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A system consists of a uniformly charged sphere of radius R and a surrounding medium filled by a charge with the volume density $ρ = \frac{α}{r},$ where $α$ is a positive constant and r is the distance from the centre of the sphere. Find the charge of the sphere for which the electric field intensity E outside the sphere is independent of R.

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$\frac{α}{2 ε_{0}}$

$\frac{2}{α ε_{0}}$

$2 π α R^{2}$

None of these

answer is C.

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Detailed Solution

Using Gauss's theorem for spherical surface of radius r outside the sphere with the charge q

$\int_{0}^{r} Eds= \frac{1}{ε_{0}} \int_{0}^{r} ρ (r) dV$

$\Rightarrow E 4 {πr}^{2} = \frac{1}{ε_{0}} (q + \int_{R}^{r} \frac{α}{r} (4 {πr}^{2}) dr)$

$\Rightarrow E 4 {πr}^{2} = \frac{(q - 2 {παR}^{2})}{ε_{0}} + \frac{4 {παr}^{2}}{2 ε_{0}}$

The intensity E does not depend on R if $\frac{q - 2 {παR}^{2}}{ε_{0}} = 0 o r q = 2 {παR}^{2}$

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