Salvato in:
| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2504.13435 |
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Sommario:
- 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 $δ= D/D_M > 0.6$, 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 $δ\to 1$ 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.