New theorem of classical electromagnetism: equilibrium magnetic field and current density are zero inside ideal conductors

TL;DR

This paper proves a new theorem in classical electromagnetism showing that in ideal conductors, the magnetic field and current density are zero inside, indicating perfect diamagnetism and surface currents.

Contribution

It introduces a novel theorem demonstrating that magnetic energy is minimized with surface currents and zero interior magnetic field in ideal conductors, extending Thomson's electric theorem.

Findings

Magnetic energy is minimized with surface currents.

Interior magnetic field in ideal conductors is zero.

Perfect conductors are perfectly diamagnetic.

Abstract

We prove a theorem on the magnetic energy minimum in a system of perfect, or ideal, conductors. It is analogous to Thomson's theorem on the equilibrium electric field and charge distribution in a system of conductors. We first prove Thomson's theorem using a variational principle. Our new theorem is then derived by similar methods. We find that magnetic energy is minimized when the current distribution is a surface current density with zero interior magnetic field; perfect conductors are perfectly diamagnetic. The results agree with currents in superconductors being confined near the surface. The theorem implies a generalized force that expels current and magnetic field from the interior of a conductor that loses its resistivity. Examples of solutions that obey the theorem are presented.