Question

Derive an expression for energy stored in a charged condenser. Obtain its different forms.

Answer 1

Let '$C$' is the capacitance of the capacitor. Let '$Q$' and '$V$' be the charge and potential difference respectively. Let '$q$' be the charge on the capacitor at some stage during charging and '$v$' is the corresponding potential difference between the plates.
so, $dw$$=$$Vdq$
but $C=\frac{q}{v}$
so $dw=\frac{q}{C}dq$
Total work done in charging the capacitor is given by
$w=\int dw={\int }_0^Q\frac{qdq}{C}=\frac{1}{C}.{\left[\frac{q^2}{2}\right]}_0^Q$
This work done is stored in the form of Potential Energy, so energy of a charged condenser

$u=\frac{Q^2}{2C}$

but $Q=CV$
so

$u=\frac{1}{2}C{V}^2$

or

$u=\frac{1}{2}QV$