This work presents an efficient method based on asymptotic models in order to solve numerically eddy current problems in a bi-dimensional setting with a high contrast of linear relative magnetic permeability mu_r between a conductor and a dielectric subdomain. We describe a magnetic skin effect by deriving a multiscale expansion for the magnetic potential in power series of a small parameter delta which represents the skin depth. We make explicit the first asymptotics up to the order three. Some numerical results based on the finite element method will be presented to illustrate the magnetic skin effect and to validate the performance of the proposed asymptotic models in the dielectric medium. We confirm that the proposed asymptotics provide reduced computational costs for a wide range of the physical parameters introduced in our problem.