Question

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

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

Answer 1

The correct option is A ${60}^{\circ }$

We know that $\overrightarrow{\tau }=\overrightarrow{p}\times \overrightarrow{E}$

It is given that, $\overrightarrow{{\tau }_1}=\overrightarrow{T_1}=\overrightarrow{\tau }\hat{k}$
when

$\overrightarrow{p}=p_x\hat{i}+p_y\hat{j}$

$=p\cos \theta \hat{i}+p\sin \theta jˆ$

and $\overrightarrow{E}={\overrightarrow{E}}_1=E\hat{i}$

Put above value in formula

$\tau \hat{k}=(p_x\hat{i}+p_{y}jˆ)\times E\hat{i}$

Since $\hat{i}\times \hat{i}=0\text{and}\hat{j}\times \hat{i}=\widehat{-k}$

$p_y=\frac{-\tau }{E}\dots \left(1\right)$

Now also given ${\overrightarrow{\tau }}_2={\overrightarrow{T}}_2=-{\overrightarrow{T}}_1=-\tau \hat{k}$

when $\overrightarrow{p}=p_x\hat{i}+p_y\hat{j}$

$=p\cos \theta \hat{i}+p\sin \theta \hat{j}$

and $\overrightarrow{E}={\overrightarrow{E}}_2=\sqrt{3}{E}_1\hat{j}$

Put above value in formula

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

Since $\hat{j}\times \hat{j}=0\text{and}\hat{i}\times \hat{j}=\hat{k}$

$p_x=\frac{-\tau }{\sqrt{3}E}\dots \left(2\right)$

Dividing equation $\left(1\right)$ and $\left(2\right)$, we have

$\frac{p_y}{p_x}=\sqrt{3}$

Also $\tan \theta =\frac{p_y}{p_x}$

$\tan \theta =\sqrt{3}$

$\theta ={60}^{\circ }$