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.

这项工作涉及二维设置中的时谐涡流问题，其中导电介质和电介质之间的磁导率具有高对比度。我们通过严格导出小参数\(\varepsilon \) 的幂级数中磁势的多尺度扩展来描述磁趋肤效应，该参数代表相对磁导率平方根的倒数。我们使第一个渐近线达到\(\varepsilon ^3\)的阶数。作为应用，我们获得了磁势近似四阶的阻抗条件。最后，我们测量这种趋肤效应 其特征长度取决于导体边界的标量曲率。