Minimal Magnetic Energy Theorem

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.