Question

The energy stored in a capacitor of capacitance $C$ having a charge $Q$ under a potential $V$ is

• A. $\frac{1}{2}{Q}^{2}V$
• B. $\frac{1}{2}{C}^{2}V$
• C. $\frac{1}{2}\frac{Q^2}{V}$
• D. $\frac{1}{2}QV$
• E. $\frac{1}{2}CV$

Answer 1

Answer: (D)

By energy stored in a charged conductor Suppose, a conductor of capacity $C$ is charged to a potential $V$ and $Q$ the charge on conductor at a instant.
The potential of the conductor
$V=\frac{Q}{C}$
Now, work done in bringing a small charge $dQ$ at this potential is,
$dW=VdQ=\frac{Q}{C}dQ$
$\therefore$ Total work done in charging it from $0$ to $Q$ is,
$W=\underset{0}{\overset{Q}{\int }}dW=\underset{0}{\overset{Q}{\int }}\frac{Q}{C}dQ$
$=\frac{1}{2}\frac{Q^2}{C}$
This work is stored as the potential energy,
$U=\frac{1}{2}\frac{Q^2}{C}(\because Q=VC)$
$U=\frac{1}{2}C{V}^2=\frac{1}{2}QV$