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

Jaap Hoekstra

Research output: Contribution to journalArticleScientificpeer-review

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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.
Original languageEnglish
Pages (from-to)13-21
Number of pages9
JournalIEEE Open Journal of Circuits and Systems
Issue number1
Publication statusPublished - 2020


  • Two-capacitor problem, two-capacitor paradox, energy paradox, energy dissipation, circuit theory, nanoelectronics, single-electron tunneling, (Dirac) delta function, ideal switch, unbounded current.


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