Simulating charged particles in electromagnetic fields

Abstract

Relativistic charged particles in electromagnetic fields follow paths that increase in complexity with an increasingly complex field. When traveling in these fields, the particles are acted upon by an electromagnetic force comprised of an electric component and a magnetic component. While solving for these paths in simple electromagnetic fields can be done analytically, the task becomes significantly more difficult when the fields get more complex. Thus, in these situations, numerical methods are required to find solutions. One of the most well-known such methods is the Boris method, which is explored in this paper.  Before this method is applied to any complex situations, its accuracy must be ensured by testing on simple cases which are solvable by hand. These cases include a 1D electric field in the direction of the initial velocity of the particle, and a 1D magnetic field perpendicular to the particle's velocity. With the accuracy proven, the method was applied to the cases of a force-free field, a dipolar field, and a quadrupolar field. In the latter two cases the method produced very interesting results that could provide significant insight that would be very difficult to achieve analytically. In the case of the force-free field, however, the method shows some limitations, as a precise cancellation of the force produced by the electric and magnetic fields is required to produce a straight line and the Boris method has some difficulty achieving this, especially when using a large time step.