(a) Define the electric displacement $\mathbf{D}(\mathbf{x}, t)$ for a medium which exhibits a linear response with polarisation constant $\epsilon$ to an applied electric field $\mathbf{E}(\mathbf{x}, t)$ with polarisation constant $\epsilon$. Write down the effective Maxwell equation obeyed by $\mathbf{D}(\mathbf{x})$ in the timeindependent case and in the absence of any additional mobile charges in the medium. Describe appropriate boundary conditions for the electric field at an interface between two regions with differing values of the polarisation constant. [You should discuss separately the components of the field normal to and tangential to the interface.]

(b) Consider a sphere of radius $a$, centred at the origin, composed of dielectric material with polarisation constant $\epsilon$ placed in a vacuum and subjected to a constant, asymptotically homogeneous, electric field, $\mathbf{E}(\mathbf{x}, t)=\mathbf{E}(\mathbf{x})$ with $\mathbf{E}(\mathbf{x}) \rightarrow \mathbf{E}_{0}$ as $|\mathbf{x}| \rightarrow \infty$. Using the ansatz

$$\mathbf{E}(\mathbf{x})= \begin{cases}\alpha \mathbf{E}_{0}, & |\mathbf{x}|<a \\ \mathbf{E}_{0}+\left(\beta\left(\widehat{\mathbf{x}} \cdot \mathbf{E}_{0}\right) \widehat{\mathbf{x}}+\delta \mathbf{E}_{0}\right) /|\mathbf{x}|^{3}, & |\mathbf{x}|>a\end{cases}$$

with constants $\alpha, \beta$ and $\delta$ to be determined, find a solution to Maxwell's equations with appropriate boundary conditions at $|\mathbf{x}|=a$.

(c) By comparing your solution with the long-range electric field due to a dipole consisting of electric charges $\pm q$ located at displacements $\pm \mathbf{d} / 2$ find the induced electric dipole moment of the dielectric sphere.