Couple Acting on an Electric Dipole in a Uniform Electric Field

Consider an electric dipole -- charges $-q$ at point A and $+q$ at point B, separated by $2l$ -- placed in a UNIFORM external electric field $\vec{E}$, with the dipole's axis making some angle $\theta$ with the direction of the field (Fig. 10.19a).

The force acting on the negative charge at A is $\vec{F}_A=-q\vec{E}$, directed OPPOSITE to the field; the force acting on the positive charge at B is $\vec{F}_B=+q\vec{E}$, directed ALONG the field. Since $\vec{F}_A=-\vec{F}_B$ (equal magnitude, opposite direction), but these two forces act at two DIFFERENT points (A and B, separated by some perpendicular distance d between their lines of action), they do NOT cancel out to give zero net effect -- instead, together they form a COUPLE (Fig. 10.19b). The turning effect (moment) of a couple is called TORQUE, defined generally as $\tau=(\text{perpendicular distance between the two forces})\times(\text{magnitude of each force}).$

Working out this perpendicular distance for the dipole geometry (using the projection of BA perpendicular to the field direction) gives $\tau=qE(2l\sin\theta)=q(2l)E\sin\theta.$ Recognising $q(2l)=p$, the dipole moment, this simplifies to $\tau=pE\sin\theta,$ or, written in full vector form using the cross product, $\vec{\tau}=\vec{p}\times\vec{E}.$ …