Past exam of the mathematics course of the University of Cambridge / 2011 / ia / Paper 3 / 11C / b / Solution

Combining the electric-field equations gives the Poisson equation ${\nabla }^2{\varphi }_i=-{\rho }_i/{\epsilon }_0$. Apply the product rule for divergence toThe two cross terms in its divergence cancel, leavingThis electrostatic reciprocity identity follows from the divergence theorem, and is exactly Green's second identity for these potentials. Integrating and rearranging givesThe boundary orientation is outward from the chosen region. For smooth charge densities this follows directly from classical differentiation. For isolated point charges, one may excise small spheres and take their radii to zero, provided the other potential is regular at each charge.