# hpfem/esco2012-boa

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 \title{About Quasi-Static Approximations of Maxwell's Equations} \tocauthor{F. Rapetti} \author{} \institute{} \maketitle \begin{center} {\large Francesca Rapetti}\\ University of Nice Sophia-Antipolis\\ {\tt francesca.rapetti@unice.fr} \end{center} \section*{Abstract} 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 micromechanical systems. %in order to achieve a preliminar accurate electromagnetic analysis of these devices, %suitable numerical approximations of Maxwell equations are carried out. 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 \cite{LL,Melcher,GM}, we discuss the fact that there exists not one but indeed two dual Galilean limits (called electric'' or EQS, and magnetic'' or MQS limits). These limits have however to be suitably coupled to model sofisticated devices such as wireless power transfer systems using resonant magnetic coupling \cite{Ka}. We thus propose to analyse a quasi-static approach for these devices that are regarded as one of the most promising methods for mid-range wireless charging systems. \begin{thebibliography}{99} \bibitem{LL} M.~Le Bellac, J.-M.~L\'evy-Leblond, Galilean Electromagnetism, \emph{Il Nuovo Cimento}, \textbf{14} (1973), 217--233. \bibitem{Melcher} J.~R.~Melcher, H.~A.~Haus, \emph{Electromagnetic Fields and Energy}, Prentice Hall (1980). \bibitem{GM} M.~de Montigny, G.~Rousseaux, On some applications of Galilean electrodynamics of moving bodies, \emph{Am. J. Phys.}, \textbf{75} (2007), 984--992. \bibitem{Ka} R.~E.~Hamam, A.~Karalis, J.D.~Joannopoulos, M.~Solja\v ci\'c, {\em Efficient weakly-radiative wireless energy transfer: An EIT-like approach}, Ann. Phys., {\bf 324} (2009), 1783--1795. \end{thebibliography}