A spherical capacitor consists of two concentric spherical conductors, held in position by suitable insulating supports (Fig. 2.34). Show that the capacitance of a spherical capacitor is given by
`C = (4 pi epsilon_(0)r_(1)r_(0))/(r_(1)-r_(2))`

where `r_(1)` and `r_(2)` are the radii of outer and inner spheres, respectively.

Explanation

By two concentric sphedcal conductors a spherical shell is formed.

Radius of the inner shell is `r_(1)` and radius of outer shell is `r_(2)` .
The outer surface of the inner shell has induced charge - Q and the inner surface of the outer shell has charge +Q .
A spherical Gaussian surface of radius r from common point centre O .
`r lt r_(1)` means inner shell E =0 and `r gt r_(2)` means outside inner shell E =0 .
Due to charges on inner shell a space between two shells have radial electric field.
According to Gauss.s law, electric flux at P at a common centre from distance r,
`E S cos 0^(@) = (Q)/(in_(0))`
`:. ES = (Q)/(in_(0))`
`:. Exx4pir^(2) = (Q)/(in_(0))` [ Area of sphere = `4pi r^(2)]`
`:. E =(Q)/(4pi in_(0)r^(2))`
p.d. between two shells
`V= int_(r_(1))^(r_(2))vecE.dvecr=-int_(r_(1))^(r_(2))E "dr cos " 180^(@)`
`=int_(r_(1))^(r_(2))E dr = int_(r_(1))^(r_(2))(Q)/(4piin_(0)r^(2))dr`
`= (Q)/(4pi in_(0))int_(r_(1))^(r_(2))(1)/(r^(2))dr`
`= (Q)/(4pi in_(0))[-(1)/(r)]_(r_(1))^(r_(2))=(Q)/(4piin_(0))[(1)/(r_(1))-(1)/(r_(2))]`
Capacitance of spherical capacitor
`C = (Q)/(V)`
`:. C (Q)/((Q)/(4piin_(0))[(1)/(r_(1))-(1)/(r_(2))])=(4piin_(0)r_(1)r_(2))/(r_(2)-r_(1))`