A Capacitance Argument for Electric Potential and Space-Charge-Limited Current in a Crossed-Field Gap

The space-charge-limited current (SCLC) is given by the Child-Langmuir law (CLL) in vacuum and the Mott-Gurney law (MGL) in a solid. The electric potential φ varies with position x by φ∝x^4/3 for the CLL and φ∝x^3/2 for the MGL¹. Without a magnetic field, we appealed to gap capacitance to show that φ∝x^ξ, where 4/3≤ξ≤3/2 for a material with intermediate collision frequency. In this presentation, we consider the behavior of φ and SCLC with a magnetic field perpendicular to the electric field¹. Assuming φ∝x^ζ, we use the anode electric field to derive an equation for ζ as a function of the electron transit time. We show that 4/3≤ζ≤2 with the higher value corresponding to the Hull cutoff magnetic field B_H at which an electron emitted from the cathode just reaches the anode. We recover the exact solution for the SCLC² all magnetic fields below B_H. At B_H, R²=0.9813 between φ(x) obtained using ζ and the exact solution obtained by parametrically² plotting φ(t) and x(t). Extensions of this approach to assess conditions for magnetic fields above B_H³ will be discussed.

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