Energy stored in the capacitor

A capacitor stores not only charge but also electrical energy. As a battery charges a capacitor, it transfers charge from one plate to the other in a series of small increments; at the moment when a charge q has already accumulated on the plates, the potential difference between them is q/C, so moving the next small increment of charge dq across requires a small amount of work dW = (q/C) dq. Summing (integrating) this small work over the whole charging process, from an initially uncharged state (q = 0) up to the capacitor's final charge Q, gives the total work done by the battery, which is stored in the capacitor as electrostatic potential energy: U = the integral from 0 to Q of (q/C) dq = Q^2/(2C). Using the relation Q = CV, this same result can equally be written as U = (1/2) C V^2, or as U = (1/2) Q V -- three equivalent forms of the same stored energy, useful in different situations depending on which of Q, V, or C is most directly known. This stored energy can also be thought of as residing in the electric field that fills the space between the capacitor's plates, rather than in the plates themselves; dividing the total energy by the volume of that field region (Ad, for a parallel-plate capacitor) gives an energy …