Nonrelativistic planar double-stream equilibria in magnetically insulated diodes

The study of magnetically insulated electron flow is a topic of relevance to the startup state of magnetrons, cross-field amplifiers, and magnetically insulated line oscillators, as well as to the operation of magnetically insulated transmission lines and ion diodes. In a recent paper, Christenson and Lau explored double-stream equilibria in magnetically insulated diodes analytically for the case of vanishing injection electron velocities at the cathode and numerically for the case of nonzero injection velocities. Here, the analytical results are generalized to the case of nonzero injection velocities to allow the calculation of the equilibrium current density and electron sheath thickness as functions of the voltage, magnetic field, and gap width. An interesting outcome of the analysis is that it is shown that space-charge limited electron emission from the cathode is no longer possible for equilibria in which the electrons are monoenergetic with vanishing canonical momenta. In addition, for the special case of zero injection energies, but not necessarily zero electric field at the cathode, closed-form expressions are derived for the orbital quantities of the electrons-positions and velocities-as a function of time measured from the moment of emission from the cathode. Unfortunately, a spatial variation is not apparently available in closed form; however, one can express these quantities alternatively in terms of a generalized potential with an implicit spatial variation.