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Question

# A sphere of radius R carries charge such that its volume charge density is proportional to the square of the distance from the centre. What is the ratio of the magnitude of the electric field at a distance 2R from the centre to the magnitude of the electric field at a distance of R2 from the centre (i.e.,Er=2R/Er=R/2)?

A
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B
2
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C
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D
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Solution

## The correct option is B 2ρ=αr2 [where α is a constant] Charge in the shell [element] dq=ρ(4πr2dr)=α(4π)r4dr ∴ Charge enclosed in sphere of radius r, q=4πα∫r0r4dr=4παr55 By Gauss's theorem at r=R2 Er=R/2=14π×4πα(R2)55∈0×1(R2)2 Er=R2=αR532∈0×4R2×5 Er=R/2=αR340∈0 Er=R/2=αR340∈0 Total charge enclosed, Q=4παR55 ∴Er=2R=4παR55∈0×14π(2R)2=αR320∈0 Er=2REr=R/2=αR320∈0×40∈0αR3 Er=2REr=R/2=4020 Er=2REr=R/2=2

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