On the role of longitudinal currents in radiating systems of charges

The time derivative of a charge density is linked to a current density by the continuity equation. However, it features only the longitudinal part of a current density, which is known to produce no radiation. This fact usually remains unnoticed and may appear puzzling at first, suggesting that the temporal variation of a charge density should be also irrelevant to radiation. We alleviate the apparent contradiction by showing that the effective longitudinal currents are not spatially confined, even when the time-dependent radiating charge density that generates them is. This enforces the co-existence of the complementary, i.e. transverse, part of the current, which, in turn, gives rise to radiation. We illustrate the necessarily delocalized nature and relevance of longitudinal currents to the emission of electromagnetic waves by a dynamic electric dipole, discussing the practical implications of that for radation in partially conducting condensed matter. More generally, we show how the connection between the longitudinal and transverse currents shapes the structure of the conventional multipole expansion and fuels the ongoing confusion surrounding the charge and toroidal multipoles.