Question

Two small identical electrical dipoles $AB$ and $CD$, each of dipole moment ${}^{\prime }{p}^{\prime }$ are kept at an angle of ${120}^{\circ }$ as shown in the figure. What is the resultant dipole moment of this combination? If this system is subjected to electric field $\left(\overrightarrow{E}\right)$ directed along $+X$ direction, what will be the magnitude and direction of the torque acting on this?

Answer 1

Resultant dipole moment

${\overrightarrow{p}}_r=\sqrt{p^2+p^2+2pp\cos {120}^{\circ }}$

$=\sqrt{2p^2+2p^2\cos {120}^{\circ }}$

$=\sqrt{2p^2+\left(2p^2\right)\times (-\frac{1}{2})=}\sqrt{2p^2-p^2}=p$,

Using law of addition of vectors, we can see that the resultant dipole makes an angle of ${60}^{\circ }$ with the $y$ axis or ${30}^{\circ }$ with $x$ - axis.

Torque, $\overrightarrow{\tau }=\overrightarrow{p}\times \overrightarrow{E}(\overrightarrow{\tan }$ is perpendicular to both $\overrightarrow{p}$ and $\overrightarrow{E}$

$=pE\sin {30}^{\circ }=\frac{1}{2}pE$.

Direction of torque is along positive $Z-direction$.