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.
(a) Determine the capacitance of the capacitor.
(b) What is the potential of the 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.

Solution

Given, a spherical capacitor consisting of two concentric sphere.

Radius of inner sphere-r₁= 12 cm=

12 \times 10^{- 2} m

Radius of ouer sphere-r₂= 13 cm =

13 \times 10^{- 2} m

Charge on inner sphere-q=

2.5 μC = 2.5 \times 10^{- 6} C

Dielctric constant of liquid-k = 32

(a)Using the formula,

C = {kC}_{0} C = k . 4 {πε}_{0} \frac{r_{1} r_{2}}{r_{2} - r_{1}}

= \frac{32 \times 13 \times 10^{- 2} \times 12 \times 10^{- 2}}{9 \times 10^{9} (13 \times 10^{- 2} - 12 \times 10^{- 2})} [∵ 4 {πε}_{0} = \frac{1}{9 \times 10^{9}}]

= \frac{32 \times 13 \times 12}{9} \times 10^{- 11} = \frac{1644}{3} \times 10^{- 11} = 5.54 \times 10^{- 9} F

(b) Potential of inner sphere,

V = \frac{q}{C} = \frac{2.5 \times 10^{- 6}}{5.54 \times 10^{- 9}} = 4.5 \times 10^{2} V

C = 4 {πε}_{0} = \frac{12 \times 10^{- 2}}{9 \times 10^{9}} = 1.33 \times 10^{- 11} F

The potential in case of two concentric spheres is distributed over both the spheres. This implies, the potential difference between the two concentric spheres becomes smaller. And, because the capacitance is inversely proportional to the potential difference (C=

\frac{Q}{V}

), the capacitance of two concentric spheres is much larger than the capacitance due to an isolated sphere.

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