Question

A spherical capacitor has an inner sphere of radius $12cm$ and an outer sphere of radius $13cm$. 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 $32$.

i) Determine the capacitance of the capacitor.
ii) What is the potential of the inner sphere?
iii) Compare the capacitance of this capacitor with that of an isolated sphere of radius $12cm$. Explain why the latter is much smaller.

Answer 1

i) Given, radius of the inner sphere, $r=12cm=0.12m$
Radius of the outer sphere, $R=13cm=0.13m$
Charge on the inner sphere $q=2.5\mu C=2.5\times {10}^{-6}C$
Dielectric constant of liquid, $K=32$
Capacitance of spherical capacitor,

$C=K4\pi {\epsilon }_0\frac{rR}{R-r}$

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

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


ii) Potential of the inner sphere is,

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

$V=\frac{2.5\times {10}^{-6}}{5.5\times {10}^{-9}}=454V$


iii) For isolated sphere, radius of outer sphere is assumed to be at
infinity.
i.e., $R\to \infty$

Radius of isolated sphere, $r=12cm=0.12m$

Capacitance 1of spherical capacitor,
$C=4\pi {\epsilon }_0\overline{(\frac{1}{r}-\frac{1}{R})}$
$C=4\pi {\epsilon }_0\frac{1}{(\frac{1}{r}-\frac{1}{R})}$

So, capacitance of isolated sphere, $C=4\pi {\epsilon }_{0}r$

$C=\frac{0.12}{9\times {10}^9}=1.33\times {10}^{-11}F$

Since outer sphere of the concentric sphere is earthed, the potential difference is less and as the capacitance is inversely proportional to potential difference, the capacitance is more than the isolated sphere.