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 densities 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

B

Explanation

The charges on the surfaces of the metallic spheres are shown in the diagram . It is given that the surface charge densities on the outer surfaces of the shells are equal . Therefore ,

`(Q_(1))/(4piR^(2)) = (Q_(1)+Q_(2))/(4pi(2R)^(2))`
`= (Q_(1)+Q_(2)+Q_(3))/(4pi(3R)^(2)) = x (say ) `
` :. " "Q_(1) = 4piR^(2)x`
`Q_(1)+Q_(2) = 4pi (2R)^(2) x = 4 [ 4piR^(2)x]`
`rArr " "Q_(2) = 4 [4piR^(2)x] -Q_(1)`
`= 4 [4piR^(2)x] - 4piR^(2) = 3 [ 4piR^(2)x] `
Also
`Q_(1)+Q_(2)+Q_(3) = 4pi (3R)^(2)`
`x = 9 [4piR^(2)x]`
` : . " " Q_(3) = 9 [4piR^(2)x] - Q_(1) - Q_(2)`
` = 9 [ 4piR^(2)x] - 4 [ 4piR^(2)x] `
` = 5 [4piR^(2) x] `
` rArr Q_(1) : Q_(2) : Q_(3) = 1:3:5 `