A generalized framework for the critical current for a one-dimensional crossed-field gap in an orthogonal coordinate system

Recent studies have applied variational calculus, conformal mapping, and point transformations to extend the one-dimensional SCLC from planar gaps to more complicated geometries. However, introducing a magnetic field orthogonal to the diode's electric field complicates these calculations due to changes in the electron trajectory. This paper extends a recent study that applied variational calculus to determine the SCLC for a cylindrical crossed-field diode to derive an equation that is valid for any orthogonal coordinate system. We then derive equations for the SCLC for crossed-field gaps in spherical, tip-to-tip, and tip-to-plane geometries that can be solved numerically. These calculations exhibit a discontinuity at the Hull cutoff magnetic field $B_H$ corresponding to the transition to magnetic insulation as observed analytically for a planar geometry. The ratio of the crossed-field SCLC to the nonmagnetic SCLC becomes essentially independent of geometry when we fix $\delta = D/D_M > 0.6$, where $D$ is the canonical gap distance accounting for geometric effects on electric potential and $D_M$ is the effective gap distance that accounts for magnetic field and geometry. The solutions for these geometries overlap as $\delta \to 1$ since the geometric corrections for electric potential and magnetic field match. This indicates the possibility of more generally accounting for the combination of geometric and magnetic effects when calculating $B_H$ and SCLC.