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Positive charge $Q$ is uniformly distributed throughout the volume of a sphere of radius $R$ . A point mass having charge $+ q$ and mass $m$ is fired towards the centre of the sphere with velocity if from a point $A$ at distance $r (r > R)$ from the centre of the sphere. Find the minimum velocity $v$ so that it can penetrate $R / 2$ distance of the sphere. Neglect any resistance other than electric interaction. Charge on the small mass remains constant throughout the motion.

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None of above

$v = \sqrt{\frac{Q}{5 π ε_{0} R} (\frac{r + R}{r} - \frac{5}{3})}$

$v = \sqrt{\frac{Q q}{2 π ε_{0} m R} (\frac{r - R}{r} + \frac{3}{8})}$

Zero

answer is B.

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Detailed Solution

From conservation of energy,

$U_{i} + K_{i} = U_{f} + K_{f}$ or $q V_{i} + K_{i} = q V_{f} + K_{f}$

or $q (\frac{Q}{4 π ε_{0} r}) + \frac{1}{2} m v^{2}$

$= q (\frac{Q}{4 π ε_{0} R^{3}}) (1.5 R^{2} - 0.5 \frac{R^{2}}{4}) + 0$

or $\frac{1}{2} m v^{2} = \frac{11 Q q}{32 π ε_{0} R} - \frac{Q q}{4 π ε_{0} r}$

or $v = \sqrt{\frac{Q q}{2 π ε_{0} m R} (\frac{r - R}{r} + \frac{3}{8})}$

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