A spherical capacitor has outer sphere of radius 5 cm and inner sphere of radius 2 cm. When the inner sphere is earthed, its capacity is `C_(1)` and when the outer sphere is earthed its capacity is `C_(2)`. Then `(C_(1))/(C_(2))` is

Options

A. `(5)/(2)`
B. `(2)/(5)`
C. `(7)/(3)`
D. `(3)/(7)`

Answer

A

Explanation

If a spherical capacitor has outer radius `R_(2)` and inner radius `R_(1)`. Then the capacitance,
(a) when outer shell is earthed,
`C = 4 pi epsilon_(0)(R_(1)R_(2))/(R_(2) - R_(1))`
(b) when inner shell is earthed
`C. = C + 4pi epsilon_(0)R_(2)`
Given, radius of outer sphere of capacitor,
`R_(1) = 2 cm` and Radius of inner sphere of capacitor, `R_(2)= 5cm`
`:. C_(1) = 4pi epsilon_(0)[(R_(1)R_(2))/(R_(2)-R_(1))+R_(2)] = 4pi epsilon_(0)[(2 xx 5)/(5-2)+5]`
`C_(1) = 4pi epsilon_(0)(25)/(3)` Farad
and `C_(2) = 4pi epsilon_(0) (R_(1)R_(2))/(R_(2)-R_(1)) = 4pi epsilon_(0)(2 xx 5)/(5-2) = 4 pi epsilon_(0)(10)/(3)`
Hence, `(C_(1))/(C_(2)) = ((25)/(3))/((10)/(3)) = (25)/(10) = (5)/(2)`
Hence, the correct option is (a).