In Section 32.2, we found that the instantaneous current to a capacitor is $i_{\text{C}}=-\omega C{V}_{\text{C}}\sin \omega t$. Thus the instantaneous energy dissipation in a capacitor is

where we used $\text{sin}\left(2x\right)=2\text{sin}\left(x\right)\text{cos}\left(x\right)$.

FIGURE 32.22 shows Equation 32.41 graphically. Energy is transferred into the capacitor (positive power) as it is charged, but, instead of being dissipated, as it would be by a resistor, the energy is stored as potential energy in the capacitor’s electric field. Then, as the capacitor discharges, this energy is given back to the circuit. Power is the rate at which energy is removed from the circuit, hence $p$ is negative as the capacitor transfers energy back into the circuit.

FIGURE 32.22 Energy flows into and out of a capacitor as it is charged and discharged.

Using a mechanical analogy, a capacitor is like an ideal, frictionless simple harmonic oscillator. Kinetic and potential energy are constantly being exchanged, but there is no dissipation because none of the energy is transformed into thermal energy. The important conclusion is that a capacitor’s average power is zero: $P_{\text{}C}=0.$

The same is true of an inductor. An inductor alternately stores energy in the magnetic field, as the current is increasing, then transfers energy back to the circuit as the current decreases. The instantaneous power oscillates between positive and negative, but an inductor’s average power is zero: $P_{\text{L}}=0.$