Derive the expression for energy stored in a charged capacitor.

Explanation

The energy stored in the charged capacitor is the total work done in charging the capacitor to a given potential, by transferring charges from one plate to another plate of the capacitor.

Consider an intermediate situation of charging the capacitor, Let `q rarr` total charge on capacitor at the intermediate situation and `V. rarr` potential difference between the two plates of capacitor so that
Now, the small amount of work done in transferring an additional charge dq from the negative plate to the positive plate is given by.
`dW= V.dq= (q)/(C )dq`
Therefore the total work done in transferring charge from 0 to Q is given by, `W= int_(0)^(Q) (q)/(C ) dq= (1)/(C ) int_(0)^(Q) q dq = (1)/(C ) [(q^(2))/(2)]_(0)^(Q)= (1)/(C )[(Q^(2))/(2)-0]`
`W = (Q^(2))/(2C)`.
This work is stored as electrostatic potential energy U in the capacitor.
`therefore U= (Q^(2))/(2C)`
Also, `U= (Q^(2))/(2C)= (1)/(2) QV= (1)/(2)CV^(2)`