Constrained motion of a charged particle in a magnetic field

Abstract

A charge that moves along a closed path is normally ascribed a magnetic dipole moment. The interaction of such a circuit with an external magnetic field is usually described in terms of the interaction of the field with the magnetic moment. Whereas this might be the appropriate approach in most cases, it is not obvious that this should always be so. In this paper we calculate the trajectory and angular momentum of a particle with charge q and mass m, constrained to move on a spherical surface of radius a, in a static magnetic field B0, and compare it with the motion of a pointlike magnetic dipole in the same field. The regime of motion is governed by the ratio between two parameters: kL/, where L = -qB0/2m is the Larmor frequency, and = v0/a, where v0 is the initial velocity of the particle, is the angular frequency of the particle in the loop. For an arbitrary value of k the relation between the spherical coordinate of the particle, , and the time is given by the elliptical integral t = (1-k2cos2)-1/2 d, which reduces to the usual relation, = 2, where is the frequency, only for k<<1. In this limit the motion of the particle becomes equivalent to the motion of the dipole moment. The problem is of interest for intermediate-level lectures on either classical mechanics or electrodynamics.