Potential Energy of a Charged Conductor

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

$U=\frac{1}{2} \frac{Q^{2}}{C}$

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

$U=\frac{1}{2} Q V$