(a) Use Green’s second identity, ∫_V d³r (f∇²g − g∇²f ) = ∫_S dS · (f∇_g − _g∇f), to prove that the potential ϕ(0) at the center of a charge-free spherical volume V is equal to the average of ϕ(r) over the surface S of the sphere. We proved this theorem in the text using Green’s reciprocity relation.

(b) Use the result of part (a) to provide an alternative to the derivation of Earnshaw’s theorem given in the text.