Derive the expression for energy stored in a charged capacitor.
To find the electrical energy stored in a capacitor, we calculate the work required to change it. Suppose charge Q′ has been deposited on the conducting plates of the capacitor, and with a constant potential difference of V′=Q′C. To add more charge dQ′, the work needed is the potential energy gained by the charge dQ′, which is dQ′×V′. Hence
dW=dQ′×V′=Q′×dQ′C.
On integrating we get
U=0∫UdW
=0∫QQ′×dQ′C
=Q22C=12CV2