathbf{k}_{T} \cdot \mathbf{x}-i \omega_{T} t\right)\right], & x>0 \end{array}\right.$$

where $\mathbf{E}_{I}, \mathbf{E}_{R}$ and $\mathbf{E}_{T}$ are constant real vectors and $\operatorname{Im}[z]$ denotes the imaginary part of a complex number $z$. Give conditions on the parameters $\mathbf{E}_{X}, \mathbf{k}_{X}, \omega_{X}$ for $X=I, R, T$, such that the above expression for the electric field $\mathbf{E}(\mathbf{x}, t)$ solves Maxwell's equations for all $x \neq 0$, together with an appropriate magnetic field $\mathbf{B}(\mathbf{x}, t)$ which you should determine.

(iii) We now parametrize the incident wave vector as $\mathbf{k}_{I}=k_{I}\left(\cos \left(\theta_{I}\right) \hat{\mathbf{i}}_{x}+\sin \left(\theta_{I}\right) \hat{\mathbf{i}}_{z}\right)$, where $\hat{\mathbf{i}}_{x}$ and $\hat{\mathbf{i}}_{z}$ are unit vectors in the $x$ - and $z$-directions respectively, and choose the incident polarisation vector to satisfy $\mathbf{E}_{I} \cdot \hat{\mathbf{i}}_{x}=0$. By imposing appropriate boundary conditions for $\mathbf{E}(\mathbf{x}, t)$ at $x=0$, which you may assume to be the same as those for the time-independent case considered above, determine the Cartesian components of the wavevector $\mathbf{k}_{T}$ as functions of $k_{I}, \theta_{I}, \epsilon_{+}$and $\epsilon_{-}$.

(iv) For $\epsilon_{+}<\epsilon_{-}$find a critical value $\theta_{I}^{\text {cr }}$ of the angle of incidence $\theta_{I}$ above which there is no real solution for the wavevector $\mathbf{k}_{T}$. Write down a solution for $\mathbf{E}(\mathbf{x}, t)$ when $\theta_{I}>\theta_{I}^{\mathrm{cr}}$ and comment on its form.