Analytic solutions to Maxwell–London equations and levitation force for a general magnetic source in the presence of a type-II superconducting sphere with a vortex line

Abstract

The interaction between a general magnetic source and a type-II superconducting sphere with a vortex line is studied within the London theory. With a single vortex line along a diameter of the sphere and a general magnetic source outside the sphere, the Maxwell-London equations are solved analytically. In the source free case, the magnetic field and its flux, the supercurrent and the magnetic moment associated with it, the London free energy of the system, etc are studied in detail. Various cases are investigated and the levitation forces are calculated when the source is a magnetic monopole, a point dipole, a current carrying loop, and a uniform magnetic field. The levitation force contains two parts, one from the stray field of the vortex line and the other from the field of the supercurrent induced by the source. In particular, for a point dipole with arbitrary location and orientation, both parts of the levitation force contain transverse as well as radial components. For a radial or transverse dipole, the levitation force due to the induced field has only a radial component and the results reduce to those previously available. When the dipole is located right above or below the vortex line, the levitation force due to the latter exhibits rather interesting features. At these positions, the two parts of the levitation force on a transverse dipole can be distinguished. Because the model for the vortex line involves a string-like core, the London free energy obtained as a series appears to be divergent in general. An improved model is suggested, where the vortex line has a conical core. Modifications to all results are obtained and the divergence problem is resolved.