Question

An electric dipole has a fixed dipole moment $\overrightarrow{p}$, which makes angle $\theta$ with respect to $x-$ axis. When system is subjected to an electric field $\overrightarrow{E_1}=\sqrt{3}E\hat{i},$ it experiences a torque $\overrightarrow{{\tau }_1}=-\tau \hat{k}$. When subjected to another electric field $\overrightarrow{E_2}=E\hat{j}$, it experiences a torque $\overrightarrow{{\tau }_2}=-\overrightarrow{{\tau }_1}$. The angle $\theta$ is

• A. ${60}^{\circ }$
  
• B. ${30}^{\circ }$
  
• C. ${90}^{\circ }$
  
• D. ${45}^{\circ }$
  

Answer 1

The correct option is B ${30}^{\circ }$


Given:
$\overrightarrow{E_1}=\sqrt{3}E\hat{i};\overrightarrow{E_2}=E\hat{j}$

$\overrightarrow{{\tau }_1}=-\tau \hat{k};\overrightarrow{{\tau }_2}=-\overrightarrow{{\tau }_1}$

Let, dipole moment, $\overrightarrow{p}=p_x\hat{i}+p_y\hat{j}$


We know that torque on a dipole in a uniform electric field is

$\tau =\overrightarrow{p}\times \overrightarrow{E}$

So, when electric field $\overrightarrow{E_1}=\sqrt{3}E\hat{i}$ is applied, then torque,

$\overrightarrow{{\tau }_1}=\overrightarrow{p}\times \overrightarrow{E_1}$

$\Rightarrow -\tau \hat{k}=(p_x\hat{i}+p_y\hat{j})\times \sqrt{3}E\hat{i}$

$\Rightarrow -\tau \hat{k}=p_y\hat{j}\times \sqrt{3}E\hat{i}$

As, $\hat{j}\times \hat{i}=-\hat{k}\text{and}{p}_y=p\cos ({90}^{\circ }-\theta )$

$\Rightarrow -\tau \hat{k}=-\sqrt{3}pE\sin \theta \hat{k}$

$\Rightarrow \tau =\sqrt{3}pE\sin \theta ........\left(1\right)$

Now, when electric field, $\overrightarrow{E_2}=E\hat{j}$ applied then torque,

$\overrightarrow{{\tau }_2}=\overrightarrow{p}\times \overrightarrow{E_2}$

$\Rightarrow -\overrightarrow{{\tau }_1}=\tau \hat{k}=pE\cos \theta \hat{k}$

$\Rightarrow \tau =pE\cos \theta .........\left(2\right)$

From eqs.$\left(1\right)$ and $\left(2\right)$

$\sqrt{3}pE\sin \theta =pE\cos \theta$

$\Rightarrow \tan \theta =\frac{1}{\sqrt{3}}$

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

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