A) \[a\log \left( 1-\frac{Q}{2\pi aA} \right)\]
B) \[\frac{a}{2}\log \left( \frac{1}{1-\frac{Q}{2\pi aA}} \right)\]
C) \[\frac{a}{2}\log \left( \frac{1}{1+\frac{Q}{2\pi aA}} \right)\]
D) \[\frac{a}{2}\log \left( 1-\frac{1}{2\pi aA} \right)\]
Correct Answer: B
Solution :
\[Q=\int{\rho 4\pi {{r}^{2}}dr}\]\[=\int\limits_{0}^{R}{\left( \frac{A}{{{r}^{2}}}{{e}^{\frac{-\,2r}{a}}} \right)}\,(4\pi {{r}^{2}})dr\]\[=4\pi A\frac{a}{2}\left( 1-{{e}^{\frac{-\,2R}{a}}} \right)\] \[\Rightarrow \] \[R=\frac{-\,a}{2}\log \,\left( 1-\frac{Q}{2\pi Aa} \right).\]
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