# 19.7 Energy stored in capacitors

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## Learning objectives

By the end of this section, you will be able to:

• List some uses of capacitors.
• Express in equation form the energy stored in a capacitor.
• Explain the function of a defibrillator.

The information presented in this section supports the following AP® learning objectives and science practices:

• 5.B.2.1 The student is able to calculate the expected behavior of a system using the object model (i.e., by ignoring changes in internal structure) to analyze a situation. Then, when the model fails, the student can justify the use of conservation of energy principles to calculate the change in internal energy due to changes in internal structure because the object is actually a system. (S.P. 1.4, 2.1)
• 5.B.3.1 The student is able to describe and make qualitative and/or quantitative predictions about everyday examples of systems with internal potential energy. (S.P. 2.2, 6.4, 7.2)
• 5.B.3.2 The student is able to make quantitative calculations of the internal potential energy of a system from a description or diagram of that system. (S.P. 1.4, 2.2)
• 5.B.3.3 The student is able to apply mathematical reasoning to create a description of the internal potential energy of a system from a description or diagram of the objects and interactions in that system. (S.P. 1.4, 2.2)

Most of us have seen dramatizations in which medical personnel use a defibrillator    to pass an electric current through a patient’s heart to get it to beat normally. (Review [link] .) Often realistic in detail, the person applying the shock directs another person 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 situation. SI units of joules are often employed. Less dramatic is the use of capacitors in microelectronics, such as certain handheld calculators, to supply energy when batteries are charged. (See [link] .) Capacitors are also used to supply energy for flash lamps on cameras.

Energy stored in a capacitor is electrical potential energy, and it is thus related to the charge $Q$ and voltage $V$ on the capacitor. We must be careful when applying the equation for electrical potential energy $\text{Δ}\text{PE}=q\text{Δ}V\phantom{\rule{0.25em}{0ex}}$ to a capacitor. Remember that $\text{Δ}\text{PE}$ is the potential energy of a charge $q$ going through a voltage $\text{Δ}V$ . But the capacitor starts with zero voltage and gradually comes up to its full voltage as it is charged. The first charge placed on a capacitor experiences a change in voltage $\text{Δ}V=0$ , since the capacitor has zero voltage when uncharged. The final charge placed on a capacitor experiences $\text{Δ}V=V$ , since the capacitor now has its full voltage $V$ on it. The average voltage on the capacitor during the charging process is $V/2$ , and so the average voltage experienced by the full charge $q$ is $V/2$ . Thus the energy stored in a capacitor, ${E}_{\text{cap}}$ , is

${E}_{\text{cap}}=\frac{QV}{2},$

where $Q$ is the charge on a capacitor with a voltage $V$ applied. (Note that the energy is not $\text{QV}$ , but $\text{QV}/2$ .) Charge and voltage are related to the capacitance $C$ of a capacitor by $Q=\text{CV}$ , and so the expression for ${E}_{\text{cap}}$ can be algebraically manipulated into three equivalent expressions:

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