Derive an expression for torque acting on an electric dipole in a uniform two dimensional electric field.
Or
Write the expression for torque `vecpi` acting on a dipole of dipole moment `vecp` placed in an electric field `vecE`.

Explanation

Consider an electric dipole consisting of two equal and opposite charges (-q and +q) separated by a small distance 2a. Let this dipole be placed in uniform electric field `vecE` at an angle `theta` with the direction of E.

Force on + q charge is given by, `vecF = q vecE` (along the direction of `vecE`)
Force on -q charge is given by, `vecF = -9 vecE` (along the direction opposite to `vecE`)
These two forces are equal and opposite and they act at different points. Hence, they form a couple whose torque is given by:
`pi=`Force xxperpendicular distance.
`pi=F xx AC`
From `Delta ABC,(AC)/(AB)=sintheta`
`implies" "AC=ABsintheta=2a sintheta`
Substituting the value of AC in equation (i), we get
`pi=qE.2a sin theta`
`implies" "pi=(q.2a)E sin theta:. pi=pE sin theta`
In terms of vectors `vecpi = vecP xx vec E" "[:.q.2a = p]`
The direction of `vecpi` is given by the right hand screw rule and is perpendicular to the plane containing `vecp` and `vecE` as shown in figure (b).