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

While space-charge limited emission current density Jcr 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 Jcr 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 (Rc) and anode (Ra) radii. VC agrees better with simulations in cylindrical geometry than LB at Rc/Ra = 0.5. The analytical VC solutions also demonstrate the asymptotic behavior for Jcr. For cylindrical geometry, Jcr ∝ 1/ R c 2 as Rc/Ra approaches zero or infinity. For spherical geometry, Jcr ∝ 1/ R c 2 as Rc/Ra → 0 and Jcr ∝ R a 2/ R c 4 as Rc/Ra → ∞.