Prove that the energy stored in a capacitor of capacitance C charged to a potential difference V is `1/2"CV"^(2)`

Explanation

Let us assume that initially both the plates of capacitor be uncharged. Now, we have to repeatedly move small positive charge from one plate to the other. Now, when an additional small charge (dp) is transferred from one plate to another plate, the small work done is given by `dW=V.dp=(q.)/(C)dq`
[Let charge on plate be q when dq charge is transferred]
The total work done in transferring charge (Q) is given by
`W=int_(0)^(Q)(q.)/(C)dq=1/Cint_(0)^(Q)q.dq=1/C[((q.)^(2))/(2)]_(0)^(Q)=(Q^(2))/(2C)`
This work is stored as electrostatic potential energy U in the capacitor, i.e.
`U=(Q^(2))/(2C)=((CV)^(2))/(2C)" "[becauseQ="CV"]`
`U=1/2CV^(2)`