(a) Show that, for $|\mathbf{x}| \gg|\mathbf{y}|$,

$$\frac{1}{|\mathbf{x}-\mathbf{y}|}=\frac{1}{|\mathbf{x}|}\left[1+\frac{\mathbf{x} \cdot \mathbf{y}}{|\mathbf{x}|^{2}}+\frac{3(\mathbf{x} \cdot \mathbf{y})^{2}-|\mathbf{x}|^{2}|\mathbf{y}|^{2}}{2|\mathbf{x}|^{4}}+O\left(\frac{|\mathbf{y}|^{3}}{|\mathbf{x}|^{3}}\right)\right]$$

(b) A particle with electric charge $q>0$ has position vector $(a, 0,0)$, where $a>0$. An earthed conductor (held at zero potential) occupies the plane $x=0$. Explain why the boundary conditions can be satisfied by introducing a fictitious 'image' particle of appropriate charge and position. Hence determine the electrostatic potential and the electric field in the region $x>0$. Find the leading-order approximation to the potential for $|\mathbf{x}| \gg a$ and compare with that of an electric dipole. Directly calculate the total flux of the electric field through the plane $x=0$ and comment on the result. Find the induced charge distribution on the surface of the conductor, and the total induced surface charge. Sketch the electric field lines in the plane $z=0$.

(c) Now consider instead a particle with charge $q$ at position $(a, b, 0)$, where $a>0$ and $b>0$, with earthed conductors occupying the planes $x=0$ and $y=0$. Find the leading-order approximation to the potential in the region $x, y>0$ for $|\mathbf{x}| \gg a, b$ and state what type of multipole potential this is.