Magnetization of Ellipsoidal Superconductors

A well-known result which can be rigorously deduced from Maxwell's equations and potential theory is that the magnetization M of any homogeneous, isotropic ellipsoid is related to the local field B and the external applied magnetic field H₀ by H=B-4πM=H₀-4πN·M(H). The configuration matrix N, whose elements are the demagnetization coefficients, depends only on the shape of the ellipsoid. In particular, this relation provides a means for computing the magnetization M (as a function of H₀), if it is known for the special case of an infinitely along cylinder whose axis is aligned with H₀. This transformation has been commonly applied to systems for which the M(H) relationship is linear: M=χH, where χ is a constant. We find, more generally, when the ellipsoid is homogeneous and isotropic, that the transformation depends only on the assumption that M is a smooth, single-valued, and otherwise arbitrary function of the local field B. We apply the above result to the well-known magnetization functions for superconductors and find for spheroids whose symmetry axis is parallel to H₀: 4πM=-H₀(1-n) (Meissnerstate) and 4πM=-[(Hc2-H₀)(γ+n)H₀]H₀ (mixedstate), where γ=(2κ₂²-1)β, and n is the element of N associated with the symmetry direction. We also compute the torque exerted on a superconducting spheroid whose symmetry axis is not aligned with H₀.