Question

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 $2.5 μC$ . The space between the concentric spheres is filled with a liquid of dielectric constant $32$ . Determine the capacitance of the capacitor. Also, what is the potential of the inner sphere. Compare the capacitance of this capacitor with that of an isolated sphere of radius 12 cm. Explain why the latter is much smaller.

Accepted Answer

Given,
The radius of the inner sphere, $r_{2} = 12 cm = 0 .12 m$ .
The radius of the outer sphere, $r_{1} = 13 cm = 0 .13 m$ .
The inner surface has a charge, $q = 2 .5 μC = 2 .5 \times 10^{- 6} C$ .
The liquid has a dielectric constant of $ε_{r} = 32$ .
The capacitance of a spherical capacitor,
$\begin{matrix} C = \frac{4 π ε_{0} ε_{r} r_{1} r_{2}}{r_{1} - r_{2}} = \frac{32 \times 0.12 \times 0.13}{9 \times 10^{9} (0.13 - 0.12)} = 5.5 \times 10^{- 9} \end{matrix} F$
Thus, the capacitance of the capacitor is $5 .5 \times 10^{- 9} F$ .

The potential of the inner sphere is,
$\begin{matrix} V = \frac{q}{C} = \frac{2.5 \times 10^{- 6}}{5.5 \times 10^{- 9}} = 4.5 \times 10^{2} V \end{matrix}$
Thus, the potential of the inner sphere is $4 .5 \times 10^{2} V$ .

Given
The radius of the isolated sphere, $r = 12 cm = 0 .12 m$ .
Capacitance of the sphere is,
$\begin{array}{l} C' = 4 π ε_{0} r \\ = 4 π \times 8.854 \times 1 0^{- 12} \times 0.12 \\ = 1.33 \times 1 0^{- 11} F \end{array}$
For an isolated sphere, the other plate is assumed to be at infinity. Therefore, the distance between the two plates of the capacitor is very large. Since the capacitance is inversely proportional to the distance between the two plates, the capacitance is very less for an isolated sphere.

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