Question

The capacitance (C) for an isolated conducting sphere of radius (a) is given by $4\pi {ϵ}_{o}a$. If the sphere is enclosed with an earthed concentric sphere. The ratio of the radii of the spheres being $\frac{n}{n-1}$ then the capacitance of such a sphere will be increased by a factor.

Answer 1

Capacity of an insulated spherical capacitor of radius $b$ is $4\pi {ϵ}_{0}b$ where there is a system of two shell with outer shell being earthed, the system behaves as a capacitor with capacitance,

$C=4\pi {ϵ}_0\left(\frac{ab}{a-b}\right)=n.4\pi {ϵ}_0$

$\left(\frac{a}{a-b}\right)=n$

$a=na-nb$

$\frac{b}{a}=\frac{n}{n-1}$