Keywords: particle accelerators, motion, relativistic speed, velocity selectors, particle trajectories, charged particles, electromagnetic fields
The trajectories of charged particles moving at relativistic speeds inside particle separators – a fully symbolic solution
Predicting the motion of charged particles through electromagnetic fields is a very important feature in the design of particle accelerators since it provides the very important function of separating the particles according to their charge, mass and velocity (Gold and Nusinovich, 1997; Chu, 2004). In our paper we consider the general case when the electric field E and magnetic field B have an arbitrary orientation with respect to each other and the particles travel at relativistic velocities v. We will show how to remove all the difficulties associated with finding a set of symbolic solutions, thus avoiding any errors in using numerical approaches and getting the most elegant and precise solutions. Our paper is organised in three parts. The first part deals with the direct approach in solving the resulting system of differential equations and with the resulting mathematical difficulties associated with such an approach. We will show that such a frontal approach leads to unsolvable equations, even for the simpler case of orthogonal orientations of E, B and v. In the second part of the paper we develop a step-by-step methodology in solving symbolically the to date unsolved general problem for the case of arbitrary orientation between E and B while considering both E and B as uniform and static. In the last section of the paper we extend the solution to time-varying fields.