A spherical ball of radius R is uniformly charged. The surrounding space of ball is charged with volume charge density $\frac{α}{r}$ , where $α$ is a constant and $r$ is the distance from the centre of the ball. The value of charge q on ball for which the strength of electric field outside the ball is independent of $r$ . (The relative permittivites of the ball and the surrounding space are assumed to be equal to unity)

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Charge on ball is $q = π α R^{2}$

Charge on ball is $q = 2 π α R^{2}$

Electric field out-side the ball is $E = \frac{2 α}{ε_{0}}$

Electric field out-side the ball is $E = \frac{α}{2 ε_{0}}$

answer is B, D.

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$\int d q = \int \frac{α}{r} 4 π r^{2} d r q = 4 π α \cdot {[\frac{r^{2}}{2}]}^{r} = 2 π α [r^{2} - R^{2}] \frac{k (Q + q)}{r^{2}} = E E = \frac{k (Q + 2 π α (r^{2} - R^{2}))}{r^{2}} E = \frac{k Q}{r^{2}} + 2 π α k - \frac{2 π α k R^{2}}{r^{2}}$