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Recent studies have extended the classical space-charge limited current (SCLC) solution in a non-magnetic, planar diode with zero injection velocity to other geometries using variational calculus (VC). We further extend VC to solve for SCLC with a non-relativistic, monoenergetic injection velocity from first principles for nonplanar diodes. By extremizing either the current or a functional of the electric field (and not its derivative), we demonstrate that VC can find either the bifurcation or the SCLC solution, respectively. The bifurcation solution is characterized by the onset of particle reflection, resulting in a singularity in the derivative of the electric field at the virtual cathode, physically analogous to the singularity at the cathode in SCLC for zero injection velocity. Alternatively, using VC to extremize a functional of the potential and its gradient (electric field) yields the maximum current SCLC result. We then derive the SCLC solutions in cylindrical and spherical diodes; additionally, we derive a method to determine SCLC numerically and the bifurcation solution exactly for any orthogonal geometry. Implications for the potential profile and virtual cathode are discussed, especially the behavior for other geometries.

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