Capacitors are important elements in electric circuits because of their ability to store energy. FIGURE 26.27 shows a capacitor being charged. The instantaneous value of the charge on the two plates is $\pm q$, and this charge separation has established a potential difference $\text{Δ}V=q\text{/}C$ between the two electrodes.

FIGURE 26.27 The charge escalator does work on charge dq as the capacitor is being charged.

An additional charge dq is in the process of being transferred from the negative to the positive electrode. The battery’s charge escalator must do work to lift charge dq “uphill” to a higher potential. Consequently, the potential energy of $dq+\text{capacitor}$ increases by

The total energy transferred from the battery to the capacitor is found by integrating Equation 26.23 from the start of charging, when $q=0$, until the end, when $q=Q$. Thus we find that the energy stored in a charged capacitor is

In practice, it is often easier to write the stored energy in terms of the capacitor’s potential difference $\text{Δ}{V}_{\text{C}}=Q\text{/}C$. This is

The potential energy stored in a capacitor depends on the square of the potential difference across it. This result is reminiscent of the potential energy $U=\frac{1}{2}k{\left(\text{Δ}x\right)}^2$ stored in a spring, and a charged capacitor really is analogous to a stretched spring. A stretched spring holds the energy until we release it, then that potential energy is transformed into kinetic energy. Likewise, a charged capacitor holds energy until we discharge it. Then the potential energy is transformed into the kinetic energy of moving charges (the current).

The usefulness of a capacitor stems from the fact that it can be charged slowly (the charging rate is usually limited by the battery’s ability to transfer charge) but then can release the energy very quickly. A mechanical analogy would be using a crank to slowly stretch the spring of a catapult, then quickly releasing the energy to launch a massive rock.

The capacitor described in Example 26.8 is typical of the capacitors used in the flash units of cameras. The camera batteries charge a capacitor, then the energy stored in the capacitor is quickly discharged into a flashlamp. The charging process in a camera takes several seconds, which is why you can’t fire a camera flash twice in quick succession.

An important medical application of capacitors is the defibrillator. A heart attack or a serious injury can cause the heart to enter a state known as fibrillation in which the heart muscles twitch randomly and cannot pump blood. A strong electric shock through the chest completely stops the heart, giving the cells that control the heart’s rhythm a chance to restore the proper heartbeat. A defibrillator has a large capacitor that can store up to 360 J of energy. This energy is released in about 2 ms through two “paddles” pressed against the patient’s chest. It takes several seconds to charge the capacitor, which is why, on television medical shows, you hear an emergency room doctor or nurse shout “Charging!”