Problem 4 – Hint 4

Suppose you have found the dipole moment p induced on the sphere, be it empty (with isolating filling) or metallic. The presence of this dipole will cause a certain voltage change between the capacitor plates. Notice that this total voltage change does not depend on the shape of the dipole (spherical or ideal point-dipole), because in both cases, the electric field due to the dipole is exactly the same at the position of the plates. Neither will the voltage change depend on displacements of the dipole parallel to the plates. Furthermore, since the Maxwell equations and boundary conditions are linear, there is a superposition: the dipole can be divided fictitiously into two pieces without changing the induced voltage. This division can be repeated ad infinitum so that together with horizontal displacements of the dipole pieces we can end up with a dipole moment being homogeneously spread over the horizontal plane. In that case, the induced voltage can be calculated relatively easily.