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.

Explanation

Here `r_(1) = 12 cm = 12 xx10^(-2)` m
`r_(2) = 13 cm = 13xx10^(-2) m`
`q = 2.5 mu C 2.5 xx10^(-6) ` C
K = 32
(A) Capacitance of spherical capacitor,
`C=(4piinr_(1)r_(2))/(r_(2)-r_(1))=(4pi in_(0)Kr_(1)r_(2))/(r_(2)-r_(1))`
`:. C = (Kr_(1)r_(2))/(k(r_(2)-r_(1)))`
`=(32xx12xx10^(-2)xx13xx10^(-2))/(9xx10^(9)(13xx10^(-2)-12xx10^(-2))`
`C = (4992xx10^(-4))/(9xx10^(+7))`
=`554.6xx10^(-9)F `
(b) Potential of the inner sphere is
`V = (q)/(C) = (2.5xx10^(-6))/(5.5xx10^(-9))`
`:. V = 0.45xx10^(3)`
`:. V = 4.5 xx10^(2)V`
(c ) Radius of an isolated sphere of radius 12 cm
`C = 4 pi in_(0) R = (R)/(k)`
`:. C = (12xx10^(-2))/(9xx10^(9))= 1.3 xx10^(-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 spheres is earthed. Hence. the potential difference is less and the capacitance is more than the isolated sphere.