Derive the expression for energy stored in a charged capacitor.

Explanation

The work done to move a small charge dq from one plate to another plate is
`dW=vdq=q/C dq`
The total work done in transferring total charge Q
`W=int_0^Q dW=int_0^Q q/C dq`
`=Q^2/(2C)=1/2 Q^2/C`
This work done is stored as energy U in the charged capacitor
`U=1/2 Q^2/C =1/2QV=1/2CV^2`
Let .C. be the capacitance of a parallel plate capaitor . By definition `C=Q/V` . By defintion amount of work done to raise the charge by .dq. is dQ =Vdq.
i.e., `dW=q/C dq`
where .q. is the instantaneous value of charge on the conductor . Total work done in charging the capacitor is given by,
`W=int_(q=0)^(q=Q)dW`
i.e., `W=int_(q=0)^(q=Q)(q/C)dq`
i.e., `W=1/C [q^2/2]_0^Q`
or `W=1/2 Q^2/C` , amount of work done is stored in the form of energy.
Therefore , `E=1/2(Q^2/C)`
Using Q=CV, `E=1/2CV^2` , where .V. is the maximum voltage supplied to a capacitor .