Two equal positive charges , `Q` , are fixed at points `(0 , d)` and `(0 , -d)` on the `y`-axis. A charged particle having charge `-q` and mass `m` is released from rest at point `(d,0)` on the `x`-axis . Discuss the motion of charged particle.

Explanation

The charge ` - q` will be attracted by charges `Q`. Let at some instant charged particle is at distance `x` from origin `O`.
The resultant force on `-q`,
`F = 2 F_(0) cos theta = 2. (1)/(4 pi in_(0)) (Qq)/((d^(2) + x^(2))) . (x)/((d^(2) + x^(2))^(1//2))`
` = (Qqx)/(2 pi in_(0) (d^(2) + x^(2))^(3//2))`
Here , the restoring force `F` is not directly proportional to displacement `x` , hence the particle will not execute `S.H.M.`
The motion of particle is oscillatory of amplitude `d`.
If initial position of charged particle is close to `O` such that `x lt lt d`.
Restoring Force `F = - (Qqx)/(2 pi in_(0) d^(3))`
Acceleration `a = (F)/(m) = - (Qqx)/( 2 pi in_(0) md^(3)) = - omega^(2) x`
`T = (2 pi)/( omega) = 2pi sqrt((2 pi in_(0) md^(3))/(Qq))`