Pedagogical study of the image of a magnetic dipole in front of a superconducting sphere

Abstract

The method of images to solve certain electrostatic boundary-value problems is taught worldwide in undergraduate-level physics courses. Though it is also possible to employ this technique for solving magnetostatic boundary value problems, examples of this usage are rarely found in textbooks or physics pedagogy literature. In particular, the problem of finding the field due to a magnetic dipole kept in front of a superconducting sphere is an interesting one, because (i) it helps the students to compare with the grounded conducting sphere image problem in electrostatics, and (ii) offers a greater degree of difficulty since the source is a dipole (vector), rather than an electric charge (scalar). The present work demonstrates an intuitive way of solving the problem. The case in which the source dipole is oriented radially with respect to the sphere is solved with a single dipole image. In the case of the transverse orientation of the source dipole, we model the dipole as a current loop. Then, we find the image of the radial and transverse current elements that satisfy the boundary conditions. Then, we show that this method can be used to deduce the form of the image dipoles when the dipole is oriented in the transverse direction. This method is very much intuitive and accessible for undergraduate-level students.