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 $u_0\le 2\mathrm{\Omega }D/\pi$, where $\mathrm{\Omega }$ is the cyclotron frequency and $D$ is the gap distance, there exists a threshold $b_{\mathrm{VC}}=B_{\mathrm{VC}}/B_H$ such that a VC forms between the cathode and anode at $J_c$ for $b<b_{\mathrm{VC}}$, but does not form for $b>b_{\mathrm{VC}}$. This is unlike the nonmagnetic case, for which a VC forms at $J_c$ whenever $u_0\ne 0$. 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 $J<J_c$. While a VC also forms at steady state for $J<J_c$ in nonmagnetic diodes with $u_0\ne 0$, VC formation is still necessary at $J=J_c$. However, for crossed-field gaps, VC formation alone is neither sufficient nor necessary for the limiting current condition.

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• Single-particle dynamics
• Electron sources
• First-principles calculations in plasma physics