Derive the expression for energy stored in a charged capacitor.

Explanation

Consider a capacitor of capacitance C. At any intermediate stage of charge, Let`Q^(1)`and `-Q^(1)` be the charge on positive and negative plate respectively and V. be the potential difference across the plates. The work done dW to move small charge `dQ^(1)` from -ve plate to +ve plate is
`dW=V^(1)dQ^(1)`
But `C=(Q)/(V^(1))impliesV^(1)=(Q)/(C)`
`therefore dW=(Q)/(C)dQ^(1)`
The total work done to transfer Q charge from -ve to +ve plate
`W=int_(0)^(0)dQ=int_(0)^(0)(Q)/(C)dQ[int QdQ=(Q^(2))/(2)]`
`W=(Q^(2))/(2C)=(1)/(2)(Q^(2))/(C)`
The work don is stored as energy U in charged capacitor.
Therefore ,`U=(1)/(2)(Q^(2))/(C)=(1)/(2)QV=(1)/(2)CV^(2)`