Average Power Associated with a Capacitor

For a purely capacitive circuit, the current leads the applied emf by a phase angle of $\pi/2$: $e=e_0\sin\omega t$ and $i=i_0\sin\left(\omega t+\dfrac{\pi}{2}\right)=i_0\cos\omega t$. The instantaneous power is $P=ei=e_0i_0\sin\omega t\cos\omega t=\dfrac12 e_0i_0\sin2\omega t$ -- by exactly the same reasoning as for the inductor (section 13.6.2), just with the opposite overall sign, this too averages to EXACTLY ZERO over one complete cycle.

So an ideal capacitor, like an ideal inductor, consumes zero net average power over a complete AC cycle: energy is drawn from the source and stored in the capacitor's electric field while it charges, and the SAME energy is returned to the source as the capacitor discharges, with nothing lost or gained overall. This is the physical basis for the statement that reactive elements (pure L or pure C) do not themselves dissipate energy as heat -- any actual energy loss in a real …