Question

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

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

Answer 1

Answer: (C)

$\vec{p}=p\cos \theta \hat{i}+p\sin \theta \hat{j}$
${\vec{E}}_1=E\hat{i}$
${\vec{T}}_1=\vec{p}\times {\vec{E}}_1$
$=(p\cos \theta \hat{i}+p\sin \theta \hat{j})\times E(\hat{i})$
$\tau \hat{k}=pE\sin \theta (-\hat{k})$ ......(i)
${\vec{E}}_2=\sqrt{3}{E}_1\hat{j}$
${\vec{T}}_2=(p\cos \theta \hat{i}+p\sin \theta \hat{j})\times \sqrt{3}{E}_1\hat{j}$
$-\tau \hat{k}=\sqrt{3}p{E}_1\cos \theta \hat{k}$ ....(ii)
From (i) and (ii)
$pE\sin \theta =\sqrt{3}pE\cos \theta$
$\tan \theta =\sqrt{3}$
$\theta =60^{\circ }$