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

Explanation

Let us 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 uniform, the total force acting on the dipole is zero. These two forces acting and the dipole experience a torque. This torque tends to rotate the dipole.

The total torque on the dipole about the point O
`vectau = vec(OA) xx (-qvecE) + vec(OB) xx qvecE`
Using right-hand coekscrew rule it is found that total torque is perpendicular to the plane of the paper and is directed into it.
The magnitude ofthe total torque
`vectau = |vec(OA)||(-qvecE)|sintheta + |vec(OB)||qvecE| sintheta`
`vectau = qE.2asintheta`
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 = vecp xx vecE`

The magnitude of this torque is `tau = pEsintheta` 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.
If the electric field is not uniform, then the force experienced by +q is different from that experienced by -q. In addition to the torque, there will be net force acting on the dipole.