Question

A cylindrical capacitor has charge $Q$ and length $L$. If both the charge and length of the capacitor are doubled, by keeping other parameters fixed, the energy stored in the capacitor

• A. remains same
• B. increases two times
• C. decreases two times
• D. increases four times

Answer 1

Answer: (B)

Energy of a charged capacitor, $E=\frac{1}{2}\frac{Q^2}{C}$
$C=\frac{2\pi {\epsilon }_{0}L}{{\log }_e\left(\frac{b}{a}\right)}$
$E=\frac{1}{2}\frac{Q^2}{2\pi {\epsilon }_{0}L}{\log }_e\left(\frac{b}{a}\right)....(i)$
for a cylindrical capacitor.
where $L=$ length of the cylinders
$a$ and $b=$ radii of two concentric cylinders
$C^{{}^{\prime }}=\frac{2\pi {\epsilon }_0(2L)}{{\log }_e\left(\frac{b}{a}\right)}$
$E^{{}^{\prime }}=\frac{1}{2}\frac{(2Q{)}^2}{C^{}}$
$=\frac{1}{2}\frac{(2Q{)}^2}{2\pi {\epsilon }_0(2L)}{\log }_e\left(\frac{b}{a}\right)...(ii)$
From Eqs. (i) and (ii), we get
$E^{{}^{\prime }}=2E$