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Question

The diagram shows a sphere of radius R that carries a charge density that is not uniform. The charge density varies according to radial distance r from the center of the sphere according to the equation =Cr. The diagram also shows the cross-section of a Gaussian sphere of radius a that is concentric with the charged sphere.
Which of the following expresses the charge enclosed in the Gaussian sphere?
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A
23πCa3
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B
43πCa4
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C
14πCa4
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D
πCa4
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E
415πCa5
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Solution

The correct option is C πCa4
The charge density of the sphere varies as ρ=Cr
The volume of a shell of infinitesimal width dV=4πr2dr
Hence the charge enclosed in Gaussian sphere Qenc=a0ρdV=a0(Cr)(4πr2dr)
Qenc=4πCa0r3dr)
Or Qenc=4πC×r44a0
We get Qenc=4πC×a44=πCa4

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