A spherical capacitor has an inner sphere of radius 12 cm and an outer sphere of radius 13 cm.

Solution:

Radius of the inner sphere, $r_{2}=12 \mathrm{~cm}=0.12 \mathrm{~m}$

Radius of the outer sphere, $r_{1}=13 \mathrm{~cm}=0.13 \mathrm{~m}$

Charge on the inner sphere, $q=2.5 \mu \mathrm{C}=2.5 \times 0^{-6} \mathrm{C}$

Dielectric constant of a liquid, $\in_{r}=32$

(a) Capacitance of the capacitor is given by the relation,

$C=\frac{4 \pi \in_{0} \in_{r} r_{1} r_{2}}{r_{1}-r_{2}}$

Where,

$\epsilon_{0}=$ Permittivity of free space $=8.85 \times 10^{-12} \mathrm{C}^{2} \mathrm{~N}^{-1} \mathrm{~m}^{-2}$

$\frac{1}{4 \pi \epsilon_{0}}=9 \times 10^{9} \mathrm{Nm}^{2} \mathrm{C}^{-2}$

$\therefore C=\frac{32 \times 0.12 \times 0.13}{9 \times 10^{9} \times(0.13-0.12)}$

$\approx 5.5 \times 10^{-9} \mathrm{~F}$

Hence, the capacitance of the capacitor is approximately $5.5 \times 10^{-9} \mathrm{~F}$.

(b) Potential of the inner sphere is given by,

$V=\frac{q}{C}$

$=\frac{2.5 \times 10^{-6}}{5.5 \times 10^{-9}}=4.5 \times 10^{2} \mathrm{~V}$

Hence, the potential of the inner sphere is $4.5 \times 10^{2} \mathrm{~V}$.

(c) Radius of an isolated sphere, r = 12 × 10−2 m

Capacitance of the sphere is given by the relation,

$C^{\prime}=4 \pi \in_{0} r$

$=4 \pi \times 8.85 \times 10^{-12} \times 12 \times 10^{-12}$

$=1.33 \times 10^{-11} \mathrm{~F}$

The capacitance of the isolated sphere is less in comparison to the concentric spheres. This is because the outer sphere of the concentric spheres is earthed. Hence, the potential difference is less and the capacitance is more than the isolated sphere.