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

Answer 1

Answer: (C)

The correct option is D $\frac{1}{2}QV$
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,
$W={\int }_0^{Q}dW={\int }_0^Q\frac{Q}{C}dQ$
$=\frac{1}{2}\frac{Q^2}{C}$
This work is stored as the potential energy.
$U=\frac{1}{2}\frac{Q^2}{C}$ $(\because Q=VC)$
$U=\frac{1}{2}C{V}^2=\frac{1}{2}QV$