Diamagnetic behavior in random classical radiation

Abstract

Calculations for diamagnetic behavior involving Faraday induction appear in classical electromagnetism textbooks. These calculations give the charged particle motions correctly but then inaccurately introduce the statement that diamagnetism is incompatible with classical thermodynamics, and that quantum theory is required for diamagnetic behavior. Actually, if classical radiative equilibrium in classical zero-point radiation holds before the application of a magnetic field, then it will hold afterwards and will preserve the diamagnetic behavior obtained by the application of Faraday's law. Here, we consider the classical diamagnetism of a charged particle in an isotropic harmonic potential which follows from the four famous spectra of random classical radiation. The zero-point radiation spectrum fully justifies the analysis appearing in the textbooks of classical electromagnetism and in the work of Langevin. The Rayleigh-Jeans spectrum gives no diamagnetic behavior, as is consistent with the Bohr-van Leeuwen theorem. The Planck spectrum without zero-point radiation (surprisingly) gives no magnetic moment at low temperature and paramagnetic behavior at high temperature! Finally, the Planck spectrum with zero-point radiation gives diamagnetic behavior at low temperature and no magnetic moment at high temperature. This last result is in agreement with elementary quantum theory. Once again the Planck spectrum with zero-point radiation provides the best classical description.