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) proportional to 1/2 as R-c/R-a approaches zero or infinity. For spherical geometry, J(cr) 2 as R-c/R-a -> 0 and J(cr) as R-c/R-a -> infinity.