Complex Harmonic Capacitors

TL;DR

This paper introduces the concept of complex harmonic potentials in doubly connected capacitors, exploring their mathematical properties and implications for energy storage and critical points, contrasting with real-valued potential theory.

Contribution

It presents a novel analogy of complex harmonic potentials in capacitors, analyzing critical points and energy interpretation, extending electrostatic potential theory into complex domains.

Findings

Identification of critical points where the Jacobian determinant vanishes.

Interpretation of complex electric capacitors in terms of hyperelastic energy.

Contrast between complex and real-valued potential behaviors.

Abstract

The concept of complex harmonic potential in a doubly connected condenser (capacitor) is introduced as an analogue of the real-valued potential of an electrostatic vector field. In this analogy the full differential of a complex potential plays the role of the gradient of the scalar potential in the theory of electrostatic. The main objective in the non-static fields is to rule out having the full differential vanish at some points. Nevertheless, there can be critical points where the Jacobian determinant of the differential turns into zero. The latter is in marked contrast to the case of real-valued potentials. Furthermore, the complex electric capacitor also admits an interpretation of the stored energy intensively studied in the theory of hyperelastic deformations.