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

B

Explanation

Simple rules for solving this question.
One is that metal surfaces facing each other must always carry equal and opposite charge and the other rule is that net charge of an isolaled object remains constant.
Net charge with each spherical shell is given. See how the charge is distributed. Surface charge densities on required surfaces can be written as follows:
`sigma_1=Q_1/(4piR^2)`….(i)
`sigma_2=(Q_1+Q_2)/(4pi(2R)^2) rArr sigma_2=(Q_1+Q_2)/(16piR^2)` …(ii)
`sigma_3=(Q_1+Q_2+Q_3)/(4pi(3R)^2)=sigma_3=(Q_1+Q_2+Q_3)/(36piR^2)`…(iii)
`sigma_1=sigma_2`
`sigma_1=sigma_2rArr Q_1/(4piR^2) =(Q_1+Q_2)/(16piR^2)`
`rArr 4Q_1=Q_1+Q_2 rArr 3Q_1=Q_2`...(iv)
`sigma_2=sigma_3rArr (Q_1+Q_2)/(16piR^2) =(Q_1+Q_2+Q_3)/(36piR^2)`
`rArr 9Q_1 +9Q_2=4Q_1+4Q_2 +4Q_3`
`rArr 5Q_1+5Q_2 =4Q_3`
Substituting `Q_2` from equation (iv) in terms of `Q_1` we get the following :
`rArr 5Q_1+15Q_1 =4Q_3`
`rArr Q_3=5Q_1`...(v)
From equation (iv) and (v) we can write the following :
`Q_1:Q_2:Q_3=1:3:5`