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

Authors:
Open Access
Publisher website
Google Scholar
Abstract
Recent studies have applied variational calculus, conformal mapping, and point transformations to extend the one-dimensional space-charge limited current (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 develop a framework 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 BH 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 δ=D/DM>0.6, where D is the canonical gap distance accounting for geometric effects on electric potential and DM is the effective gap distance that accounts for magnetic field and geometry. The solutions for these geometries overlap as δ→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 BH and SCLC.
Funding Information
  • Air Force Office of Scientific Research (FA9550-22-1-0434)

This publication has 48 references indexed in Scilit: