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Q.

A system consist of 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 $r (> R)$ is the distance from the centre of the sphere. The charge of the sphere for which electric field intensity $E$ outside the sphere is independent of $r$ is:

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a

$α R^{2}$

b

$\frac{α}{2 ε_{0}}$

c

$2 π α R^{2}$

d

$\frac{2}{α ε_{0}}$

answer is C.

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Detailed Solution

$\begin{array}{l} \int E \cdot ds = \frac{q_{i n}}{ε_{0}} \Rightarrow E (4 π r^{2}) = \frac{Q + \int_{R}^{r} \frac{α}{r} 4 π r^{2} d r}{ε_{0}} \\ = \frac{Q + 2 π α (r^{2} - R^{2})}{ε_{0}} \Rightarrow E = \frac{α}{2 ε_{0}} + \frac{Q - 2 π α R^{2}}{4 π ε_{0} r^{2}} \end{array}$

To have an electric field that is independent of the charge density

$\Rightarrow Q = 2 π α R^{2}$

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A system consist of uniformly charged sphere of radius R and a surrounding medium filled by a charge with the volume density ρ=αr, where α is a positive constant and r(>R) is the distance from the centre of the sphere. The charge of the sphere for which electric field intensity E outside the sphere is independent of r is: