On a magnetic skin effect in eddy current problems: the magnetic potential in magnetically soft materials

This work is concerned with the time-harmonic eddy current problem in a bi-dimensional setting with a high contrast of magnetic permeabilities between a conducting medium and a dielectric medium. We describe a magnetic skin effect by deriving rigorously a multiscale expansion for the magnetic potential in power series of a small parameter $$\varepsilon $$ which represents the inverse of the square root of a relative permeability. We make explicit the first asymptotics up to the order $$\varepsilon ^3$$ . As an application, we obtain impedance conditions up to the fourth order of approximation for the magnetic potential. Finally, we measure this skin effect with a characteristic length that depends on the scalar curvature of the boundary of the conductor.