The Magnetic Interaction Energy between an Infinite Solenoid and a Passing Point Charge

Abstract

The standard expression for the magnetic interaction energy used in the study of the Aharonov-Bohm effect is investigated. We calculate the magnetic interaction energy between a point charge and an infinite solenoid from first principles. Two alternative expressions are used: the scalar products of the currents with the vector potentials and the scalar product of the magnetic fields. The alternatives are seen to agree. The latter approach also involves taking into account surface integrals at infinity, which are shown to be zero. Our model problem indicates no classical Aharonov-Bohm effect, but we also discuss the normally neglected fact of energy non-conservation. The problem is treated from the point of view of Lagrangian and Hamiltonian mechanics.