Meissner state properties of a superconducting disk in a non-uniform magnetic field

We describe a formalism for studying a superconducting disk shielding an arbitrary magnetic field. The theory is developed for problems with rotational symmetry, and the results are applied to investigating various experimental situations. In particular, we have considered a cylindrical permanent magnet which is brought over an ideal superconductor. The levitation force, F, and the induced supercurrent density, j, have been calculated and the results put into closed forms. Both magnitudes are analyzed in terms of suitable dimensionless parameters related to the relevant length scales: the distance between the magnet and the superconductor, the disk radius and the magnet transversal dimensions. We demonstrate that both F and j provide a very precise description in the whole range of parameters by means of simple expressions which strictly correspond to the limits. In the specific range of magnetic force microscopy, we show how to approximate F to the desired accuracy in terms of the reduced penetration dep...