Comment on “London model for the levitation force between a horizontally oriented point magnetic dipole and superconducting sphere”

Abstract

In a recent paper the magnetostatic boundary-value problem for a magnetic dipole with transverse direction in the presence of a superconducting sphere was solved in both cases when the London penetration depth is zero and finite. It was concluded that the levitation force on the transverse magnetic dipole is exactly half that for a magnetic dipole with radial direction. We show that this conclusion is incorrect in either case. In the former case it is due to an incorrect boundary condition. In the latter case it is caused by calculational errors. Corrected results are presented. The distribution of supercurrent and the associated magnetic moment are also calculated.In a recent paper Coffey solved the magnetostatic boundary-value problem for a magnetic dipole with transverse direction in the presence of a superconducting sphere in both cases when the London penetration depth is zero and finite. 1 The result for the former case was also published in a separate paper. 2 The latter case involves some mathematical difficulty and is interesting. From these studies it was concluded that the levitation force on the transverse magnetic dipole is exactly half that for a magnetic dipole with radial direction. Unfortunately, this conclusion appears to be incorrect in either case. In the former case it is due to an incorrect boundary condition employed. In the latter case it is caused by calculational errors. Because the conclusion is impressive it deserves some clarification. In addition to discussions of the errors, the corrected results are presented here. We also calculate the distribution of supercurrent and the associated magnetic moment.For the convenience of comparison, we will use similar notations as in Ref. 1. We use both the rectangular coordinates ͑x , y , z͒ and the spherical ones ͑r , , ͒. The unit vectors of these coordinate systems are denoted by ͑e x , e y , e z ͒ and ͑e r , e , e ͒, respectively. The position vector is denoted by r. As in Ref. 1 we use MKS units.Consider a superconducting sphere with radius b, whose center is located at the origin of the coordinate system. There is a point magnetic dipole located at the position d = de z where d Ͼ b ͑d is denoted by a in Ref. 1͒, the magnetic dipole moment being m 0 = e x m 0 cos 0 + e y m 0 sin 0 . Here we are considering a somewhat more general case; when 0 = 0 it reduces to the case in Ref. 1. ͑m 0 is denoted by m in Ref. 1. We change the notation to avoid confusion with the angular eigenvalue in the spherical harmonics.͒ The problem is to find the magnetic induction in the whole space.The magnetic induction outside the sphere is B = B 1 + B 2 , where B 1 is the field of m 0 in free space, and B 2 is the induced field produced by the supercurrent in the sphere. B 1 can be described by a scalar potential B 1 =− 0 ١⌽ 1 , whereThe Maxwell equation for B 2 is obviouslyTherefore, B 2 can also be described by a scalar potential B 2 =− 0 ١⌽ 2 , where ⌽ 2 satisfies the Laplace equation ١ 2 ⌽ 2 = 0, and thus can be expanded aswhere A lm Ͼ are constants to be dete...

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“…Here the Legendre polynomial P 1 (µ) = cos θ. This wellknown result corresponds to a constant magnetic induction of strength H applied along the z-direction in the presence of a magnetic sphere [2,14]. The image system in the exterior phase for constant magnetic induction consists of a dipole of strength In this case the magnetic permeabilities of the exterior and interior phases coincide.…”

“…• Superconducting sphere limit k = 1: In this limit, (23) becomes which represents the solution for the Neumann problem involving a superconducting sphere in a source field. After performing the integration in (25), the result can be shown to be equivalent to the form given in [2]. Here again, the interior potential field is zero.…”

“…is the nondimensional magnetic permeability parameter with 0 ≤ k ≤ 1. For k = 0, the above coefficients reduce to that of Dirichlet problem and when k = 1 the constant B n yields the multipole coefficient for the Neummann boundary value problem for a superconducting sphere (in Meissner state [2]). Now the analytic solutions for the scalar magnetostatic potentials in the two regions due to a monopole in the presence of the magnetic sphere r = a for arbitrary k are now given by…”

“…The magnetic interaction force acting on the magnetic sphere of radius a due to the field generated by a magnetic pole can be found from Newton's third law as the inverse of the (more easily calculated) force that the sphere exerts on the monopole. For the magnetic pole located at (0, 0, c), using the approach adopted in [2,26], one can calculate the force from 0 0,0, ,…”

“…Exact results for the magnetostatic scalar potentials, interactions, levitation forces and moments on the magnets are available for a spherical superconductor [8][9][10] in the homogeneous case. A typical physical problem reduces to a Neumann boundary value problem for the Laplace equation for which closed form analytical solutions are tractable via spherical harmonic methods [2,8,9] or using analogous hydrodynamical approach [3,11]. For permanent magnets and materials with dissimilar magnetic permeability constants compared to that of the surrounding host medium the magnetic field calculations are more challenging due to 'heterogeneity'.…”