Question

The capacitance of a cylindrical capacitor of length $72\text{mm}$ is $2\text{pF}$. If the ratio of radii of its outer cylindrical conductor to that of its inner cylindrical conductor is $e^x$, then $x$ is

• A. $2$
• B. $4$
• C. $5$
• D. $3$

Answer 1

The correct option is A $2$

Let inner radius be $R_1$ and outer radius be $R_2$

Given:
$l=72\text{mm}=72\times {10}^{-3}\text{m}$


For a cylindrical capacitor

$C=\frac{2\pi {ϵ}_{0}l}{\ln \left(\frac{R_2}{R_1}\right)}$

$\Rightarrow \ln \frac{R_2}{R_1}=\frac{2\pi {ϵ}_{0}l}{C}=\frac{4\pi {ϵ}_{0}l}{2C}$

$\Rightarrow \ln \frac{R_2}{R_1}=\frac{72\times {10}^{-3}}{9\times {10}^9\times 2\times 2\times {10}^{-12}}$

$\Rightarrow \ln \frac{R_2}{R_1}=2$

$\because \ln a=b\Rightarrow a=e^b$

So, $\frac{R_2}{R_1}=e^2$

$\therefore x=2$

Hence, option $\left(b\right)$ is correct.