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A charged +Q is uniformly distributed in a spherical volume of radius R . A particle of charge +q and mass m projected with velocity $\;v_{0}\;$ from the surface of the spherical volume to its centre inside a smooth tunnel dug across the sphere . The minimum value of $\;v_{0}\;$ such that it just reaches the centre of the spherical volume is


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Answer : (d) $\;\sqrt{\large\frac{kQq}{mR}}$
Explanation :
Work done by electric field on particle = Charge in kinetic energy of particle
Electric field at a distance r from the centre is E = $\;\large\frac{kQr}{R^3}$
Work done by electric field from r to rdr is
$dW=qE dr$
$W=\int_{R}^{0}\;qE\; dr$
$W=-\large\frac{kqQ}{2R^3} \;R^2=-\large\frac{kqQ}{2R}$
For $\;v_{0}\;$ to be minimum velocity on surface velocity at centre should be zero
$\bigtriangleup K.E=\large\frac{1}{2} m (0)^2 - \large\frac{1}{2} mv_{0}^2=-\large\frac{1}{2} mv_{0}^2$
$W=\bigtriangleup K.E$
answered Feb 13, 2014 by yamini.v

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