An isolated conducting sphere of radius carries a fixed total charge
PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Electricity, Magnetism & Electromagnetism · Conductors, capacitance and boundary-value electrostatics
The Boundary-Value Problem, Ch Uniqueness, images and the Green function of a region
Problem
An isolated conducting sphere of radius a=3.0cm carries a fixed total charge Q=5.0nC. A point charge q=2.0nC is held fixed at a distance b=9.0cm from the sphere's centre, and the whole arrangement is far from any other conductor, so V→0 at infinity. Someone argues: "The second uniqueness theorem only tells us that the field is fixed by the given charges q and Q; since no piece of the boundary carries a stated potential, the sphere's own potential is left undetermined."
((a)) Explain what is wrong with this argument.
((b)) Find the sphere's potential, using the image construction (an off-centre image plus a charge at the centre that restores the total charge Q).
((c)) The images used in part (b) are not real charges. State, with a one-sentence reason, whether the electrostatic interaction energy of q with the sphere may be found by treating q, the off-centre image and the centre charge as three real point charges and summing 4πε01∑i<jqiqj/rij.