The two capacitor problem revisited: simple harmonic oscillator model approach

TL;DR

This paper uses a mechanical harmonic oscillator model to explain the energy loss in the two-capacitor problem, showing that exactly half of the energy is dissipated regardless of the dissipation mechanism, providing a simple undergraduate-level explanation.

Contribution

It introduces a harmonic oscillator approach to clarify the energy dissipation in the two-capacitor problem, offering a simple and general explanation applicable at undergraduate level.

Findings

Exactly half of the energy is dissipated during capacitor connection.

The dissipation occurs regardless of the mechanism, such as Joule heat or radiation.

The approach simplifies understanding of energy loss in capacitor systems.

Abstract

The well-known two-capacitor problem, in which exactly half the stored energy disappears when a charged capacitor is connected to an identical capacitor is discussed based on the mechanical harmonic oscillator model approach. In the mechanical harmonic oscillator model, it is shown first that \emph {exactly half} the work done by a constant applied force is dissipated irrespective of the form of dissipation mechanism when the system comes to a new equilibrium after a constant force is abruptly applied. This model is then applied to the energy loss mechanism in the capacitor charging problem or the two-capacitor problem. This approach allows a simple explanation of the energy dissipation mechanism in these problems and shows that the dissipated energy should always be \emph {exactly half} the supplied energy whether that is caused by the Joule heat or by the radiation. This paper which provides a simple treatment of the energy dissipation mechanism in the two-capacitor problem is suitable for all undergraduate class level.