Question

Derive the expression for the torque acting on an electric dipole, when it is held in a uniform electric field. Identify the orientation of the dipole in the electric field, in which it attains a stable equilibrium.

Answer 1

Consider a permanent dipole, of dipole moment $\overrightarrow{p}$ in a uniform external field $\overrightarrow{E}$, as shown in Fig.



There is a force $q\overrightarrow{E}$ on $q$ and a force $–q\overrightarrow{E}$ on $–q$. The net force on the dipole is zero since $\overrightarrow{E}$ is uniform.

Magnitude of torque $=qE\times 2a\sin \theta$

$=2qaE\sin \theta$

Its direction is normal to the plane of the paper, coming out of it.

The magnitude of $\overrightarrow{p}\times \overrightarrow{E}$ is $pE\sin \theta$ and its direction is normal to the paper, coming out of it. Thus, $\overrightarrow{\tau }=\overrightarrow{p}\times \overrightarrow{E},$ where $p=2qa$

When the dipole is placed in an electric field in the direction of the field i.e., when $\theta =0^{\circ }$, it attains a stable equilibrium.