The Hull cutoff represents the maximum magnetic field in a vacuum crossed-field gap (CFG) such that an electron emitted from the cathode reaches the anode. Prior studies demonstrated that introducing ions into a CFG always causes increased excursion of electrons toward the anode. In this paper, we assess a collisional CFG by incorporating collision frequency into the electron force law. The theoretical electron trajectories agree well with a one-dimensional particle-in-cell simulation and demonstrate that emitted electrons always cross a collisional CFG. We derive a modified Hull cutoff condition for a collisional CFG corresponding to an electron reaching the anode with zero velocity in the direction of the electric field. Rather than representing the threshold for magnetic insulation, this condition gives the maximum magnetic field and maximum collision frequency for which an electron reaches the anode without turning around; higher magnetic fields and/or collision frequencies cause the electron to turn around before crossing the gap. Further increasing either quantity causes the electron to change direction more frequently as it crosses the gap, noticeably increasing the transit time with each change in electron direction. In the limit of high collision frequency, the electron velocity across the gap approaches a constant, meaning that electrons will reach the anode at nonzero velocity. The transit time above this condition increases smoothly and monotonically with increasing magnetic field or collision frequency. These results elucidate the implications of collisions on magnetic insulation for future assessments of the limiting current in a collisional CFG.

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