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 surface 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 charge distribution in the surfaces of the concentric spherical shells is shown in the figure.
Charge density, `sigma = ("Charge")/("Area")`
Since the surface charge densities are equal
`:. (Q_(1))/(4pi R^(2)) = (Q_(1) + Q_(2))/(4pi)(4R^(2)) = (Q_(1) + Q_(2) + Q_(3))/(4pi (9R^(2)))`
`Q_(1) = (Q_(1) + Q_(2))/(4)` ...(i)
and `Q_(1) = (Q_(1) + Q_(2) + Q_(3))/(9)` ....(ii)
From (i), we get `3Q_(1) = Q_(2)` ....(iii)
From (ii), we get
`8Q_(1) - Q_(2) = Q_(3)`
`8Q_(1) - 3Q_(1) = Q_(3)` Using (iii)
`5Q_(!) = Q_(3)`
`:. (Q_(1))/(1) = (Q_(2))/(3) = (Q_(3))/(5)` or `Q_(1) : Q_(2) : Q_(3) = 1 : 3 : 5`