An electric dipole has fixed dipole moment $\vec p$, which makes angle $\theta$ 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 $\vec E_2 = \sqrt3 E_1\hat j$, it experiences a torque $\vec T_2 = -\vec T_1$. The angle $\theta$ is:
Answer: (D) $60^\circ$
Write $\vec p = p(\cos\theta\,\hat i + \sin\theta\,\hat j)$. Then
$$\vec T_1 = \vec p\times E\hat i = -pE\sin\theta\,\hat k$$
$$\vec T_2 = \vec p\times\sqrt3E\hat j = \sqrt3\,pE\cos\theta\,\hat k$$
The torques are equal and opposite, so their magnitudes are equal:
$$pE\sin\theta = \sqrt3\,pE\cos\theta \;\Rightarrow\; \tan\theta = \sqrt3 \;\Rightarrow\; \theta = 60^\circ$$
Solution by Sreeraj P, M.Sc Physics