A capacitance-based approach for determining the space-charge limited current density and electric potential in a crossed-field gap

Introducing a magnetic field perpendicular to the electric field in a diode turns electrons as they cross the gap and alters the electric potential’s spatial profile and the space-charge limited current density (SCLCD) compared to a nonmagnetic diode. In this paper, we apply gap capacitance to crossed-field gaps with magnetic fields below and above the Hull cutoff magnetic field BH for magnetic insulation to assess these properties. Combining capacitance and the force law gives the same relationship for the limiting current for B<BH and B>BH as the exact approach that extremizes the current density, while simplifying the mathematics required. Although the crossed-field geometry prohibited a direct solution for the electric field E as a function of position across the gap x for B<BH, we derived a closed form solution for x as a function of E. Using the ansatz that the electric potential ϕx∝xζ, where 4/3≤ζ≤2 accounts for 0≤B≤BH, we calculated ϕ in excellent agreement with the exact solution up to B≈0.999BH. Finally, we used this ansatz to derive a correction factor to convert the current density obtained using the charge-free capacitance to the exact solution for the SCLCD. In summary, we demonstrate the feasibility of estimating ϕx based on B/BH at the space-charge limit and the utility of charge-free capacitance for characterizing the SCLCD of a crossed-field gap.