Whereas the above reciprocity theorems were for oscillating fields, Green's reciprocity is an analogous theorem for electrostatics with a fixed distribution of electric charge (Panofsky and Phillips, 1962).
In particular, let denote the electric potential resulting from a total charge density . The electric potential satisfies Poisson's equation, where is the vacuum permittivity. Similarly, let denote the electric potential resulting from a total charge density, satisfying . In both cases, we assume that the charge distributions are localized, so that the potentials can be chosen to go to zero at infinity. Then, Green's reciprocity theorem states that, for integrals over all space:
This theorem is easily proven from Green's second identity. Equivalently, it is the statement that, i.e. that is a Hermitian operator (as follows by integrating by parts twice).
Read more about this topic: Reciprocity (electromagnetism)
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