The motion of a superconducting loop in an homogeneous magnetic field: the harmonic oscillator equation in an unfamiliar setting

Abstract

If a conducting loop is allowed to fall under the influence of gravity in an inhomogeneous magnetic field, Faraday's law and Newton's second law together lead to differential equation for the vertical velocity of the loop. A simple case is analysed in which this equation turns out to be just the familiar second-order differential equation for damped harmonic motion. The same equation applies no matter what the resistance of the loop, but the resulting motion of a superconducting loop is dramatically different from that of a loop with small but non-zero resistance: the magnetic flux through a superconducting loop is time independent, and the result is that in the absence of damping, both the vertical position of the loop and the current in the loop oscillate indefinitely, with a frequency determined by the properties of the loop and the strength of the magnetic field.