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