An electric dipole has a fixed dipole moment \(\vec P\) 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.......\(^o\)

\(\quad \vec{T}=\vec{P} \times \vec{E}\)

\(\vec{p}=p \cos \theta \hat{i}+p \sin \theta \hat{j}\)

\(\vec{E}_{1}=\overrightarrow{E i}\)

\(\vec{T}_{1}=\vec{p} \times \vec{E}_{1}=(p \cos \theta \hat{i}+p \sin \theta \hat{j}) \times E(\hat{i})\)

\(\tau \hat{k}=p E \sin \theta(-\hat{k}).........(i)\)

\(\vec{E}_{2}=\sqrt{3} E_{1} \hat{j}\)

\(\left.\vec{T}_{2}=p \cos \theta \hat{i}+p \sin \theta \hat{j}\right) \times \sqrt{3} E_{1} \hat{j}\)

\(\tau \hat{k}=\sqrt{3} p E_{1} \cos \theta \hat{k}.........(ii)\)

From eqns. ( \(i\) ) and \((ii )\)

\(p E \sin \theta=\sqrt{3} p E \cos \theta\)

\(\tan \theta=\sqrt{3} \quad \therefore \quad \theta=60^{\circ}\)