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The importance of a well-known theorem, originally due to Larmor, is emphasized. It enables a definition of "momentum" and "moment of momentum" for electrons in a magnetic field, hence the possibility of writing the conservation of these quantities when the geometry of the structure is convenient. As typical examples of the method, two special cases are discussed: a plane electron beam and a cylindrical electron beam with longitudinal magnetic field. In both cases it is found that the space-charge density of the beam is entirely controlled by the magnetic field and that the maximum current is obtained for a suitable optimum magnetic field.
Léon Brillouin (1945) studied this question.