Abstract
Exact solutions are constructed for the butt-end region of electron beams in the form of finite-length cylinders; truncated cones; cylindrical and conical rings; toroidal, spherical, and ellipsoidal Brillouin formations; a cylindrical magnetron in the subcritical mode; and a cylindrical diode. Problems for finite-length cylinders with elliptic and circular cross sections are investigated. The cylinders are considered in the presence of a uniform longitudinal magnetic field for the case when the longitudinal velocity in a flow exhibits a cycloidal variation and the Cauchy conditions are exactly fulfilled on the flow’s boundary. Relativistic and nonrelativistic flows are analyzed. In the latter case, both the electrostatic and magnetostatic problems are solved. Solution of the magnetostatic problem provides for the continuous transition from the eigenfield to the external magnetic field on a beam’s boundary.
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