A particle of mass m and charge q is located midway between two fixed charged particles each having a charge q and a distance 2l apart. Prove that the motion of the particle will be SHM if it is displaced slightly along the line connecting them and released. Also find its time period.

Explanation

Let x-y direction is taken as :

Particle is shifted along y-axis by a small displacement x.
After resolving component of forces between q and -q charges :
By figure. `F_("net")` in x-axis =0 [`F_("net")=` net force on -q charge]
Net force on -q charge in y direction `=-2F cos theta=-2. (kq q)/((x^(2)+l^(2))). x/((x^(2)+l^(2))^(1//2))`
`|vec(F)|=(2Kq^(2)x)/((x^(2)+l^(2))^(3//2))" "implies" "ma=(2Kq^(2)x)/l^(3)" "("for "x lt lt l)` (a=acceleration of -q charge)
`implies a=(2Kq^(2))/(ml^(2)). x ("downwards")`
This is equation of S.H.M. `(a=-omega^(2) x)`
So, time period of this charge (-q) :
`T=2pi sqrt((ml^(3))/(2Kq^(2)))`