Classical Electrodynamics · Problem 1.12

Boundary-Value Problems in Electrostatics: I — John David Jackson

Physics — E&M · greens-reciprocation-theorem

Problem

Prove Green's reciprocation theorem: If Φ\Phi is the potential due to a volume-charge density ρ\rho within a volume VV and a surface-charge density σ\sigma on the conducting surface SS bounding the volume VV, while Φ′\Phi' is the potential due to another charge distribution ρ′\rho' and σ′\sigma', then ∫VρΦ′ d3x+∫SσΦ′ da=∫Vρ′Φ d3x+∫Sσ′Φ da\int_V \rho\Phi'\, d^3x + \int_S \sigma\Phi'\, da = \int_V \rho'\Phi\, d^3x + \int_S \sigma'\Phi\, da

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