Two concentric rings placed in a gravity free region in $y z$ -plane one of radius $R$ carries a charge $Q$ and second of radius $4 R$ and charge $- 8 Q$ distributed uniformly over it. Find the minimum velocity with which a point charge of mass $m$ and charge $+ q$ should be projected from a point at distance $3 R$ from the centre of rings on its axis so that it will reach to the centre of the rings.

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$\sqrt{\frac{Q q}{2 π ε_{0} m R} (\frac{3 \sqrt{10} - 5}{5 \sqrt{10}})}$

None o of above

$\sqrt{\frac{2 Q q}{2 π ε_{0} b R R} (\frac{3 \sqrt{10} - 5}{5 \sqrt{10}})}$

$\frac{1}{3} \sqrt{\frac{Q q}{5 π ε_{0} b R R} (\frac{3 \sqrt{20} - 5}{5 \sqrt{20}})}$

answer is A.

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Detailed Solution

From energy conservation principle,

$K_{i} + U_{i} = K_{f} + U_{f} or \frac{1}{2} m v^{2} + (+ q) V_{i} = 0 + (+ q) V_{f} or v = \sqrt{\frac{2 q}{m} (- V_{i} + V_{f})} = \sqrt{\frac{2 g}{m} [\frac{- Q}{\sqrt{10} R} + \frac{8 Q}{5 R} + \frac{Q}{R} - \frac{8 Q}{4 R}] \frac{1}{4 π E_{0}}} = \sqrt{\frac{Q q}{2 π ε_{0} m R} (\frac{3 \sqrt{10} - 5}{5 \sqrt{10}})}$