show an electric dipole formed by two
. particles fixed at the ends of a light rod of length l. The
. mass of each particle is m and the charges are -q and
. +q . The system is placed in such a way that the dipole
. axis is parallel to a uniform electric field E that exist
. in the region. The dipole is slightly rotated abut its
. centre and released. Show that for small angular
. desplacement, the motin is anguler sumple harmonic
. and find its time period.

Explanation

Let an instant rod makes small angle `theta` with the field.
Restoring torque `tau = -p E sin = - ql E theta ,(since sin theta = theta)`
`tau = I alpha , I = m ((l)/(2))^(2) xx 2 = (ml^(2))/(2)`
`alpha = (tau)/(I) = (qEl)/(ml^(2)//2) theta = -(2qE)/(ml) theta`
`alpha alpha - theta` (case of `S.H.M.`)
`alpha = -omega^(2) theta`
`omega^(2) = (2qE)/(ml) rArr omega = sqrt((2qE)/(ml))`
Time period `T = (2pi)/(omega) = 2pi sqrt((ml)/(2qE))`