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

Engineering

Physics

Charge and Coulombs Law

Question

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

$alog (1 - \frac{Q}{2 πaA})$

$\frac{a}{2} \log (\frac{1}{1 - \frac{Q}{2 πaA}})$

$alog (\frac{1}{1 - \frac{Q}{2 πaA}})$

$\frac{a}{2} \log (1 - \frac{Q}{2 πaA})$

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$Q = \int ρ 4 π^{2} dr = \int_{0}^{R} (\frac{A}{r^{2}} e^{- \frac{2 r}{a}}) (4 {πr}^{2}) dr$

$= 4 πA \frac{a}{2} (1 - e^{\frac{- 2 R}{a}})$

$\Rightarrow R = \frac{- a}{2} \log (1 - \frac{Q}{2 πAa})$