Motion of a Charged Particle in an Axisymmetric Magnetic Field Inversely Proportional to the Radius

Abstract:

We discuss an exact solution of the Lorentz equations of motion of a charged particle in an axisymmetric magnetic field inversely proportional to the distance from the axis of symmetry. This solution involves only elementary mathematical functions, however, it requires finding a root of a transcendental equation numerically. The solution requires considering several distinct types of trajectories determined by the initial conditions as well as developing procedures for selecting the correct branches of the inverse trigonometric functions at multiple turning points. While this exact solution has been mentioned in the literature, and it appears to be relatively unknown and its detailed description has been lacking. We also discuss comparison of the exact solution with the guiding center approximation. The results are relevant to the motion of equatorially mirroring particles in dipole magnetic field.