In Section 3.1.4, I proved that the electrostatic potential at any point P in a charge-f...

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_ave(R) be the average; first show that

(note that the R² in da cancels the 1/R² 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_ave(R) = V_ave(0) = V(P), for all R.¹⁸