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

Answer: (A)

Capacitance of isolated Conducting sphere
$=4 \pi \varepsilon_0 R_1$
By enclosing inside another sphere of radius
$R_2 \text {, new capacitance }=\frac{4 \pi \varepsilon_0 R_1 R_2}{\left(R_2-R_1\right)}$
Given : $\frac{4 \pi \varepsilon_0 R_1 R_2}{\left(R_2-R_1\right)}=n \times 4 \pi \varepsilon_0 R_1$
$ \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$
$\Rightarrow \frac{ R _2}{ R _1}= n \frac{ R _2}{ R _1}- n \Rightarrow \frac{ R _2}{ R _1}=\frac{ n }{( n -1)}$