Suppose the region $z<0$ is occupied by an earthed ideal conductor.

(a) Derive the boundary conditions on the tangential electric field $\mathbf{E}$ that hold on the surface $z=0$.

(b) A point charge $q$, with mass $m$, is held above the conductor at $(0,0, d)$. Show that the boundary conditions on the electric field are satisfied if we remove the conductor and instead place a second charge $-q$ at $(0,0,-d)$.

(c) The original point charge is now released with zero initial velocity. Ignoring gravity, determine how long it will take for the charge to hit the plane.