Carter separable electromagnetic fields

The purely electromagnetic analogue in flat space of Kerr’s metric in general relativity is only rarely considered. Here we carry out in flat space a programme similar to Carter’s investigation of metrics in general relativity in which the motion of a charged particle is separable. We concentrate on the separability of the motion (be it classical, relativistic or quantum) of a charged particle in electromagnetic fields that lie in planes through an axis of symmetry. In cylindrical polar coordinates ( t , R , φ , z ) the four‐vector potential takes the formis the unit toroidal vector. The forms of the functions Φ( R , z ) and A ( R , z ) are sought that allow separable motion. This occurs for relativistic motion only when A R ,Φ and A 2 −Φ 2 are all of the separable form ζ ( λ )− η ( μ )]/( λ − μ ), where ζ and η are arbitrary functions, and λ and μ are spheroidal coordinates or degenerations thereof. The special forms of A and Φ that allow this are deduced. They include the Kerr metric analogue, with E +i B =−∇{ q [( r −i a )·( r −i a )] −1/2 }. Rather more general electromagnetic fields allow separation when the motion is non‐relativistic. The investigation is extended to fields that lie in parallel planes. Connections to Larmor’s theorem are remarked upon.