Virtual cathode behavior in a crossed-field gap

While the behavior of virtual cathodes (VCs) in a space-charge limited gap in the absence of a magnetic field is well understood, it is less characterized for crossed-field gaps with perpendicular magnetic and electric fields. Thus, this study examines VC formation at steady state in a crossed-field gap with magnetic field B below the Hull cutoff magnetic field B_{H} for magnetic insulation. VC formation at the critical current density J_{c} depends strongly on the initial electron velocity u_{0} and b=B/B_{H}. For u_{0}≤2ΩD/π, where Ω is the cyclotron frequency and D is the gap distance, there exists a threshold b_{VC}=B_{VC}/B_{H} such that a VC forms between the cathode and anode at J_{c} for bb_{VC}. This is unlike the nonmagnetic case, for which a VC forms at J_{c} whenever u_{0}≠0. When no VC forms in a crossed-field gap, the electrons accelerate upon emission from the cathode and decelerate near the anode, causing a peak in electron density. Thus, the limiting current no longer corresponds to the buildup of space-charge at or near the cathode, but instead to buildup at the anode. We also show VC formation for steady-state current densities J