Magnetic levitation of superconductors in the critical state

The vertical levitation force in a cylindrically symmetric system composed of a permanent magnet and a type-II superconductor is studied assuming the latter to be in a critical state. In a first step, a constant-field-gradient approximation along the length of the superconductor is used to see the effects of the dependence of the critical current density on the internal magnetic field. In a second step, the theory is generalized in order to study the case when the field gradient is not constant, as is usually found in levitation experiments with bulk superconducting samples. The model results are discussed in relation to recent experimental data on magnetic levitation of high-$T_c$ superconductors, explaining the observed behavior and making some predictions for future experiments.