The Paths of Ions and Electrons in Non-Uniform Crossed Electric and Magnetic Fields

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Abstract
The integration of the force equations for a charged particle moving in the presence of particular types of crossed, non-uniform, electric, and magnetic fields is shown to be possible in a very simple manner. The cases which admit of the integration are those such that: the magnetic field is a function of only one coordinate (either Cartesian or the radius vector in a polar system); the electric field is a function of only one coordinate which for any particular case is the same coordinate with which the magnetic field varies; the electric field has a component only in the direction of the variable with which it varies; and the two fields are orthogonal. Since these conditions, in most cases, are only met on a median plane symmetrically situated relative to magnetic pole faces and electrostatic electrodes, the calculations refer to the motion in this plane. The equations are solved and discussions made of the orbits for several different field arrangements and for one a new type of perfect focusing. The method can be used with numerical integration when the analytical difficulties are too great or when the fields are only known empirically.