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