Introduction to Electrodynamics · Problem 3.1

Potentials — David J. Griffiths

Physics · average-potential-spherical-surface

Problem

Find the average potential over a spherical surface of radius

R

due to a 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.) Show that, in general,

V_{\text{ave}} = V_{\text{center}} + \frac{Q_{\text{enc}}}{4\pi\epsilon_0 R},

where

V_{\text{center}}

is the potential at the center due to all the external charges, and

Q_{\text{enc}}

is the total enclosed charge.