The electromagnetic fields of a moving anapole

We examine the electromagnetic potential and field of a small toroidal solenoid moving non-relativistically at constant velocity. In the rest frame of the solenoid, the electric field vanishes everywhere, and the magnetic field is confined to the interior of the solenoid. However, the vector potential of the solenoid is non-trivial everywhere. Moving relative to this rest frame, we discover that the scalar potential of the toroidal solenoid is that of an electric quadrupole. Despite this non-trivial scalar potential, the electric field outside the moving torus vanishes by virtue of the time dependence of the vector potential. We briefly discuss the relevance of these features within the context of particle physics. Spin-12 fermions can possess a non-zero anapole moment, which is classically likened to a toroidal solenoid. As with the classical analogue, the particle's electromagnetic fields that are sourced by the anapole moment vanish outside of a contact interaction; however, the particle's electromagnetic four-potential is everywhere non-trivial. Because of this, scattering from the anapole moment would be an example of the Aharonov–Bohm effect in which the scattering is caused by the electromagnetic potential in a region where the electromagnetic fields vanish. From the rest frame of the anapole moment, scattering of a charged particle would be due to the anapole moment's vector potential, but from the rest frame of the charged particle, scattering would be caused by the moving anapole's scalar potential.