Question

A spherical capacitor has an inner sphere of radius 12 cm and anouter sphere of radius 13 cm. The outer sphere is earthed and theinner sphere is given a charge of 2.5 μC. The space between theconcentric 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 anisolated sphere of radius 12 cm. Explain why the latter is muchsmaller.

Answer 1

Given: The radius of inner sphere of a spherical capacitor is 12 cm, the radius of outer sphere is 13 cm, the charge on the inner sphere is 2.5 μC and the dielectric constant of a liquid is 32.

a)

The capacitance of the capacitor is given as,

C= 4π ε 0 ε r r 1 r 2 ( r 1 − r 2 )

Where, the radius of the inner sphere is r 2 , the radius of the outer sphere is r 1 , the dielectric constant of a liquid ε r , the permittivity of free space is ε 0 .

By substituting the given values in the above formula, we get

C= 32×0.12×0.13 9× 10 9 ×( 0.13−0.12 ) =5.5× 10 −9  F

Thus, the capacitance of the capacitor is 5.5× 10 −9  F

b)

The potential of the inner sphere is given as,

V= q C

Where, the charge on the inner sphere is q.

By substituting the given values in the above formula, we get

V= 2.5× 10 −6 5.5× 10 −9 =4.5× 10 2  V

Thus, the potential of the inner sphere is 4.5× 10 2  V.

c)

Given: The radius of an isolated sphere is 12 cm.

Since, 1 m=100 cm

The capacitance of the sphere is given as,

C ′ =4π ε 0 r

Where, the radius of an isolated sphere is r.

By substituting the given values in the above equation, we get

C=4π( 8.85× 10 −12 )( 12× 10 −2 ) =1334.5× 10 −14 =1.33× 10 −11  F

Thus, the capacitance of the isolated sphere is less as compared to the concentric spheres. This is due to the outer sphere of the capacitance is more than the isolated sphere so the potential difference is less and the capacitance is more than the isolated sphere.