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Question #43847

Difficulty: 8/10 | 🟢 Free

The Question

A polar molecule with a constant dipole moment $\vec{p} = p \cos\theta \hat{i} + p \sin\theta \hat{j}$ is initially located at $x = 0$ in a depletion region. The electric field in this region is non-uniform and given by $\vec{E}(x) = E_0 \left(1 - \frac{x}{2W}\right) \hat{i}$ for $0 \le x \le W$ . Initially, the dipole is aligned with the electric field ( $\theta = 0$ ). An external agent moves the molecule quasi-statically from $x = 0$ to $x = W$ . During this translation, the agent simultaneously rotates the dipole such that its angle $\theta$ with the $+x$ -axis increases linearly with $x$ , reaching $\theta = \pi/2$ at $x = W$ . What is the work done by the electric torque on the dipole during this process?

Options

A.
$-p E_0 \left( 1 - \frac{1}{\pi} \right)$
B.
$-p E_0$
C.
$-\frac{3}{4} p E_0$
D.
$-p E_0 \left( 1 + \frac{1}{\pi} \right)$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Depletion Region Dynamics|Work Done in Rotating Dipole|Reference Potential Convention|Vector Cross-Products

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to decouple the work done by electric torque from the total electrostatic potential energy change in a non-uniform field.

A common trap is assuming that the work done by the electric torque simply equals the negative change in potential energy ( $W = -\Delta U = -p E_0$ ), ignoring that $\Delta U$ accounts for both rotational work and translational work ( $W_{trans} = \int F_x dx$ ).

Another trap is treating the electric field as uniform or using its average value ( $-\frac{3}{4} p E_0$ ) for the torque integral.

By forcing a specific kinematic coupling between translation and rotation ( $\theta \propto x$ ), the problem requires a rigorous path integral of the torque $\vec{\tau} = \vec{p} \times \vec{E}(x)$ , penalizing heuristic shortcuts.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Semiconductor Electronics: Materials, Devices, and Simple Circuits - Class 12
Concept	Depletion Region Dynamics
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free