Closed-Form Solutions for the Trajectories of Charged Particles in an Exponentially Varying Magnetostatic Field

We present a new reference solution for charged particle motion in a strongly inhomogeneous magnetostatic field. The solution describes both bound and unbound particle motion, which can be split into three regimes, the deflection, loop-deflection, and drift regime. We calculate the trajectory in terms of trigonometric and hyperbolic functions, resulting in simple analytical expressions for the particle position and del B-drift velocity. This reference solution is useful to verify and compare the performance of kinetic and guiding-center charged particle pushers in inhomogeneous fields by verifying the conservation of two constants of motion, as well as the exact trajectory at any time.