Question

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

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

Answer 1

Answer: (D)

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