A spherical capacitor has an inner sphere of radius $12 \;cm$ and an outer sphere of radius $13\; cm .$ The outer sphere is earthed and the inner sphere is given a charge of $2.5\; \mu \,C .$ The space between the concentric spheres is filled with a liquid of dielectric constant $32$

$(a)$ Determine the capacitance of the capacitor.

$(b)$ What is the potential of the inner sphere?

$(c)$ Compare the capacitance of this capacitor with that of an isolated sphere of radius $12 \;cm .$ Explain why the latter is much smaller.

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

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

Charge on the inner sphere, $q=2.5\, \mu\, C=2.5 \times 10^{-6}\, C$

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

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

$C=\frac{4 \pi \epsilon_{0} \epsilon_{r} r_{1} r_{2}}{r_{1}-r_{2}}$ Where,

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

$\frac{1}{4 \pi \epsilon_{0}}=9 \times 10^{9} \,N\, m ^{2}\, C ^{-2}$

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

$=5.5 \times 10^{-9}\, F$

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

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

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

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

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

$(c)$ Radius of an isolated sphere, $r=12 \times 10^{-2} \,m$

Capacitance of the sphere is given by the relation, $C^{\prime}=4 \pi \in_{0} r$

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

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

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.