Non-linear oscillation of the fluid in a plate capacitor

Employing the principle of virtual work the equation of motion for the liquid between two electrode walls of a charged parallel plate capacitor is established. Analysis in phase plane stands out the motion is non-linear oscillation. The phase orbit is determined then the oscillation period is expressed with elliptic integrations. Besides the intrinsic parameters in the system, the initial disturbing conditions affect the oscillation period, which readily demonstrates the non-linearity of oscillation. The characters of the period are discussed and the approximate computation is presented. Further, the threshold of control voltage under which the results can be still used for the capacitor with an inclined plate is achieved.