Question

The capacitance $\left(C\right)$ for an isolated conducting sphere of radius $\left(a\right)$ is given by $4\pi {\epsilon }_{0}a$. This sphere is enclosed within an earthed concentric sphere. The ratio of the radii of the spheres being $\frac{n}{(n-1)}$, then the capacitance of such will be increased by a factor

• A. $an$
• B. $\frac{(n-1)}{n}$
• C. $\frac{n}{(n-1)}$
• D. $n$

Answer 1

The correct option is D $n$

Capacitance of concentric spheres having outer and inner radii as b and a is given as

$\frac{4\pi {\epsilon }_{0}ab}{b-a}$

In the given condition, let $b$ be the radius of outer sphere

$C=\frac{4\pi {\epsilon }_{0}ab}{b-a}=\frac{4\pi {\epsilon }_{0}a}{1-\frac{a}{b}}=\frac{4\pi {\epsilon }_{0}a}{1-\left(\frac{n-1}{n}\right)}$
$=n\left(4\pi {\epsilon }_{0}a\right)$

Thus, resultant capacitance is increased by the factor $\left(n\right)$
Hence, option (a) is correct.