Levitation Force Between a Point Magnetic Dipole and Superconducting Sphere

The magnetic interaction between a point dipole and a superconducting sphere in the Meissner state is investigated in detail. It is demonstrated that the special case of perfect flux exclusion can be solved by the method of images when the dipole is vertically magnetized. The incorporation of a nonzero penetration depth is then made by developing the solution within London theory. The analytic solution, given in terms of an infinite series representation, is refined by exactly summing the perfectly diamagnetic response. Throughout, special cases of physical interest are discussed and applications to magnetic force microscopy and levitation are described.     

Levitation Force Between a Small Magnet and Superconducting Sphere

Based on the Meissner effect and the image method, we studied the interaction between a magnetic point dipole and a superconducting sphere. We obtained analytical expressions for the interaction energy and the levitation force when the dipole is in an arbitrary orientation. Our calculations show the validity of using the image method for a antisymmetrical superconducting sphere-magnet system. Results show that the energy and the force are maxima when the magnetic dipole is perpendicular to the surface of superconductor and minima when it is tangent to the surface. Furthermore, the force acting on the tip of a magnetic force microscope above a superconducting sphere was derived as a generalization of the levitation force acts on a point dipole.      

Calculation of Levitation Force between Small Superconducting Cylinder and Magnetic Ring in the Critical State

Levitation force between a small superconducting cylinder in the mixed state and a magnetic ring was calculated using the critical state model. The dependence of the levitation force on the size of the superconductor as well as the magnet was investigated. The finite size effect of the superconductor is included by demagnetizing factor approximation. The stiffness for the levitating system was calculated. The position of the maximum levitation force depends on the size of the superconductor and the magnet. Trend of the calculated force is in agreement with the experimental results.      

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.     

An Inverse Problem for the ac Magnetic Force Microscopy of Superconductors

A basic inversion problem for the magnetic force microscopy (MFM) of a semiinfinite superconductor in the Meissner state is formulated, ac Detection is assumed, with the MFM tip oscillating at angular frequency Ω. The coupling of all electrodynamic fields is treated, including a normal-current density in the superconductor. Under certain assumptions on the tip and superconductor geometry, a unique penetration depth λ(z) can be recovered from onedimensional force gradient measurements. This development opens new possibilities for the nondestructive evaluation of superconducting crystals and films.