The figure below shows a hollow metal sphere with a positive point charge $Q$ sitting outside it. What is the electric field at the center of sphere ? The answer is zero (look at here at the beginning of page 4), but I do not understand why?
Suppose electric field inside the sphere is nonzero. Then since there is no charge inside the sphere and since electric lines of force do not form closed loops so we should be able to find two points A and B on the surface of sphere such that a line of force starts from A and ends at B, thus causing a potential difference between these points. But since the sphere is made of metal (which are usually good conductors) so there will be a flow of current between these two points until the potential difference between them vanishes. So in equilibrium i.e. when no current is flowing, electric field inside sphere should be zero.
Since (1) the metallic sphere is an equipotential surface and (2) the potential inside the sphere must satisfy Laplace's equation, it follows, by the uniqueness theorem, that the potential inside the sphere is constant and thus, that the electric field inside the sphere is zero.