Chapter 3.1, Problem 3.1P

Find the average potential over a spherical surface of radius R due toa point charge q located inside (same as above, in other words, only with $z<R$ ).(In this case, of course. Laplace’s equation does not hold within the sphere.) Showthat, in general,

   $V_{\text{ave}}=V_{\text{center}}+\frac{Q_{\text{enc}}}{4\pi {\in }_{0}R},$

where V_centeris the potential at the center due to all the external charges, and Q_encisthe total enclosed charge.