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We investigate the coupling of the electromagnetic sources (charge andcurrent densities) to the scalar and vector potentials in classicalelectrodynamics, using Green function techniques. As is well known, the scalarpotential shows an action-at-a-distance behavior in Coulomb gauge. Theconundrum generated by the instantaneous interaction has intrigued physicistsfor a long time. Starting from the differential equations that couple thesources to the potentials, we here show in a concise derivation, using theretarded Green function, how the instantaneous interaction cancels in thecalculation of the electric field. The time derivative of a specific additionalterm in the vector potential, present only in Coulomb gauge, yields asupplementary contribution to the electric field which cancels the gradient ofthe instantaneous Coulomb gauge scalar potential, as required by gaugeinvariance. This completely eliminates the contribution of the instantaneousinteraction from the electric field. It turns out that a careful formulation ofthe retarded Green function, inspired by field theory, is required in order tocorrectly treat boundary terms in partial integrations. Finally, compactintegral representations are derived for the Lienard-Wiechert potentials(scalar and vector) in Coulomb gauge which manifestly contain two compensatingaction-at-a-distance terms.

Sources, Potentials and Fields in Lorenz and Coulomb Gauge: Cancellation of Instantaneous Interactions for Moving Point Charges