Electric Dipole in an External Uniform Electric Field ( Torque on a Dipole in uniform Electric Field)
Consider an electric dipole in a uniform electric field $\vec{E}$ such that the dipole moment ($\vec{p}$) of the electric dipole makes an angle $\theta$ with the electric field $\vec{E}$
Charge +q and -q constituting the dipole , experience equal and opposite forces given by $\vec{F} = q \vec{E}$ and $\vec{F}= -q \vec{E}$ respectively due to electric field $\vec{E}$
Therefore , the net force on the electric dipole is given by
$\vec{F_{net}} = q \vec{E} - q\vec{E} = 0$
This , Net force on an electric dipole in a uniform electric field is zero.
Therefore , electric dipole placed in uniform electric field does not undergo any translatory motion.In other word , electric dipole is in translational equilibrium in an uniform electric field.
But two equal and opposite forces acting on the dipole constitute a couple. This couple tends to rotate the dipole in the clockwise direction and hence tries to align the dipole along the direction of the electric field.
Torque ( $\tau$) = moment of couple
= Magnitude of either force \times Perpendicular distance between two forces.
$\tau= qE \times AC$ ....(i)
From $\triangle ABC$
$\frac{AC}{AB} = \sin \theta$
$AC = AB \sin \theta = 2l \sin \theta$
Hence, eqn. (i) becomes,
$\tau = qE \times 2l \sin \theta$
$\tau= (q \times 2l) E \sin \theta$
But $q \times 2l = p$(dipole moment)
$\tau = pE \sin \theta$ ..... (ii)
In vector form, eqn. (ii) can be written as : $\vec{\tau} = \vec{p} \times \vec{E}$ ...(iii)
The direction of $\vec{\tau}$ is given by right handed screw rule. In the above case, direction of $\vec{\tau}$ is into the plane of the paper perpendicular to the plane containing $\vec{p}$ and $\vec{E}$.
Special Cases:
1. If $\theta = 0^\circ$, then $\tau = pE \sin 0^\circ = 0$
Thus, when dipole moment ($\vec{p}$) is parallel to the electric field $\vec{E}$, no torque acts on the dipole as shown in figure 52(A). Thus, electric dipole is in stable equilibrium.
2. If $\theta = 30^\circ$, then $\tau = pE \sin 30^\circ = \frac{pE}{2}$ (i.e., half the maximum value)
Thus, when dipole moment ($\vec{p}$) makes an angle of $30^\circ$ with the uniform electric field ($\vec{E}$), the torque acting on the dipole is half the maximum value of the torque acting on the dipole.
3. If $\theta = 90^\circ$, then $\tau = pE \sin 90^\circ = pE$ (maximum)
Thus, when dipole moment ($\vec{p}$) is perpendicular to the electric field $\vec{E}$, maximum torque acts on the dipole.
4. If $\theta = 180^\circ$, then $\tau = pE \sin 180^\circ = 0$
It means when dipole moment $\vec{p}$ is anti-parallel to the electric field $\vec{E}$, no torque acts on the dipole. In this case, electric dipole is in unstable equilibrium.