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Use Green’ reciprocity theorem to prove “mean value theorem”: the average of \(\phi(\vec{r})\) over a spherical surface \(S\) that encloses a charge-free volume is equal to the potential at the center of the sphere:\[ \frac{1}{4\pi R^2} \iint \phi(\vec{r}) dS = \phi(0) \]
Hint:
Apply reciprocity theorem considering two systems
(i) an empty sphere with charge spread uniformlyover its surface and
(ii) the given potential \(\phi(\vec{r})\) and its source \(\rho(\vec{r})\).
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