Three-dimensional calculations of the magnetic fields in a finite superconducting hollow cylinder in an applied axial magnetic field

In this study, a set of self-consistent coupled-integral equations for the local magnetic flux and current distributions in a finite superconducting hollow cylinder under an axial magnetic field has been directly derived by using the Biot-Savart law within the framework of the critical-state model. The equations were first solved numerically in the three-dimensional space before obtaining the hysteresis loops for the Kim and Exponential models. We haveassumed the contribution of the flux penetration from the inner surface of the sample to be higher than that of other surfaces. It is found that the variation in the area of lateral surface changes the magnitude of the magnetic moment of the finite hollow cylinder in the applied magnetic field. The obtained results are in good agreement with calculates. The formalism presented here can be used for an arbitrary shape of the superconducting system in the presence of any magnetic field dependence of the critical current density
 in an external magnetic field of arbitrary direction.