On a magnetic skin effect in eddy current problems: the magnetic potential in magnetically soft materials (Q2044752)

The authors study the time-harmonic eddy current problem in \(\mathbb{R}^2\) and derive an asymptotic expansion for the magnetic potential with high relative permeability. The main result is a description of the magnetic skin effect for the magnetic potential with a multiscale expansion. The power series expansion is done in terms of a small parameter \(\varepsilon\) which is the inverse of the square root of the relative permeability. First order asymptotics up to order \(\varepsilon^3\) are obtained explicitly. The skin effect is measured with a characteristic length which depends on the scalar curvature of the boundary of the conductor. Asymptotic behavior of this characteristic length function is obtained when \(\varepsilon \to 0\).