Magnetic Interaction Force and a Couple on a Superconducting Sphere in an Arbitrary Dipole Field

The calculation of the magnetostatic potential and levitation force due to a point magnetic dipole placed in front of a superconducting sphere in the Meissner state is readdressed. Closed-form analytical expression for the scalar potential function that yields the image system for an arbitrarily oriented magnetic dipole located in the vicinity of a superconducting sphere is given. Analytic expression for the lifting or levitation force acting on the sphere is extracted from the solution for a general dipole. A special case of our expression where the initial magnetic dipole makes an angle with the z-axis is derived. Our expression for the force in this particular case shows that a recently obtained result (J. Supercond. Nov. Magn. 21:93–96, 2008) for an arbitrary dipole is incorrect. A brief discussion of another erroneous result (J. Supercond. Nov. Magn. 15:257–262, 2002) for a transverse/tangential dipole–sphere configuration, corrected elsewhere recently, is reproduced. Correct expressions for the interaction energy with some limiting cases are also provided. The result derived here demonstrates that the value of the levitation force for a dipole that makes an angle with z-axis lies between the values for a radial dipole–sphere and transverse dipole–sphere configurations providing upper and lower bounds. It is found that for a magnetic dipole making an angle with z-axis, there exits a second force component along the negative y-direction, which influences a couple acting on the superconducting sphere. It is also shown that the couple is proportional to the second force component and that both the couple and second force components vanish for a radial dipole–sphere and transverse dipole–sphere configurations, respectively. These results appear to be new and have not had received due attention in the context of superconductivity.