An approximate calculation of the non-adiabatic motion of a charged particle in an inhomogeneous rotationally symmetric magnetic field

Abstract

A study is made of the motion of a particle injected longitudinally into a rotationally symmetric magnetic field increasing linearly in an arbitrarily sharp way on the axis of symmetry. By expanding the magnetic potential in a series in the vicinity of the line of force from which the particle starts, and by restricting our considerations to linear terms, we arrive at a single linearized equation of motion for the radial motion, as we assume the longitudinal motion to be uniform. The linearized equation can be solved exactly. Numerical evaluation is carried out (with the linearized line of force) for several different slopes of the field, and compared for similar cases with a numerical evaluation of an exact nonlinear problem. The method described in the paper can also be applied to cases with more general initial conditions, when the particle also moves with an azimuthal velocity.