Classical Electrodynamics · Problem 2.28

Boundary-Value Problems in Electrostatics: II — John David Jackson

Physics — E&M · mean-value-theorem-polyhedron

Problem

A closed volume is bounded by conducting surfaces that are the

n

sides of a regular polyhedron (

n = 4, 6, 8, 12, 20

). The

n

surfaces are at different potentials

V_i

i = 1, 2, \ldots, n

. Prove in the simplest way you can that the potential at the center of the polyhedron is the average of the potential on the

n

sides. This problem bears on Problem 2.23b, and has an interesting similarity to the result of Problem 1.10.