Conditions for a pure toroidal dipole source

Abstract

Recently, the physical significance of dynamic toroidal multipoles in the context of electrodynamics has been put under discussion. Indeed, the latter can be shown to arise simply from a Taylor series of the exact source (Cartesian) multipole moments. The split into elementary and toroidal parts was demonstrated to lead to an unphysical result were forbidden components of the momentum transform of the current could radiate into free space. In this contribution, we elaborate the conditions that a current distribution must necessarily satisfy to be considered a ‘pure’ toroidal dipole source. We demonstrate for the first time that symmetry prevents such current distribution to radiate as an elementary electric dipole moment, without leading to an unphysical result. Thus, while both elementary electric dipole and toroidal dipoles are indistinguishable outside the source, they display topologically distinct characteristics within the smallest spherical surface enclosing the source itself and have different physical origin. Based on our results, a pure ‘toroidal’ source can be designed. We believe the outcome of our investigations will help clarify further the formal meaning of the toroidal multipoles.