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Capacitance of an isolated conducting sphere of radius $\mathbf{R}_{1}$ becomes $\mathrm{n}$ times when it is enclosed by a concentric conducting sphere of radius $\mathbf{R}_{2}$ connected to earth. The ratio of their radii $\mathrm{\left(\frac{R_{2}}{R_{1}}\right)}$ is :

Option: 1
$\mathrm{\frac{n}{n-1} \\ }$

Option: 2
$\mathrm{\frac{2n}{2n+1}}$

Option: 3
$\mathrm{\frac{n+1}{n}}$

Option: 4
$\mathrm{\frac{2n+1}{n}}$

Answers (1)

best_answer

Case I

Capacitance of an isolated conducting sphere of radius $\mathrm{R_{1}}$

$\mathrm{C_{1}=4\pi \varepsilon _{0}R_{1}}\rightarrow (1)$

Case II

$\mathrm{C_{2}=\frac{4\pi \varepsilon _{0}}{(\frac{1}{R_{1}}-\frac{1}{R_{2}})}}\rightarrow (2)$

It is given that ,

$\mathrm{C_2 =n C_1 }$

$\mathrm{\frac{4 \pi \varepsilon_0}{\frac{1}{R_1}-\frac{1}{R_2}} =n 4 \pi \varepsilon_0 R_1 }$

$\mathrm{\frac{R_1 R_2}{R_2-R_1} =n R_1 }$

$\mathrm{R_2 =n R_2-n R_1 }$

$\mathrm{n R_1 =(n-1) R_2 }$

$\mathrm{ \frac{R_2}{R_1}=\frac{n}{n-1}}$

Hence (1) is correct option

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