Import Question JSON

$\text{Hint: } \tau = qE \sin \theta$ $\text{Step 1: Find the moment of inertia about centre of mass i.e., } I_{CM}. \text{The moment of inertia of a given configuration about centre of mass is given by;}$ $I_{CM} = \frac{m}{2} r_1^2 + mr_2^2$ $\text{Where, } r_1 = \frac{l \times m}{m + m} = \frac{2l}{3}, r_2 = \frac{l \times \frac{m}{2}}{\frac{m}{2} + m} = \frac{l}{3}$ $\Rightarrow I_{CM} = \frac{m}{2} \left( \frac{2l}{3} \right)^2 + m \left( \frac{l}{3} \right)^2$ $\Rightarrow I_{CM} = \frac{ml^2}{3}$ $\text{Step 2: Find the angular frequency of oscillation.}$ $\text{We know that when a dipole is placed in a uniform electric field then torque acting on the dipole is given by;}$ $\vec{\tau} = \vec{p} \times \vec{E} = pE \sin \theta$ $\text{As } \theta \text{ is very small, } \sin \theta \approx \theta$ $\Rightarrow \vec{\tau} = qlE \theta$ $\Rightarrow I_{CM} \times \alpha = qlE \theta$ $\Rightarrow \alpha = \frac{qlE}{I_{CM}} \theta \quad \cdots (1)$ $\text{The general equation of angular SHM is given by;}$ $\alpha = \omega^2 \theta \quad \cdots (2)$ $\text{From equation (1) and (2) we get;}$ $\omega^2 = \frac{qlE}{I_{CM}}$ $\Rightarrow \omega = \sqrt{\frac{qlE}{\frac{ml^2}{3}}} = \sqrt{\frac{3qE}{ml}}$ $\text{Hence, option (2) is the correct answer.}$