Oscillating Currents

Abstract

Here, we shall consider what happens when the current is repeatedly turned on and off. We begin by ramping the current up linearly during a time Δr ≥ 2 (in units of R∕c) and then (after an elapsed time of 2R∕c for the bubble to leave the sphere) ramping it down again over an equal time Δr. During the turn-on, a total energy of 3∕2 + (6∕5)( Δr)⁻² units is required for the linear ramp: 1 unit for the sphere, 1/2 a unit for the external dipole field, and (6∕5) ( Δr)⁻² of a unit for the escaping bubble. During the turn-off, 3∕2 − (6∕5)( Δr)⁻² is regained from the field, but (6∕5)( Δr)⁻² escapes with the bubble. Thus, in the process of gradually turning the current on, waiting momentarily, and then gradually turning it off again, a total energy of (12∕5)( Δr)⁻² is lost to the escaping bubbles.