17 Energy Stored in Capacitors

Most of us have seen dramatizations in which medical personnel use a defibrillator to pass an electric current through a patient’s heart to restore a normal rhythm. (Review Figure 17.2.) Often, the person operating the device instructs another to “make it 400 joules this time.” The energy delivered by the defibrillator is stored in a capacitor and can be adjusted to fit the patient's needs. The SI unit of energy is the joule (J).

Less dramatic applications of capacitors are found in microelectronics, where they temporarily supply energy to preserve memory in devices such as calculators while batteries are being replaced (see Figure 17.1). Capacitors also power camera flash units and many other electronic devices. In every application, the underlying principle is the same: a capacitor stores separated electric charge, and because separated charges possess electric potential energy, the capacitor can store energy and release it rapidly when needed.

Energy stored in a capacitor is a form of electrical potential energy, so it is naturally related to the charge [latex]Q[/latex] stored on the capacitor and the voltage [latex]V[/latex] across its plates. However, we must be careful when applying the familiar expression for electric potential energy, [latex]\Delta PE=q\Delta V[/latex]. That equation describes the energy change of a specific amount of charge moving through a fixed potential difference. During the charging of a capacitor, the potential difference is not constant—it increases continuously as charge accumulates.

A capacitor begins uncharged, so its voltage is initially zero. As more charge is transferred onto the plates, the electric field between them grows stronger and the voltage rises. In other words, the "electrical hill" that later charges must climb becomes progressively steeper as the capacitor charges.

One way to visualize this is to imagine adding the charge in many tiny increments. At the beginning, when the capacitor's voltage is very small, very little work is required to move an additional bit of charge onto the plates. Later, when the capacitor already holds substantial charge, each additional bit of charge must be moved against a much larger potential difference. Consequently, the work required to charge the capacitor steadily increases throughout the charging process.

The first charge placed on the capacitor experiences a voltage difference of essentially zero, while the last charge experiences the full voltage [latex]V[/latex]. Since the voltage increases uniformly from 0 to [latex]V[/latex], the average voltage during charging is

The total energy stored therefore equals the total charge multiplied by this average voltage:

where [latex]Q[/latex] is the final charge stored on the capacitor and [latex]V[/latex] is the final voltage across it. The factor of one-half appears because the capacitor is charged gradually—it does not remain at its final voltage throughout the charging process.

A useful analogy is stretching a spring. As the spring is stretched, the required force increases from zero to its final value. Consequently, the energy stored is one-half the product of the final force and displacement. Capacitors behave similarly: the voltage increases from zero to its final value as charge accumulates.

Because charge and voltage are related by

the energy equation can be written in several equivalent forms. Substituting [latex]Q=CV[/latex] into [latex]E_{\text{cap}}=\frac{QV}{2}[/latex] gives

Alternatively, eliminating the voltage using [latex]V=\frac{Q}{C}[/latex] gives

Thus, the energy stored in a capacitor may be expressed as

where [latex]Q[/latex] is the stored charge, [latex]V[/latex] is the voltage across the capacitor, and [latex]C[/latex] is its capacitance. The energy is measured in joules when charge is in coulombs, voltage in volts, and capacitance in farads. Each form is useful in different situations: [latex]\frac{QV}{2}[/latex] emphasizes the charging process, [latex]\frac{CV^2}{2}[/latex] is convenient when the capacitance and voltage are known, and [latex]\frac{Q^2}{2C}[/latex] is useful when the stored charge is known directly.