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  4. A capacitor of capacitance C is charged to potential V. If it carries a charge Q, then the energy stored in it is

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Q. A capacitor of capacitance C is charged to potential V. If it carries a charge Q, then the energy stored in it is

KEAMKEAM 2009Electrostatic Potential and Capacitance

Solution:

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 o to $Q$ is,
$
\begin{array}{l}
W =\int_{0}^{ Q } dW =\int_{0}^{ Q } \frac{ Q }{ C } dQ \\
=\frac{1}{2} \frac{ Q ^{2}}{ C }
\end{array}
$
This work is stored as the potential energy.
$
U =\frac{1}{2} \frac{ Q ^{2}}{ C } (\because Q = VC )
$
$
U =\frac{1}{2} CV ^{2}=\frac{1}{2} QV
$