A coordinate system invariant formulation for space-charge limited current in vacuum

While space-charge limited emission current density J_cr is calculated exactly for one-dimensional (1D) planar geometry, 1D cylindrical and spherical geometries require approximations such as the Langmuir-Blodgett (LB) equations or nonphysical assumptions. Using variational calculus (VC), we derive a differential equation from first principles to calculate J_cr for any geometry. This yields exact, closed-form analytical solutions for 1D coaxial cylindrical and concentric spherical geometries that approach LB for sufficiently close cathode (R_c) and anode (R_a) radii. VC agrees better with simulations in cylindrical geometry than LB at R_c/R_a = 0.5. The analytical VC solutions also demonstrate the asymptotic behavior for J_cr. For cylindrical geometry, J_cr $\propto$ 1/$R_c^2$ as R_c/R_a approaches zero or infinity. For spherical geometry, J_cr $\propto$ 1/$R_c^2$ as R_c/R_a → 0 and J_cr $\propto$ $R_a^2$ /$R_c^4$ as R_c/R_a → ∞.