Obtain the expression for the energy stored in a capacitor connected across a dc battery. Hence define energy density of the capacitor.

Explanation

Let at a particular instant charge on the plate of capacitor be q and its potential difference be `(q)/(C )` . If an additional charge dg is given to the capacitor plate, work done for it is given by
`dW = (q)/(C ).qq`
Therefore, whole process of charging from 0 to Q requires a work
`W= int_(0)^(Q)(qdq)/(C )=(1)/(C) [(q^(2))/(2)]_(0)^(Q)=(Q^(2))/(2C)`
This work done is stored as the electrostatic potential energy U of the charged capacitor. Moreover, Q=CV, where V be the potential difference between the plates of capacitor, hence
`U= (Q^(2))/(2C)= (1)/(2)QV=(1)/(2)CV^(2)`
Energy per unit volume of the capacitor is called its energy density u. It can be easily shown that u
`=(1)/(2)in_(0)E^(2)` where E = electric field between the plates of capacitor.