Hanging a capacitor plate from a spring: a coupled mechanical–electromagnetic oscillator

Abstract

A pair of coupled nonlinear differential equations describes the oscillations in the position and charge of one of the disks of a parallel-plate capacitor hanging from a spring, if the capacitor is initially charged and then connected to an inductor. For realistic values of the parameters, the results are well-approximated by Mathieu functions wherein the rapid charge oscillations are simultaneously modulated in amplitude and frequency by the slow position oscillations of the plate.