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(A) The capacitance of an isolated sphere of radius $r_1$ is given by $C_1 = 4 \pi \varepsilon_0 r_1$.When this sphere is enclosed by an earthed conce

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$\frac{4}{5}$

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(A) The capacitance of an isolated sphere of radius $r_1$ is given by $C_1 = 4 \pi \varepsilon_0 r_1$.When this sphere is enclosed by an earthed concentric sphere of radius $r_2$,the capacitance becomes $C_2 = 4 \pi \varepsilon_0 \left( \frac{r_1 r_2}{r_2 - r_1} \right)$.According to the problem,$C_2 = 5 C_1$.Substituting the expressions,we get $4 \pi \varepsilon_0 \left( \frac{r_1 r_2}{r_2 - r_1} \right) = 5 (4 \pi \varepsilon_0 r_1)$.Canceling $4 \pi \varepsilon_0 r_1$ from both sides,we get $\frac{r_2}{r_2 - r_1} = 5$.This implies $r_2 = 5(r_2 - r_1) = 5r_2 - 5r_1$.Rearranging the terms,$4r_2 = 5r_1$.Therefore,the ratio $\frac{r_1}{r_2} = \frac{4}{5}$.

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 capacitance of an isolated sphere of radius $r_1$ is increased by $5$ times,when it is enclosed by an earthed concentric sphere of radius $r_2$. The ratio of their radii is

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