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 .The

Options

A. If q gt 0 and is displaced away from the centre in the plane of the ring, it will be pushed back towards the centre.
B. If q lt 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.
C. If q lt 0, it will perform S.H.M for small displacement along the axis.
D. q at the centre of the ring is in an unstable equilibrium within the plane of the ring for qgt0.

Answer

A, B, C

Explanation

If positive charge kept at the centre is displaced in the plane of ring away from centre then increased repulsion due to decrease in distance from some portion of ring and charge pushes it back towards the centre, hence option (a) is correct.
In case of negative charge at the centre it will be force of attraction and negative charge will be further moved away from the centre, hence option (b) is correct.
If the negative charge is releasod at some distance from the centre of ring on its axis then electric field of positively charged ring will move the negative charge towards the centre, hence negative charge experiences restoring force when placed on axis. Electric field intensity on the axis of ring is given by `E=(Qx)/(4piepsilon_0(R^2+x^2)^(3//2))` and the same can be approximated as proportional to x for small values of x. Acceleration that the negative charge acquires will be proportional to x for small values of x. Hence, it will perform S.H.M for small x. We can see that option (c) is also correct.
Option (d) is contsadicting option (a) and it is not correct, because positive charge kept at the centre of positively charged ring is in stable equilibrium at the centre within the plane of ring.