Question

# Charge is distributed within a spehere of radius R with a volume charge density ρ(r)=Ar2e−2r/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⎛⎜ ⎜ ⎜⎝11−Q2πaA⎞⎟ ⎟ ⎟⎠Consider an elemantal shell of radius r and thickness dr Q=∫ρdv=∫R0Ar2e−2r/a(4πr2dr) Q=4πA∫R0e−2r/adr Q=4πA⎛⎜ ⎜⎝e−2r/a−2a⎞⎟ ⎟⎠R0Q=4πA(−a2)(e−2R/a−1) Q=2πaA(1−e−2R/a) R=a2log⎛⎜ ⎜ ⎜⎝11−Q2πaA⎞⎟ ⎟ ⎟⎠

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