Question

Capacitance of an isolated conducting sphere of radius $R_1$ becomes $n$ times when it is enclosed by a concentric conducting sphere of radius $R_2$ connected to earth The ratio of their radii $\left(\frac{ R _2}{ R _1}\right)$ is:

• A. $\frac{n}{n-1}$
• B. $\frac{2 n}{2 n+1}$
• C. $\frac{ n +1}{ n }$
• D. $\frac{2 n+1}{n}$

Answer 1

The Correct Option is A

Capacitance of isolated Conducting sphere
$=4\pi {\epsilon }_0{R}_1$
By enclosing inside another sphere of radius
$R_2\text{, new capacitance}=\frac{4\pi {\epsilon }_0{R}_1{R}_2}{\left(R_2-R_1\right)}$
Given : $\frac{4\pi {\epsilon }_0{R}_1{R}_2}{\left(R_2-R_1\right)}=n\times 4\pi {\epsilon }_0{R}_1$
$\Rightarrow \frac{R_2}{\left(R_2-R_1\right)}=n\Rightarrow \frac{\frac{R_2}{R_1}}{\left(\frac{R_2}{R_1}-1\right)}=n$
$\Rightarrow \frac{R_2}{R_1}=n\frac{R_2}{R_1}-n\Rightarrow \frac{R_2}{R_1}=\frac{n}{(n-1)}$