Question

A charged particle q of mass m is in wquilibrium at a height h from a horizontal infinite line charge wit uniform linear charge density `lambda`. The charge lies in the vertical plane containing the line charge. If the particle is displaced slightly (vertically) prove that the motion of the charged particle will be simple harmonic. Also find its time period.

Answer 1

Correct Answer - `2 pi sqrt((mh^(2))/(2kq lambda))`
At equilibrium `mg=qE`
`mg=(q2k lambda)/(h)`

when particle is displaced to distance x in upwar direction. Then net force in upward direction is
`F=q((2k lambda)/(h+x))-mg`
`=(2kq lambda)/(h)(1+x/h)^(-1)-mg=(2kq lambda)/(h)(1+x/h)-mg`
`=-((2kq lambda)/(h^(2)))x=-K_(0)x`
`:. F prop-x` (Motion is SHM)
Time period of motion is `T=2pisqrt((m)/(K_(0)))=2 pi sqrt((mh^(2))/(2kq lambda))`.