Question

Two small identical electrical dipoles AB and CD, each of dipole moment 'p' is kept at an angle ${120}^{\circ }$. What is the resultant dipole moment of this combination? If this system is subjected to an electric field (E vector) directed along the + x-direction, what will be the magnitude and direction of the torque acting on this?

Answer 1

The resultant dipole moment of the arrangement will be given as,
$P_R^2=P_{AB}^2+P_{CD}^2+2P_{AB}{P}_{CD}cos{120}^{\circ }$
As, $P_{AB}=P_{CD}=p$, therefore we get,
$P{R}^2=p^2+p^2+2p^2(-\frac{1}{2})$
$P{R}^2=2p^2-p^2$
$P{R}^2=p^2$
PR = p
And the magnitude of the torque when the system is subjected to the electic field E along the +x-direction will be,
$\tau =pEsin\theta$
As, the electric field is applied along x – axis, therefore the angle between the dipole and x–axis will be
$\theta ={120}^{\circ }-{90}^{\circ }={30}^{\circ }$
On substituting the values in the question we get,
$\tau =\frac{pE}{2}$
Thus, the magnitude of the torque will be and its direction will be clockwise when it is viewed from the above that is $\overrightarrow{\tau }$ is perpendicular to both $\overrightarrow{p}and\overrightarrow{E}$.