Charge is distributed within a sphere of radius R with a volume c… - Vidyadip

MCQ

Charge is distributed within a sphere of radius $R$ with a volume charge density $\rho (r) = \frac{A}{{{r^2}}}{e^{ - 2r/a}}$ where $A$ and $a$ are constants. If $Q$ is the total charge of this charge distribution, the radius $R$ is.

A
$a\,\log \,\left( {1 - \frac{Q}{{2\pi aA}}} \right)$
✓
$\frac{a}{2}\,\log \,\left( {\frac{1}{{1 - \frac{Q}{{2\pi aA}}}}} \right)$
C
$a\,\log \,\left( {\frac{1}{{1 - \frac{Q}{{2\pi aA}}}}} \right)$
D
$\frac{a}{2}\,\log \left( {1 - \frac{1}{{2\pi aA}}} \right)\,$

✓

Answer

Correct option: B.

$\frac{a}{2}\,\log \,\left( {\frac{1}{{1 - \frac{Q}{{2\pi aA}}}}} \right)$

b
$Q = \int \rho 4\pi {r^2}dr = \int_0^R {\left( {\frac{A}{{{r^2}}}{e^{ - \frac{{2r}}{a}}}} \right)} \left( {4\pi {r^2}} \right)dr$

$ = 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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