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

Analytical solutions to the electromagnetic field in a thinconductive plate with an elliptical hole are derived by meansof complex potentials and conformal mapping techniques. Thesteady-state current field in a thin conductive plate is twodimensional (2D) and is explored by a standard complex variabletechnique. 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 magneticfield can be challenging. The magnetic boundary conditions takedifferent forms for the soft ferromagnetic and the para- ordiamagnetic materials under consideration. A simplified analysistaking account of the magnitude of the magnetic permeabilityof the magnetic material and air surrounding the material isproposed to reduce the magnetic field in a thin plate to 2Dcalculations. The magnetic field distributions are derived foreach material and the equations of the magnetic components atthe tip of elliptical hole are presented.