A thin fixed ring of radius a has a positive charge q uniformly distributed over it.A particle of mass m having a negative charge Q, is placed on the axis at a distance of `x(xltlta)` form the center of the ring. Show that the motion of the negatively charged particle is approximately simple harmonic. Calculate the time period of oscillation.

Explanation

When the negative charge is shifted at a distance x from the centre of the ring along its axis then force acting on the point charge due to the ring :
`F_(E)=qE` (towards centre) `=q[(Kqy)/((a^(2)+y^(2))^(3//2))]`
If `a gt gt y` then `a^(2)+y^(2) cong a^(2)`
`:. F_(E)=1/(4pi epsi_(0)) (Qqy)/a^(3)` (Towards centre)
Since, restoring force `F_(E) prop y`, therefore motion of charge the particle will be S.H.M. Time period of SHM
`T=2pi sqrt(m/k)=2 pi sqrt(m/((Qq)/(4pi epsi_(0) a^(3))))=[(16 pi^(3) epsi_(0) ma^(3))/(Qq)]^(1//2)`