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
. displacement, the motion is anguler simple harmonic
. and find its time period.

Explanation

Suppose, the dipole axis makes an angle `theta` with
the electric field at an instant. The magnitude of the
torque on it is
`|tau| =| vecp xx vecE|`
`=qlE sin theta.`
This torque will tend to rotate the dipole back towards
the electric field. Also, for small angular displacement
`sin theta =~~ theta so that`
`tau = -qlEtheta.`
The moment of inertia of the system about the axis of
rotation is
`I = 2 xx m ((l)/(2))^2 = (ml^2)/(2).`
Thus the angular acceleration is
`alpha = (tau)/(I) = - (2qE)/(ml) theta = - omega ^2theta`
. where `omega^2 = (2qE)/(ml)`
Thus, the motion is angular simple harmonic and the
time period is `T = 2 pi sqrt (ml)/(2qE)`.