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 `2.5muC`. The space between the concentric spheres is fillled 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_(a) = 12 cm = 1.2 xx 1^(-2) m`
`r_(b) = 13 cm = 12 xx 10^(-2) m`
`q = 2.5 mu C = 2.5 xx 10^(-6) C E_(r ) = 32`
(a) C = ?
As `C = 4 pi epsilon_(0) = (epsilon_(r ) r_(a) r_(b))/(r_(b) - r_(a))`
`= (1)/(9 xx 10^(9)) (32 xx 12 xx 10^(-2) xx 13 xx 10^(-2))/((13 - 12)10^(-2))`
` = 5.5 xx 10^(-9)` Farad,
(b) V = ?
`V = (q)/(C ) = (2.5 xx 10^(-6))/(5.5 xx 10^(-9)) = 4.5 xx 10^(2)` volt
(c ) Capacity of an isolated sphere of radius
`R = 12 xx 10^(-2) m` is
`C^(1) = 4 pi epsilon_(0) R = (1)/(9 xx 10^(9)) xx 12 xx 10^(-12)`
`= 1.33 xx 10^(-11)` Farad.
The capacity of an isolated sphere is much smaller because in a capacitor outer shpere is earthed difference decreases and capacitor increases.