Electromagnetic fields induced by electric current in an infinite plate with an elliptical hole

Analytical solutions to the electromagnetic field in a thin conductive plate with an elliptical hole are derived by means of complex potentials and conformal mapping techniques. The steady-state current field in a thin conductive plate is two dimensional (2D) and is explored by a standard complex variable technique. The current is disturbed around the elliptical hole, and produces a three dimensional magnetic field. In this case, using the complex variable method to solve the real magnetic field can be challenging. The magnetic boundary conditions take different forms for the soft ferromagnetic and the para- or diamagnetic materials under consideration. A simplified analysis taking account of the magnitude of the magnetic permeability of the magnetic material and air surrounding the material is proposed to reduce the magnetic field in a thin plate to 2D calculations. The magnetic field distributions are derived for each material and the equations of the magnetic components at the tip of elliptical hole are presented.