A Solution of the Two-Capacitor Problem Through its Similarity to Single-Electron Electronics

The purpose of this paper is to investigate the two-capacitor paradox using circuit models developed in the analysis of circuits that include nanoelectronic single-electron tunneling devices. The two-capacitor paradox, in which it seems that energy is not conserved in a simple circuit consisting of two capacitors in parallel separated by an ideal switch, is resolved by applying linear circuit theory utilizing a current—described by a (Dirac) delta function—and stepping voltages across all three elements. Based on a similar description, successfully used for tunneling of electrons through metal junctions in nanoelectronics, the switch is modeled as a device across which—upon closing—the voltage steps down while the current through it is an impulse. The model distinguishes three intervals in describing the ideal switch: t < 0, t = 0, and t > 0. As a consequence, the ideal switch dissipates energy during the switching action at t = 0 in zero time. Although the solution of the two-capacitor problem looks like a theoretical curiosity, the application of nanoelectronic concepts allow a physical explanation based on electron tunneling; it shows that the ideal switch is best described by the tunneling of many electrons. In such a context, some of those electrons loose energy and the v-i characteristic shows Ohm’s law.