(a) Define the polarisation of a dielectric material and explain what is meant by the term bound charge.

Consider a sample of material with spatially dependent polarisation $\mathbf{P}(\mathbf{x})$ occupying a region $V$ with surface $S$. Show that, in the absence of free charge, the resulting scalar potential $\phi(\mathbf{x})$ can be ascribed to bulk and surface densities of bound charge.

Consider a sphere of radius $R$ consisting of a dielectric material with permittivity $\epsilon$ surrounded by a region of vacuum. A point-like electric charge $q$ is placed at the centre of the sphere. Determine the density of bound charge on the surface of the sphere.

(b) Define the magnetization of a material and explain what is meant by the term bound current.

Consider a sample of material with spatially-dependent magnetization $\mathbf{M}(\mathbf{x})$ occupying a region $V$ with surface $S$. Show that, in the absence of free currents, the resulting vector potential $\mathbf{A}(\mathbf{x})$ can be ascribed to bulk and surface densities of bound current.

Consider an infinite cylinder of radius $r$ consisting of a material with permeability $\mu$ surrounded by a region of vacuum. A thin wire carrying current $I$ is placed along the axis of the cylinder. Determine the direction and magnitude of the resulting bound current density on the surface of the cylinder. What is the magnetization $\mathbf{M}(\mathbf{x})$ on the surface of the cylinder?