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Question #42472

Difficulty: 8/10 | 🟢 Free

The Question

A short electric dipole of constant dipole moment $\vec{p} = p_0 \hat{i}$ is placed in a region where the electrostatic field is given by $\vec{E} = E_0 \left( \frac{x^3}{a^3} - \frac{3x}{a} \right) \hat{i}$ , where $E_0$ and $a$ are positive constants. The dipole is constrained to move only along the $x$ -axis while maintaining its orientation. If the dipole is released from rest from its unstable equilibrium position and given an infinitesimal nudge towards the origin, what is the work done by the electric field on the dipole by the time it reaches its stable equilibrium position?

Options

A.
4 p_0 E_0
B.
-4 p_0 E_0
C.
0
D.
2 p_0 E_0

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Variable Force Integration|Equilibrium Stability|Potential Gradient Direction

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to correctly model the force on a dipole in a non-uniform electric field and analyze equilibrium stability.

A common trap is assuming the force is proportional to the electric field ( $F \propto E$ ) and integrating $E(x)$ directly, which yields zero work since $E(x)$ is an odd function.

Another trap is failing to use the second derivative test ( $\frac{d^2U}{dx^2}$ ) to correctly identify $x=a$ as the unstable and $x=-a$ as the stable equilibrium, leading to a reversed integration limit and a sign error in the work-energy theorem application.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Work, Energy, and Power - Class 11 Part 1
Concept	Variable Force Integration
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free