Response to 'Comment on 'A coordinate system invariant formulation for space-charge limited current in vacuum'' [Appl. Phys. Lett. 118, 266101 (2021)]

摘要

The preceding Comment1 raises concerns regarding our derivation of a space-charge limited current (SCLC) using variational calculus (VC)2 by arguing that the results for concentric cylinders and spheres seem to violate steady-state continuity, as defined by △·J=-∂ρ/∂t =0, (1) with current density J, charge density ρ, and time t. Here, we reconcile the VC solutions with continuity by examining the assumptions inherent in (1).Fundamentally, the steady-state continuity demands that the amount of charge q in a source-free region remains constant with time dq/dt + ∮∮_SJ·dS = 0, (2) with differential area vector S for the closed surface S. Equation (2) commonly reduces to (1) using divergence theory; however, this assumes constant differential volume. This constraint always holds in a three-dimensional (3D) space; however, the VC derivation uses symmetric assumptions and only considers a one-dimensional (1D) geometry. When translating from 3D to 1D, the variation of differential volume for interpreting (2) must accurately account for the coordinates orthogonal to J. Translating from 3D to 1D in cylindrical and spherical coordinates is complicated since the coordinates are not independent-the angular components depend on the radial coordinate r, which must be explicitly considered.