From Thermostatistics to Maxwell's Equations: A Variational Approach of Electromagnetism

The Maxwell equations are derived from thermodynamic principles. While flux density divergence-free is obtained everywhere from the stationary condition on the Gibbs' free energy, the Maxwell-Faraday equation and the Ohm's law with motion are obtained, in conductors, by assuming an adiabatic and reversible evolution of the field. Hence, the Maxwell-Faraday equation may be extended in the dielectric region for any time-varying excitation. Besides, magnetic- and dielectric-behavior laws result from the convexity of the magnetic and electrostatic Gibbs' potentials. Furthermore, the conservation of the power yields the so-called Lorentz force from virtual work principle. Extension to high frequency is also proposed beyond the plasma pulsation of metal. To sum up, the approach is shown to be: 1) consistent with the finite element method; 2) coherent with a coarse graining optimization, from "scratch" to the design scale; and 3) suitable to consolidate energy processes involved in electromagnetic and electromechanical conversion.