Derive an expression for the torque experienced by a dipole due to a uniform electric field .

Explanation

Torque experienced by an electric dipole in the uniform electric field : Consider an electric dipole of dipole moment `vecp` placed in a uniform electric field `vecE` whose field lines are equally spaced and point in the same direction . The charge +q will experience a force q `vecE` in the direction of the field and charge -q will experience a force -q `vecE` in a direction opposite to the field . Since the external field `vecE` is unform the total force acting on the dipole is zero . These two forces acting at different points will constitute a couple and the dipole experience a torque . This torque tends to rotate the dipole . (Note that electric field lines of a uniform field are equally spaced and point in the same direction). The total torque on the dipole about the point O
`vectau=vec(OA)xx(-qvecE)+vec(OB)xxqvecE`
Using right -hand corkscrew rule it is found that total torque is perpendicular to the plane of the paper and is directed into it. The magnitude of the total torque `tau=|vec(OA)|(-qvecE)sintheta+|vec(OB)||qvecE|sintheta `
`vectau =q E 2 a sin theta `

where `theta ` is the angle made by `vecp` with `vecE` . Since p =2aq the torque is written in terms of the vector product as `vectau=vecpxxvecE`
The magnitude of this torque is `tau =pE` sin `theta` and is maximum when `theta=90^(@)`
This torque tends to rotate the dipole and align it with the electric field`vecE` . Once `vecp` is aligned with `vecE` the total torque on the dipole becomes zero.