Definition:
The work done in charging the conductor is stored as potential energy in the electric field in the vicinity of the conductor is called the potential energy of a charged conductor.
Derivation →
Let us consider, a conductor of capacitance $C$. If charge $+Q$ is given into small amount of $dq$ to the surface of the conductor. Then the work done will be
$dw=V\:dq \qquad(1) $
$dw=\frac{q}{C} \:dq
\qquad \left (\because V=\frac{q}{C} \right)$
Therefore, as the amount of charge on the conductor will increase from $0$ to $Q$ that causes also increase in work done. So integrate the above equation for total work done
$ \int_{0}^{W} dw=\frac{1}{C} \int_{0}^{Q} q dq$
$W=\frac{1}{C} \int_{0}^{Q} q dq$
$W=\frac{1}{C} \left[ \frac{q^{2}}{2} \right]^{Q}_{0}$
$W=\frac{1}{2} \frac{Q^{2}}{C}$
This work is stored in the form of electric potential energy $U$. Then
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$U=\frac{1}{2} \frac{Q^{2}}{C}$
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