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(B) Let the radius of the isolated sphere be $a$ and the radius of the earthed outer sphere be $b$.The capacitance of an isolated sphere is given by $

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$\frac{n}{n - 1}$

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(B) Let the radius of the isolated sphere be $a$ and the radius of the earthed outer sphere be $b$.The capacitance of an isolated sphere is given by $C = 4 \pi \epsilon_0 a$.When this sphere is enclosed by an earthed concentric sphere of radius $b$,the new capacitance $C'$ is given by $C' = \frac{4 \pi \epsilon_0 a b}{b - a}$.According to the problem,the capacity is increased $n$ times,so $C' = nC$.Substituting the expressions,we get $n(4 \pi \epsilon_0 a) = \frac{4 \pi \epsilon_0 a b}{b - a}$.Dividing both sides by $4 \pi \epsilon_0 a$,we get $n = \frac{b}{b - a}$.Rearranging the equation: $n(b - a) = b$,which implies $nb - na = b$.Grouping the terms with $b$: $nb - b = na$,or $b(n - 1) = na$.Therefore,the ratio of their radii is $\frac{b}{a} = \frac{n}{n - 1}$.

Capacitor Type	Capacitance Formula
$A$. Cylindrical capacitor	$i$. $4\pi \epsilon_0 R$
$B$. Spherical capacitor	$ii$. $\frac{K A \epsilon_0}{d}$
$C$. Parallel plate capacitor with dielectric	$iii$. $\frac{2\pi \epsilon_0 \ell}{\ln(r_2/r_1)}$
$D$. Isolated spherical conductor	$iv$. $\frac{4\pi \epsilon_0 r_1 r_2}{r_2 - r_1}$

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

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