Question

Two small identical dipoles $\text{AB}$ and $\text{CD}$, each having dipole moment of magnitude $p$ are kept at an angle of ${120}^{\circ }$ as shown in the figure. If this system is subjected to an uniform electric field of magnitude $E$ directed along $+x$ direction, then the magnitude of torque acting on this is

• A. $\frac{pE}{2}$
• B. $\frac{pE}{4}$
• C. $2pE$
• D. $pE$

Answer 1

The correct option is A $\frac{pE}{2}$

In the figure the two dipole moments act at an angle of ${120}^{\circ }$. As they are equal the resultant dipole moment will be at an angle of ${60}^{\circ }$ from the $y-$ axis.


So, Resultant dipole moment will be

$|{\overrightarrow{p}}_R|=\sqrt{p_1^2+p_2^2+2p_1{p}_2\cos 120}$

from the question, $p_1=p,p_2=p$

$\therefore |{\overrightarrow{p}}_R|=\sqrt{p^2+p^2+2pp\times \left(\frac{-1}{2}\right)}$

$\Rightarrow |{\overrightarrow{p}}_R|=\sqrt{p^2}=p$


Now electric field acts along the $x-$ axis

$\Rightarrow \angle XOR=\theta ={30}^{\circ }$

So, torque acting on dipole due to uniform electric field will be

$\tau =pE\sin \theta$

$\Rightarrow \tau =pE\sin {30}^{\circ }=\frac{pE}{2}$

Hence, $\left(c\right)$ is the correct option.