Minimal Magnetic Energy Theorem

TL;DR

This paper proves Thomson's electrostatic energy minimization theorem using variational principles and introduces a magnetic analogue showing that surface currents minimize magnetic energy, aligning with superconductor behavior.

Contribution

It provides a variational proof of Thomson's theorem and introduces a magnetic energy minimization theorem for surface currents, linking to superconductor properties.

Findings

Electrostatic energy is minimized by surface charge distributions.

Magnetic energy is minimized by superficial current distributions.

Superconductors' surface currents align with the magnetic energy minimization principle.

Abstract

Thomson's theorem states that static charge distributions in conductors only exist at the conducting surfaces in an equipotential configuration, yielding a minimal electrostatic energy. In this work we present a proof for this theorem based on the variational principle. Furthermore, an analogue statement for magnetic systems is also introduced and proven: the stored magnetic field energy reaches the minimum value for superficial current distributions so that the magnetic vector potential points in the same direction as the surface current. This is the counterpart of Thomson's theorem for the magnetic field. The result agrees with the fact that currents in superconductors are confined near the surface and indicates that the distinction between superconductors and hypothetical perfect conductors is fictitious.