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Question #42566

Difficulty: 8/10 | 🟢 Free

The Question

An ideal electric dipole of dipole moment $p$ is constrained to move slowly in the $xy$ -plane within a non-uniform electric field $\vec{E} = \alpha x \hat{i}$ , where $\alpha$ is a positive constant. The dipole is moved from $x = d$ to $x = 2d$ along the x-axis. During this translation, an external agent simultaneously rotates the dipole such that the angle $\theta$ between the dipole moment and the positive x-axis continuously satisfies $\cos\theta = \frac{d}{x}$ . What is the work done strictly by the external torque during this entire process?

Options

A.
$p \alpha d \ln 2$
B.
$-p \alpha d \ln 2$
C.
$p \alpha d$
D.
$0$

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

Work Done in Non-PV Diagrams|Work Done in Rotating Dipole|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the critical distinction between the total work done (change in potential energy) and the specific work done by rotational versus translational forces in a non-uniform field.

Students often incorrectly assume $W = \Delta U$ applies to the torque alone.

Here, the potential energy $U = -\vec{p}\cdot\vec{E} = -p\alpha x \cos\theta = -p\alpha d$ is constant throughout the path, meaning $\Delta U = 0$ .

The lethal trap is choosing 0 (Option D), ignoring that the external agent does positive work via torque ( $p\alpha d \ln 2$ ) and an equal amount of negative work via translational force ( $-p\alpha d \ln 2$ ) to maintain the kinematic constraint.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Thermodynamics - Class 11 Part 2
Concept	Work Done in Non-PV Diagrams
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free