∇²V = −ρ/ε A single operator applied to the electrostatic potential recovers, up to a constant, the charge density that produces it. Where the density vanishes the same equation collapses to Laplace’s and the potential becomes harmonic: its value at any interior point equals its average over every surrounding sphere. Poisson wrote the inhomogeneous form in 1813 after observing that Laplace’s equation holds only outside matter. With a change of constants the identical relation governs Newtonian gravity and steady heat flow.

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