On streams of charged particles in an electromagnetic field

Abstract

Problems of two kinds arise in the consideration of streams of charged particles in an electromagnetic field. In the first kind of problem (direct problems) the magnetic and electric fields are given, and it is required to study the trajectories which are possible in these fields. This type of problem reduces to the examination of the solutions of the following system of equations with partial derivatives: (v▽)v = e m E + e mc v × H .(1) The electric and magnetic fields in these equations are taken to be known solutions of the Maxwell equations ▽ × H = 4π c ϱτ , ▽ × E = 0, ▽ H = 0, ▽ E = 4 πϱ. (2) For dense streams, the effect of the space charge is taken into account (the effect of the magnetic proper field is not usually considered) and the problem is much more complicated, since equations (1) must be considered together with equations (2) in which the first equation is replaced by ▽( ϱv) = 0. (3) However, in constructing electronic instruments it is often necessary to create a stream of charged particles with given properties. In problems of this kind the stream of charged particles must be considered to be given and the solutions of Maxwell's equations (2) which correspond to such a stream must be examined. This problem is the converse of the previous problem. It has been studied in its application to electron optics in [1] and [2], where in [1] streams close to an arbitrary curve are considered, and in [2] streams close to an arbitrary surface. The problem has been studied in [3] for streans in an electric field, with the restriction that the lines of flow must be coordinate lines of some orthogonal curvilinear coordinate system. An equation is obtained which the Lamb coefficients of this coordinate system must satisfy, and is then solved. The theory of the formation of streams in an arbitrary electromagnetic field is constructed in [4], with similar restrictions. In this work a method is devised to determine the fields from the properties of the stream, given the condition that the considered stream exists. In this paper we consider the following converse problem: we are given a two-parameter family of curves; it is required to find when this family can be the lines of flow of a stream of charged particles in an electromagnetic field.