A spherical capacitor consists of two concentric spherical conductors, held in position by suitable insulating supports (Fig). Show that the capacitance of a spherical capacitor is given by `C=(4piepsi_0r_1r_2)/(r_1-r_2)`

where `r_1 and r_2` are the radii of outer and inner spheres , respectively.

Explanation

As is clear from figure + Q charge spreads uniformly on innersurface of outer sphere of radius `r_(1)`. The induced charge-Q spreads uniformly on the outer surface of inner sphere of radius `r_(2)`.The outer surface of outer sphere is earthed. Due to electro static shielding E = 0 for `r lt r_(2)` and E = 0 for `r gt r_(1)`
In the space between the two spheres electric intensity E exists as shown. Potential difference between the two spheres

`V = (Q)/(4 pi epsilon_(0) r_(2)) - (Q)/(4 pi epsilon_(0) r_(1))`
`phi = (Q)/(4 pi epsilon_(0)) [(1)/(r_(2)) - (1)/(r_(1))] = (Q)/(4 pi epsilon_(0)) ((r_(1) - r_(2)))/(r_(1) r_(2))`
`C = (Q)/(V)`
`:. C = (4 pi epsilon_(0) r_(1) r_(2))/(Q(r_(1) - r_(2))) = (4 pi epsilon_(0) r_(1)r_(2))/(r_(1) - r_(2))`