Question

Show that the torque on an electric dipole placed in a uniform electric field is
$\overrightarrow{\tau }=\overrightarrow{P}\times \overrightarrow{E}$

Answer 1

Consider a dipole with charges +q and –q with a distance d away from each other. The dipole is placed in a uniform electric field E such that the axis of the dipole forms an angle $\theta$ with the electric field.

The force on the charges is
${\overrightarrow{F}}_{+}=+q\overrightarrow{E}$
${\overrightarrow{F}}_{-}=-q\overrightarrow{E}$

The components of force perpendicular to the dipole are:
$F_{+}^{\perp }=+qEsin\theta$
$F_{-}^{\perp }=-qEsin\theta$

Since the force magnitudes are equal and are separated by a distance d, the torque on the dipole is given by:
$Torque\left(\tau \right)=Force\times Distance$
$\tau =\left(qE\right)dsin\theta$

Now, the dipole moment is given by
$p=qd$

The direction of the dipole moment is from the positive to the negative charge. Notice that the torque is in the clockwise direction (hence negative) in the above figure if the direction of Electric field is positive.
Thus, $\tau =pEsin\theta$
$\tau =\overrightarrow{p}\times \overrightarrow{E}$