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A small charged bead can slide on a circular frictionless, insulating wire frame. A point like dipole of dipole moment $\vec{P}$ is fixed at the centre of circle. Initially, the bead is on the plane of symmetry of the dipole. Bead is released from rest. Ignore the effect of gravity. Select the correct options: ( $θ$ -angle made by position vector with dipole moment)

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Magnitude of velocity of bead as a function of its angular position $\sqrt{\frac{- Q P \cos θ}{2 π ε_{0} m r^{2}}} where {\frac{π}{2} \leq θ \leq π}$

Normal force exerted by the wire frame on bead is zero.

If the wire frame were not present, bead executes circular motion and returns to initial point after tracing a complete circle

Bead executes periodic motion.

answer is A, B, D.

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Detailed Solution

$\frac{1}{2} m v^{2} + Q (\frac{p \cos θ}{4 π ε_{0} r^{2}}) = 0 O R v = \sqrt{\frac{- 2 Q P \cos θ}{4 π ε_{0} m r^{2}}} {\frac{π}{2} \leq θ \leq π}$

Circular motion of bead requires a centripetal force

$E_{r} = - \frac{\partial V}{\partial r} = \frac{2 P \cos θ}{4 π ε_{0} m r^{3}}$

Note that $Q E_{r} = \frac{m v^{2}}{r}$

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