a positively charged thin metal ring of radius R is fixed in the X-Y plane with its centre at the origin O.A negatively charged particle P is released fron rest at the point (0,0,z₀) where z₀>0. then the motion of P is-

a) periodic,for all values of z₀ satisfying 0<z₀<infinity

b) simple harmonic,for all values of z0 satisfying 0<z₀<=R

c)   approximately simple harmonic ,provided z₀<<R

d) such that P crosses O and continues to move along the negative Z-axis towards z= -infinity

how is it an SHM

k*qx/(a^2+x^2)^3/2

so force on a particle of say -Q is given by

-k*qQx(a^2+x^2)^3/2

now if x<<a then force = -kqQx/a^3 which is an SHM

so c) is true

but for x------>inf force becomes inverse square so it is not SHM

so only c) is true

but jinu as x becomes large it becomes an inverse square force so how can it be periodic as attractive force decreases with distance ?

I think both options a and c. jinu.coolboy  has shown that it is SHM for displacement of particle << radius of ring.

Option a is right as force is always towards the centre of the ring and of equal magnitude for same distance from centre of ring so it is periodic.