Question

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

• A. $a\log (1-\frac{Q}{2\pi aA})$
• B. $\frac{a}{2}\log \left(\frac{1}{1-\frac{Q}{2\pi aA}}\right)$
• C. $a\log \left(\frac{1}{1-\frac{Q}{2\pi aA}}\right)$
  
• D. \dfrac{a}{2}\log\left ( 1-\dfrac{Q}{2 \pi a A} \right )

Answer 1

The correct option is B $\frac{a}{2}\log \left(\frac{1}{1-\frac{Q}{2\pi aA}}\right)$

Consider an elemantal shell of radius $r$ and thickness $dr$


$Q=\int \rho dv={\int }_0^R\frac{A}{r^2}{e}^{-2r/a}\left(4\pi r^{2}dr\right)$

$Q=4\pi A{\int }_0^R{e}^{-2r/a}drQ=4\pi A{\left(\frac{e^{-2r/a}}{-\frac{2}{a}}\right)}_0^{R}Q=4\pi A(-\frac{a}{2})(e^{-2R/a}-1)$
$Q=2\pi aA(1-e^{-2R/a})$
$R=\frac{a}{2}\log \left(\frac{1}{1-\frac{Q}{2\pi aA}}\right)$