Derive an expression for the energy stored in a charged parallel plate 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 dq is given to the capacitor plate, work done for it is given by
`dW=(q/C).dq`
Therefore, whole process of charging from 0 to Q requires a work
`W = int_0^Q (qdp)/C = 1/C [q^2/2]_0^Q = Q^2/(2C)`
This work done is stored as the electrostatic potential energy of the charged capacitor. Hence, potential energy of charged capacitor
`u = Q^2/(2C)`
But Q=CV, where V be the potential difference between the plates of capacitur, hence
`u = (Q^2)/(2C) =1/2QV =1/2 CV^2`