2.30 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 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.

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alok kumar singh | Contributor-Level 10

3 months ago

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

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

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

Dielectric constant of the liquid, k = 32

Capacitance of the capacitor is given by the relation, C = $\frac{4 π ε_{0} {k r}_{1} r_{2}}{(r_{1} - r_{2})}$

$ε_{0}$ = permittivity of free space = 8.854 $\times 10^{- 12}$ $C^{2} N^{- 1}$ $m^{- 2}$

$C =$ $\frac{4 π \times 8.854 \times 10^{- 12} \times 32 \times 0.13 \times 0.12}{(0.13 - 0.12)}$ = 5.55 $\times 10^{- 9}$ F

Potential of the inner surface is given by

V = $\frac{q}{C}$ = $\frac{2.5 \times 10^{- 6}}{5.55 \times 10^{- 9}}$ = 450 V

Radius of the isolate sphere, r = 12 cm = 12 $\times 10^{- 2}$ m

Capacitance on the isolated sphere is given by C’ = 4 $π ε_{0}$ r

= 4 $\times π \times$ 8.854 $\times 10^{- 12} \times 12 \times 10^{- 2}$ F

= 1.34 $\times 10^{- 11}$ F

The capacitance of the iso

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

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

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

Dielectric constant of the liquid, k = 32

Capacitance of the capacitor is given by the relation, C = $\frac{4 π ε_{0} {k r}_{1} r_{2}}{(r_{1} - r_{2})}$

$ε_{0}$ = permittivity of free space = 8.854 $\times 10^{- 12}$ $C^{2} N^{- 1}$ $m^{- 2}$

$C =$ $\frac{4 π \times 8.854 \times 10^{- 12} \times 32 \times 0.13 \times 0.12}{(0.13 - 0.12)}$ = 5.55 $\times 10^{- 9}$ F

Potential of the inner surface is given by

V = $\frac{q}{C}$ = $\frac{2.5 \times 10^{- 6}}{5.55 \times 10^{- 9}}$ = 450 V

Radius of the isolate sphere, r = 12 cm = 12 $\times 10^{- 2}$ m

Capacitance on the isolated sphere is given by C’ = 4 $π ε_{0}$ r

= 4 $\times π \times$ 8.854 $\times 10^{- 12} \times 12 \times 10^{- 2}$ F

= 1.34 $\times 10^{- 11}$ F

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

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