Coordinate System Invariant Formulation of Fractional‐Dimensional Child‐Langmuir Law for a Rough Cathode

Abstract This work presents an exact closed‐form analytical model of the space charge limited (SCL) current density for planar, cylindrical, and spherical geometries in fractional dimensional spaces (Fα) where α represents the fractional dimensions having a range 0 < α ⩽ 1. Decreasing value of α shows an increasing degree of surface roughness. This analytical model approaches Langmuir–Blodgett equations for sufficiently close anode (Ra) and cathode (Rc) radii for integer dimensions (α = 1). To calculate SCL current density for any geometry, variational calculus (VC) is used to derive the differential equations in fractional (α) dimensions. The effect of electrode radius ratios on space charge limited current is investigated here considering the surface roughness.