A ring of radius R carries a uniformly distributed charge +Q. A point charge `-q` is placed on the axis of the ring at a distance 2R from the centre of the ring and released from rest. The particle executes a ismple harmonic motion along the axis of the ring.

Explanation

Electric field intensity at a distance x from the centre of ring on its axis is given by the following relation :
`E=(Qx)/(4piepsilon_0(R^2+x^2)^(3//2))`
Electric field of ring will attract the charge q towards its centre. This electric force will be restoring force for oscillation.
Acceleration of particle, when at a distance x from the centre :
`a=(qE)/m=(Qqx)/(4piepsilon_0m(R^2+x^2)^(3//2))`
If x < < R , then we may neglect `x^2` in comparison to `R^2` in denominator of above equation . Acceleration may be approximately written as :
`a=(Qq)/(4piepsilon_0mR^3)x`
On comparing it with `a=omega^2x`, we can say that it is S.H.M and we get the angular frequency of oscillation :
`omega=sqrt((Qq)/(4piepsilon_0mR^3))`
Time period of S.H.M. is given by `T=(2pi)/omega`, hence the time period of oscillation is :
`T=2pisqrt((4piepsilon_0mR^3)/(Qq))`