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# 5.10: Energy Stored in a Capacitor

Let us imagine (figure $$V.$$10) that we have a capacitor of capacitance $$C$$ which, at some time, has a charge of $$+q$$ on one plate and a charge of $$-q$$ on the other plate. The potential difference across the plates is then $$q/C$$. Let us now take a charge of $$+\delta q$$ from the bottom plate (the negative one) and move it up to the top plate. We evidently have to do work to do this, in the amount of $$\frac{q}{C}\delta q$$.

$$\text{FIGURE V.10}$$

The total work required, then, starting with the plates completely uncharged until we have transferred a charge $$Q$$ from one plate to the other is

$\frac{1}{C}\int_0^Q q\,dq=Q^2/(2C)\label{5.10.1}\tag{5.10.1}$

This is, then, the energy $$U$$ stored in the capacitor, and, by application of $$Q = CV$$ it can also be written $$U=\frac{1}{2}QV$$, or, more usually,

$U=\frac{1}{2}CV^2\label{5.10.2}\tag{5.10.2}$

Verify that this has the correct dimensions for energy. Also, think about how many expressions for energy you know that are of the form $$\frac{1}{2}ab^2$$. There are more to come.