In the figure shown, an electric dipole is placed at a distance `x` from an infinitely long rod of linear charge density `lambda`.
(`a`) Find the net force acting on the dipole ?
(`b`) What is the work done in rotating the dipole through `180^(@)` ?
(`c`) If the dipole is slightly rotated about its equilibrium position, find the time period of oscillation. Assume that the dipole is linearly restrained.

Explanation

(`a`) The electric field at a distance `x` is
`E=(lambda)/(2piepsilon_(0)x)`
The net force on the dipole is
`F=p(dE)/(dx)=(2a)q[(-lambda)/(2piepsilon_(0)x^(2))]=(-lambdaqa)/(piepsilon_(0)x^(2))`
The negative sign indicates that the force is attractive.
(`b`) `W=DeltaU`
`U_(i)=-vecp*vecE=-pEcos0^(@)=-pE`
`U_(f)=-vecp*vecE=-pEcos180^(@)=+pE`
`W=2pE=2(2a)q(lambda)/(2piepsilon_(0)x)=(2lambdaqa)/(piepsilon_(0)x)`
(`c`) Restoring torque
`tau=-pEsintheta`
`tau=I(d^(2)theta)/(dt^(2))=(2ma^(2))(d^(2)theta)/(dt^(2))`
`:. 2ma^(2)(d^(2)theta)/(dt^(2))+[(lambdaqa)/(piepsilon_(0)x)]theta=0`
or `(d^(2)theta)/(dt^(2))+[(lambdaq)/(2piepsilon_(0)max)]theta=0`
`T=2pisqrt((2piepsilon_(0)max)/(lambdaq))=sqrt((8pi^(2)epsilon_(0)max)/(lambdaq))`