A ring of josephson junctions in a magnetic field

We have studied the magnetic properties of the ring of Josephson junctions in a magnetic field at different capacitance of superconducting regions and temperatures. We show that the capacitance of junctions does not change qualitatively the magnetic properties of the ring of junctions although quantitatively the properties are strongly changed. The decrease of the size increases the charging energy and destroys the superconducting fermions, decreasing the diamagnetic current (Coulomb blockade). At half-quantum of the flux through the ring there appears a magnetic phase transition which shows up as a singularity in the magnetization, in the susceptibility and in the specific heat. At the transition there is the trapping of the one quantum of flux by the ring. The interpretation of experiments, describing the inverse kinetic inductance as a function of the flux per plaquette of a lattice of two-dimensional arrays of the Josephson junctions and the relation with granular high-temperature superconductors are proposed. New singularities in the inverse kinetic inductance are predicted.