= Electrostatic reciprocity identity
{title2=$\int_V(\phi_1\rho_2-\phi_2\rho_1)\,dV=\varepsilon_0\int_{\partial V}(\phi_2\nabla\phi_1-\phi_1\nabla\phi_2)\cdot d\mathbf S$}
For two <electrostatic potentials> satisfying the <Poisson equation> $\Delta\phi_i=-\rho_i/\varepsilon_0$, <Green's second identity> yields the displayed reciprocity formula. The <product rule for divergence> applied to $\phi_1\nabla\phi_2-\phi_2\nabla\phi_1$ cancels the cross-gradient terms. The normal points outward from the integration region; for an exterior region this means inward toward the excluded object on its inner boundary. Boundary terms may be removed only under suitable decay or boundary conditions.
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