The region $z<0$ is occupied by an ideal earthed conductor and a point charge $q$ with mass $m$ is held above it at $(0,0, d)$.

(i) What are the boundary conditions satisfied by the electric field $\mathbf{E}$ on the surface of the conductor?

(ii) Consider now a system without the conductor mentioned above. A point charge $q$ with mass $m$ is held at $(0,0, d)$, and one of charge $-q$ is held at $(0,0,-d)$. Show that the boundary condition on $\mathbf{E}$ at $z=0$ is identical to the answer to (i). Explain why this represents the electric field due to the charge at $(0,0, d)$ under the influence of the conducting boundary.

(iii) The original point charge in (i) is released with zero initial velocity. Find the time taken for the point charge to reach the plane (ignoring gravity).

[You may assume that the force on the point charge is equal to $m d^{2} \mathbf{x} / d t^{2}$, where $\mathbf{x}$ is the position vector of the charge, and $t$ is time.]