In this paper, the relativistic Brillouin flow in a crossed-field gap is investigated. For this, the case of a planar magnetron is considered. In contrast to previous studies, it is assumed that the electron discharge occurs in a timescale that is long compared to the magnetic diffusion time in the metal. It is found that the Brillouin flow properties and the overall scenario for the loss of magnetic insulation are different from the short pulse case. In particular, it is shown that two branches of equilibrium Brillouin flow solutions can coexist inside the gap region: one linearly stable and the other linearly unstable. As the parameters are varied, the two branches coalesce and cease to exist in a bifurcation that leads to a complete loss of magnetic insulation. Nevertheless, the mere existence of the unstable solution inside the gap is shown to affect the electron dynamics causing cathode–anode currents. An expression for the onset of the unstable solution is obtained and compared to the relativistic Hull cutoff condition for the short pulse case. It is found that the loss of magnetic insulation occurs for lower accelerating potentials in the present case. This effect is noticeable even for weakly relativistic cases.

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