Space Charge Limited Current for a Nonzero Injection Velocity in Multiple Geometries

Space charge limited current (SCLC), first derived at vacuum by Child and Langmuir [1], is the maximum current that can be transmitted across a diode at steady state. Several subsequent studies have derived SCLC for planar diodes with a nonzero initial velocity [2,3]. Recent work has applied variational calculus (VC) to derive SCLC for one-dimensional planar, cylindrical, and spherical geometries [4]. In this study, we extend VC to derive SCLC for these geometries for a nonzero injection velocity. We compare the results to other planar diodes with nonzero injection velocity. Consequences regarding the virtual cathode and future applications of this method and extensions using conformal mapping [5] to more complicated non-planar diodes will be discussed.