question_answer

Capacity of an isolated sphere is increased n times when it is enclosed by an earthed concentric sphere. The ratio of their radii is-

A) \[\frac{{{n}^{2}}}{n-1}\]                    

B) \[\frac{n}{n-1}\]

C) \[\frac{2n}{n+1}\]                    

D) \[\frac{2n+1}{n+1}\]

Correct Answer: B

Solution :

\[{{C}_{1}}\,\,=\,\,4\pi {{\varepsilon }_{0}}{{R}_{1}}\] \[and\,\,\,\,\,\,\,{{C}_{2}}\,\,=\,\,4\pi {{\varepsilon }_{0}}\,\left( \frac{{{R}_{1}}{{R}_{2}}}{{{R}_{2}}-{{R}_{1}}} \right)\] Given that \[{{C}_{2}}=n{{C}_{1}}\] \[or\,\,\,\,\,\frac{{{R}_{2}}{{R}_{1}}}{{{R}_{2}}-{{R}_{1}}}=n{{R}_{1}}\] \[or\,\,\,\,\,\frac{{{R}_{2}}/{{R}_{1}}}{{{R}_{2}}/{{R}_{1}}-1}=n\] \[or\,\,\,\,\frac{{{R}_{2}}}{{{R}_{1}}}=\frac{n}{n-1}\]