Oscillation and radiation of a superconducting ring in a Helmholtz coil

Abstract

We discuss oscillation and radiation of a superconducting ring in a nonuniform magnetic field produced by a Helmholtz coil. This is a simple example of the oscillation of a superconductor between two magnets. Two oscillation modes (the parallel mode and the antiparallel mode) exist, depending on the directions in which the currents flow in the two coils of the Helmholtz coil. Calculations show the following facts. (i) A characteristic radius-to-height ratio of the Helmholtz coil where a maximum frequency can be produced exists. (ii) As the temperature increases, both the parallel and the antiparallel mode frequencies are fixed, but the parallel mode frequency starts to decrease and falls to 0 in a temperature region below the critical temperature, indicating an oscillation in the intermediate state. (iii) When the superconducting ring oscillates with trapped flux in it, as the trapped flux increases, the parallel mode frequency decreases, but the antiparallel mode frequency increases. When the frictional force is not present, radiation alone reduces the motion energy. We show that the oscillations in both modes stop as time approaches infinity. Over time, the amplitude decreases in the form of a root inverse for the parallel mode and decreases exponentially for the antiparallel mode. The half-lives of the amplitudes have been calculated as 2.1 × 1023 years for the parallel mode and 6.9 × 1023 years for the antiparallel mode.