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 qe in the direction of the field and charge -q will experience a force `vecE` in a direction opposite to the field. Since the external field E is uniform, the total force acting on the dipole is zero. Torque la into the paper These two forces acting at different points will constitute Torque on dipole 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
`tau=vec(OA)xx(-qvecE)+vec(OB)xxqvecE`
Using right-hand corkscrew rule, it is found that total torque is perpendicular to the plane ol the paper and is directed into it.
The magnitude of the total torque
`vectau=|(vecOB)||sintheta+|vec(OB)||qvecE|sintheta`
`vectau=qE2asintheta` ...(2)
where is the angle made by `vecp` with `vecĒ`. Since p = 2aq, the torque is written in terms of the vector product as
`vectau=vecpxxvecE`
The magnitude of this torque is t = pe sin 0 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.