Three concentric metallic spherical shells of radii R, 2R, 3R, are given charges `Q_1`, `Q_2`, `Q_3`, respectively. It is found that the surface charge denisties on the outer surfaces of the shells are equal. Then, the ratio of the charges given to the shells, `Q_1:Q_2:Q_3`, is

Options

A. `1 : 2 : 3`
B. `1 : 3 : 5`
C. `1 : 4 : 9`
D. `1 : 8 : 18`

Answer

2

Explanation

The facing surfaces have equal and opposite charges.

`(Q_(1))/(4 pi R^(2)) = (Q_(1) + Q_(2))/(4 pi (2 R)^(2)) = (Q_(1) + Q_(2) + Q_(3))/(4 pi (3 R)^(2))`
`Q_(1) = (Q_(1) + Q_(2))/(4) = (Q_(1) + Q_(2) + Q_(3))/(9)`
`36 Q_(1) = 9 Q_(1) + 9 Q_(2) = 4 Q_(1) + 4 Q_(2) + 4Q_(3)`
`36 Q_(1) = 9 Q_(1) + 9 Q_(2) rArr 27 Q_(1) = 9 Q_(2) rArr Q_(2) = 3Q_(1)`
`36 Q_(1) = 4 Q_(1) + 4 Q_(2) + 4 Q_(3)`
`32 Q_(1) = 4 Q_(2) + 4 Q_(3) = 12 Q_(1) + 4 Q_(3)`
`4 Q_(3) = 20 Q_(1) rArr Q_(3) = 5 Q_(1)`
`Q_(1) : Q_(2) : Q_(3) = 1 : 3 : 5`