A positive charge Q is uniformly distributed along a circular ring of radius R. A small test charge q is placed at the centre of the ring in figure. Then

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if q > 0 and is displaced away from the centre in the plane of the ring, it will be pushed back towards the centre

if q < 0 and is displaced away from the centre in the plane of the ring, it will never return to the centre and will continue moving till it hits the ring

if q < 0, it will perform SHM for small displacement along the axis

q at the centre of the ring is in an unstable equilibrium within the plane of the ring for q > 0

answer is A, B, C.

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Detailed Solution

The positive charge Q is uniformly distributed over the cicumference of the ring . At the centre of the ring electric field is zero but at any other point inside the ring and in the plane of the ring , resultant electric field is not zero and directed towards the centre of the ring .

So, the effect of electric field on charge q due to the positive charge Q is zero.

Now, the only governing factor is the attractive and repulsive forces between charges (Q and q) there are two cases arise.

Case l When charge q> 0, i.e. q is a positive charge, they creates a repulsive force between charge q and Q.

The repulsive forces of charge Q from all around the charge q will push it towards the centre if it is displaced from the centre of the ring.

Case II When charge q < 0, i.e. q is a negative charge, then there is an attractive force between charge Q and q.

If q is shifted from the centre, then the positive charges nearer to this charge will attract it towards itself and charge q will never return to the centre.

q is in stable equilibrium in the plane of the ring .

Electric field , produced by Q ,at any point on the axis of the ring is directed away from the centre of the ring . So if q $< 0 a n d t h e c h a r g e q i s s l i g h t l y d i s p l a c e d f r o m t h e c e n t r e a n d r e l e a s e d,$ it will be pushed back towards the centre . So q is in stable equilibrium along the axis of the ring and for small displacement q will perform SHM.