Energy Stored in a Capacitor

Charging a capacitor means gradually building up charge on its plates against the electrostatic field of the charge already present -- and, exactly as with any electrostatic process, the work done in that process is not lost, but stored as genuine electrostatic potential energy in the charged capacitor, available to be released again (as, for instance, light and heat) when the capacitor is later discharged.

Deriving the energy formula. Suppose a capacitor of capacitance $C$ is being charged gradually, and at some intermediate instant it already carries a charge $q$ (so the potential difference across it, at that instant, is $q/C$). Transferring a further small increment of charge $dq$ from one plate to the other, against this existing potential difference, requires an incremental amount of work

The total work done charging the capacitor from $q=0$ all the way up to its final charge $Q$ is found by integrating this expression over the whole charging process:

By definition, this work done IS the electrostatic potential energy $U$ stored in the fully charged capacitor. Using $Q = CV$, this same result can equally be written in two other, equally common, equivalent forms:

Any one of these three forms may be the most convenient depending on which two of $Q$, $C$, $V$ happen to be known in a given problem.

Energy density. For the specific case of a parallel plate capacitor (Section 2.13), the volume of space between the plates, where essentially the entire field is confined, is $\text{Volume} = Ad$. Substituting $C = \epsilon_0 A/d$ and $V = Ed$ into $U = \tfrac{1}{2}CV^2$,

so the electrostatic energy PER UNIT VOLUME stored between the plates is

$u = \frac{U}{\text{Volume}} = \frac{1}{2}\epsilon_0 E^2$ …