Question

Consider an electric dipole comprising two charges \(+q\) and \(-q\), each with mass \(m\), separated by a fixed distance \(d\) and initially at rest with its dipole moment pointing along \(\hat{i}\). A uniform electric field \(E\hat{j}\) is turned on at time \(t=0\) and it is turned off at \(t=t_f\), when the dipole moment makes an angle \(\theta_f\) with \(\hat{i}\). Neglecting any sources of energy loss, correct option(s) is/are:

• A. The center of mass of the dipole is deflected towards $\hat{j}$ in the presence of the field.
• B. If the magnitude of the final angular velocity $\omega_f = \sqrt{\frac{2qE}{md}}$, then $\theta_f = \frac{\pi}{6}$.
• C. If $\theta_f = \pi/3$, then the change in kinetic energy of the dipole is given by $2\sqrt{3}qEd$.
• D. For $\theta_f = \pi/4$, the dipole rotates around its center of mass with a constant angular velocity after $t > t_f$.

Hint:

Dipoles in uniform fields are purely rotational systems. Always use $W = -\Delta U$ where $U = -\vec{p} \cdot \vec{E}$ to find the kinetic energy gained during rotation.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
A dipole in a uniform field experiences no net force but does experience a torque. The energy supplied by the field is converted into rotational kinetic energy.

Step 2: Key Formula or Approach:

• Net force $F_{net} = 0$, so CM remains stationary.

• Torque $\tau = |\vec{p} \times \vec{E}| = pE \cos\theta$ (where $\theta$ is angle with $\hat{i}$).

• Work-Energy: $\Delta K = \int \tau d\theta = pE \sin\theta_f$.

• Moment of inertia $I = 2 \times m(d/2)^2 = \frac{md^2}{2}$.

Step 3: Detailed Explanation:

• Statement (A): Since the field is uniform, $F_{net} = qE - qE = 0$. CM does not move. (A) is incorrect.
• Statement (B): $K = \frac{1}{2} I \omega_f^2 = \frac{1}{2} (\frac{md^2}{2}) \omega_f^2 = \frac{md^2 \omega_f^2}{4}$. Equating work done by field: $qdE \sin\theta_f = \frac{md^2 \omega_f^2}{4}$. Substitute $\omega_f^2 = \frac{2qE}{md}$: $qdE \sin\theta_f = \frac{md^2}{4} \frac{2qE}{md} = \frac{1}{2} qdE \implies \sin\theta_f = 1/2 \implies \theta_f = \pi/6$. (B) is correct.
• Statement (D): Once the field is off, the net torque is zero. The dipole will continue to rotate with the angular velocity it had at $t = t_f$. (D) is correct.

Step 4: Final Answer:
The correct options are (B) and (D).