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Three concentric metallic spherical shel...

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

A

(a) 1:2:3

B

(b) 1:3:5

C

(c) 1:4:9

D

(d) 1:8:18

Text Solution

Verified by Experts

The correct Answer is:
B
The charges on the surfaces of the metallic spheres are shown in the diagram. It is given that the surface charge denisties 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^2x`
`Q_1+Q_2=4pi(2R)^2k=4[4piR^2x]`
`implies Q_2=4[4piR^2x]-Q_1`
`=4[4piR^2x]-4piR^2x=3[4piR^2x]`
Also `Q_1+Q_2+Q_3=4pi(3R)^2x=9[4piR^2x]`
`:.` `Q_3=9[4piR^2x]-Q_1-Q_2=9[4piR^2x]-[4piR^2x]-3[4piR^2x]=5[4piR^2x]`
`implies Q_1:Q_2:Q_3=1:3:5`
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