Magnetic Field Expulsion in Perfect Conductors-The Magnetic Equivalent of Thomson's Theorem

摘要

Thomson's theorem states that electric charge density on metal conductors, at static equilibrium, is distributed on the surface of the conductors in such a way that the interior electric field is zero, and the electric field on the surface must be perpendicular to the surface. In this paper, we present a theorem for static magnetic fields, analogous of the Thomson's theorem of electrostatics. We prove, by making use of a variational principle, that the minimization of the magnetic field energy corresponds to the magnetic field expulsion of perfect conductive systems through surface currents. As a result, the current density distributes itself on the surface of the ideal conductor so that the interior magnetic field becomes zero, and all current flows on its surface. This result is put into the context of superconductivity, and leads us to conclude the Meissner effect is not a pure quantum effect, restricted to superconductors, but rather a magnetostatic equilibrium state as a consequence of zero resistivity. In addition, the London equations are derived following an approach by Pierre-Gilles de Gennes where "the superconductor finds an equilibrium state where the sum of the kinetic and magnetic energies is minimum, and this state, for macroscopic samples, corresponds to the expulsion of magnetic flux". For further confirmation, the same result is also derived in the classical limit of the Coleman-Weinberg model, the most successful quantum macroscopic theory of superconductivity. A specific example is presented to corroborate the result of our theorem. In particular, an explicit solution for a minimal energy magnetic field configuration is analyzed, and found to be in agreement with our statement.