The capacitance (C) of an isolated conducting | vedclass

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(B) The capacitance of an isolated conducting sphere of radius $a$ is $C_0 = 4\pi \varepsilon_0 a$.When this sphere is enclosed by an earthed concentr

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

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(B) The capacitance of an isolated conducting sphere of radius $a$ is $C_0 = 4\pi \varepsilon_0 a$.When this sphere is enclosed by an earthed concentric sphere of radius $b$ $(b > a)$,the system forms a spherical capacitor.The capacitance of a spherical capacitor is given by $C = 4\pi \varepsilon_0 \left( \frac{ab}{b-a} \right)$.We can rewrite this as $C = 4\pi \varepsilon_0 a \left( \frac{b}{b-a} \right)$.Dividing the numerator and denominator by $a$,we get $C = 4\pi \varepsilon_0 a \left( \frac{b/a}{b/a - 1} \right)$.Given the ratio $\frac{b}{a} = n$,we substitute this into the expression:$C = (4\pi \varepsilon_0 a) \left( \frac{n}{n-1} \right) = C_0 \left( \frac{n}{n-1} \right)$.Thus,the capacitance is increased by a factor of $\frac{n}{n-1}$.

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