Eddy Current Approximation of Maxwell Equations

Continuamentenasconoifatti 1 aconfusionedelleteorie 2 Carlo Dossi Electromagnetism is withoutany doubt a fascinating area of physics engineering and mathematics. Since the early pioneeringworks ofAmpere Faraday and Maxwell the scienti?cliteratureon this subject has become immense and books devoted to almost all of its aspects have been published in the meantime. However webelievethatthereisstillsomeplacefornew booksdealingwithel- tromagnetism particularly if they are focused on more speci?c models or try to mix different levels of analysis: rigorous mathematical results sound numerical appro- mation schemes real-life examples from physics and engineering. The complete mathematical description of electromagnetic problems is provided by the celebrated Maxwell equations a system of partial differential equations - pressed interms ofphysical quantitiesliketheelectric?eld themagnetic?eld and the currentdensity.Maxwell scontributiontotheformulationofthese equationsisrelated to the introductionof a speci?c term called displacement current that he proposed to add to the set of equations generally assumed to hold at that time in order to ensure the conservation of the electric charge. The presence of the displacement current permits to describe one of the most - portant phenomenon in electromagnetism namely wave propagation; however in many interesting applications the propagation speed of the wave is very high with respect to the ratio of some typical length and time scale of the considered device and therefore the dominant aspect becomes the diffusionof the electromagnetic ?elds. When the focus is on diffusioninstead of propagation from the modelingpointof view this corresponds to neglecting the time derivative of the electric induction (i.e. thedisplacement current introducedby Maxwell)or alternatively neglectingthe time derivative of the magnetic induction.