PDE seminar by Prof. Francesca Rapetti (Lab. J.-A. Dieudonne, CNRS & Univ. Nice Sophia-Antipolis)
Comments on the Galilean limits of Maxwell's equations
Maxwell’s equations are fundamental for the description of electromagnetic phenomena and valid over a wide range of spatial and temporal scales. The static limit of the theory is well defined and much easier. The electric and magnetic fields are given by the laws of Coulomb and Biot-Savart. As soon as there is any time dependence, we should in principle use the full set of Maxwell’s equations with all their complexity. However, a broad range of important applications are described by some particular models, as the ones in the low frequency range, emerging from neglecting particular couplings of electric and magnetic field related quantities. These applications include motors, sensors, power generators, transformers and microme- chanical systems. Note also that the quasi-static models are useful for a better understanding of both low frequency electrodynamics and the transition from statics to electrodynamics. We thus present a wider frame to treat the quasi-static (QS) limit of Maxwell equations. Following [1, 2, 3], we discuss the fact that there exists not one but indeed two dual Galilean limits (called “electric” or EQS, and “magnetic” or MQS limits). As a consequence, one has to be careful when investigating non-relativistic limits. We start by a re-examination of the gauge conditions and their compatibility with Lorentz and Galilean covariance. By means of an adimensional analysis, first on the fields and secondly on the potentials, we emphasize the correct scaling yielding the two (limit) sets of Maxwell equations . This work has been done in collaboration with Germain Rousseaux now at the Pprime Institute in Poitiers.
Keywords: Maxwell’s equations, quasi-static approximations, gauge conditions, dimensional analysis
 M. Le Bellac, J.-M. Levy-Leblond, Galilean Electromagnetism, Il Nuovo Cimento, 14 (1973), 217–233.
 J. R. Melcher, H. A. Haus, Electromagnetic Fields and Energy, Prentice Hall (1980).
 M. de Montigny, G. Rousseaux, On some applications of Galilean electrodynamics of moving bodies, Am. J. Phys., 75 (2007), 984–992.
 F. Rapetti, G. Rousseaux, On quasistatic models hidden in Maxwell’s equations Applied Numerical Mathematics, 79 (2014), 92–106.