question_answer

An electric dipole has a fixed dipole moment\[\vec{p},\], which makes angle \[\theta \] with respect to x-axis. When subjected to an electric\[{{\vec{E}}_{1}}=E\hat{i},\] field It experiences a torque \[\vec{T}=\tau \hat{k}.\] When subjected to another electric field \[{{\vec{E}}_{2}}=\sqrt{3}{{E}_{1}}\hat{j}\] it experiences torque\[{{\vec{T}}_{2}}=-{{\vec{T}}_{1}}.\] The angle \[\theta \] is:                             [JEE Main 2017]

A)  \[{{60}^{o}}\]                                   

B)  \[{{90}^{o}}\]

C)  \[{{30}^{o}}\]                                   

D) \[{{45}^{o}}\]

Correct Answer: A

Solution :

 So from\[\vec{\tau }=\vec{p}\times \vec{E}\] \[\tau \hat{k}-\tau \hat{k}=\left( {{p}_{x}}\hat{i}+{{p}_{y}}\hat{j} \right)\times \left( E\hat{i}+\sqrt{3}E\hat{j} \right)\] \[={{p}_{x}}\sqrt{3}E\hat{k}+{{p}_{y}}E\left( -\hat{k} \right)\] \[0=E\hat{k}\left( \sqrt{3}{{p}_{x}}-{{p}_{y}} \right)\] \[\frac{{{p}_{y}}}{{{p}_{x}}}=\sqrt{3}\] \[\therefore \]  \[\tan \theta =\sqrt{3}\] \[\theta ={{60}^{o}}\]