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The test-particle formalism of Rostoker is applied to electrons which move only in one dimension through randomly distributed, singly charged ions. This model is appropriate to the limit of infinite magnetic field. The test particle equations are solved to find the conductivity 0.(k, co),for arbitrary wave number k, neglecting electron-electron correlations. In the long-wavelength limit k= 0 our result agrees with previous work. We then find explicit expressions for 0.(~), using degenerate statistics for the electrons. Contributions emerge both from excitation of undamped normal modes and from nonresonant individual particle scattering. For finite magnetic fields the cyclotron motion of the electrons is important, and we explicitly analyze the corrections to the one-dimensional model when co is comparable to Further corrections for higher frequencies or lower magnetic fields may be obtained by a straightforward procedure.

The purpose of this paper is to show in detail how the electron energy-eigenvalue spectrum changes from the form eB = (N+12)ħωc+Pz2/2M in a finite magnetic field to the free-particle form e0 = P2/2M in zero magnetic field. We consider an electron confined to a “box” of finite volume and calculate the exact eigenstates for arbitrary magnetic field strengths. When the electron cyclotron radius rc = (2ħ/Mωc)1/2 is small compared to the dimensions of the box, we get the spectrum eB, and when it is large we get e0. The key to this demonstration is a mathematical identity which relates the magnetic eigenfunctions to the free-particle eigenfunctions. Magnetic corrections to the spectrums e0 are obtained without the use of quantum mechanical perturbation theory.

Self‐consistent solutions to the Vlasov‐Maxwell equations are given for three types of relativistic beams: (a) The current density is hollowed out on axis; (b) the current density is peaked on axis; and (c) the current density is uniform. The models used here give beams with sharp radial boundaries.