Introduction to Electrodynamics · Problem 3.37

Potentials — David J. Griffiths

Physics · laplace-equation-average-value

Problem

In Section 3.1.4, I proved that the electrostatic potential at any point

P

in a charge-free region is equal to its average value over any spherical surface (radius

R

) centered at

P

. Here's an alternative argument that does not rely on Coulomb's law, only on Laplace's equation. We might as well set the origin at

P

. Let

V_{\text{ave}}(R)

be the average; first show that

\frac{dV_{\text{ave}}}{dR} = \frac{1}{4\pi R^2}\oint \nabla V \cdot d\mathbf{a}

(note that the

R^2

d\mathbf{a}

cancels the

1/R^2

out front, so the only dependence on

R

is in

V

itself). Now use the divergence theorem, and conclude that if

V

satisfies Laplace's equation, then

V_{\text{ave}}(R) = V_{\text{ave}}(0) = V(P)

, for all

R