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Charge is distributed within a spehere of radius R with a volume charge density ρ(r)=Ar2e2r/a, where A and a are constants. If Q is the total charge of this charge distribution, the radius R is

A
alog(1Q2πaA)
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B
a2log⎜ ⎜ ⎜11Q2πaA⎟ ⎟ ⎟
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C
alog⎜ ⎜ ⎜11Q2πaA⎟ ⎟ ⎟
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D
\dfrac{a}{2}\log\left ( 1-\dfrac{Q}{2 \pi a A} \right )
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Solution

The correct option is B a2log⎜ ⎜ ⎜11Q2πaA⎟ ⎟ ⎟
Consider an elemantal shell of radius r and thickness dr


Q=ρdv=R0Ar2e2r/a(4πr2dr)

Q=4πAR0e2r/adr Q=4πA⎜ ⎜e2r/a2a⎟ ⎟R0Q=4πA(a2)(e2R/a1)
Q=2πaA(1e2R/a)
R=a2log⎜ ⎜ ⎜11Q2πaA⎟ ⎟ ⎟

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