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 muC`. 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 later is much smaller ?

Explanation

Radius of the inner sphere, `r_(2)=12 cm =0.12 m `
Radius of the outer sphere , `r_(1)=13 cm =0.13 m `
charge on the inner sphere , `q=2.5 muC=2.5xx10^(-6)C`
Dielectric constant of a liquid , `in_(r)=32`
(a) capcitances of the capacitors is given by the relation
`C=(4pi in_(0)in_(r)r_(1)r_(2))/(r_(1)-r_(2))`
where, `in_(0)=`permitivity of free space =`8.85xx10^(-12)C^(2)N^(-4)m^(-2)`
`1/(4piin_(0))=9xx10^(9)Nm^(2)C^(-2)`
`:. C=(32xx0.12xx0.13)/(9xx10^(9)xx(0.13-0.12))`
`=5.5xx10^(-9)F`
Hence, the capacitance of the capacitors is approximately `5.5xx10^(-9) F`. (b)potential of the inner sphere is given by
`V=q/C`
`=(2.5xx10^(-6))/(5.5xx10^(-6))=4.5xx10^(2)V`
Hence, the potential of the inner sphere is `4.5xx10^(2)V`
(c) radius of an isolated sphere `, r=12xx10^(-2)m`
capcaitance of the sphere is given by the relation.
`C'=4pi in_(0)r`
`=4pixx8.85xx10^(-12)xx12xx10^(-12)`
`1.33xx10^(-11)F`
The capacitance of the isolated sphere is less in comparison to the concentric spheres. thie is because the outer sphere of the concentric sphers is eartherd . hence, the potential difference is less and the capacitances is more than the issolated sphere.