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Question #42210

Difficulty: 8/10 | 🟢 Free

The Question

An ideal elastic string of natural length $L$ and stiffness $k$ lies along the $x$ -axis with one end fixed at the origin. A small electric dipole of moment $p$ is attached to the free end of the string. The region contains a non-uniform electric field $\vec{E} = \alpha x^2 \hat{i}$ (where $\alpha > 0$ ). Initially, the dipole is held at $x = L$ with its moment oriented along the $y$ -axis ( $\vec{p} = p \hat{j}$ ). An external agent slowly rotates the dipole until it is aligned with the positive $x$ -axis ( $\vec{p} = p \hat{i}$ ) and simultaneously allows the string to reach its new equilibrium length. Assuming $k > 2p\alpha$ and all quantities are in SI units, what is the work done by the external agent during this entire process?

Options

A.
$-\frac{p \alpha k L^2}{k - 2 p \alpha}$
B.
$-p \alpha L^2$
C.
$-\frac{p \alpha k^2 L^2}{(k - 2 p \alpha)^2}$
D.
$\frac{p \alpha k L^2}{k - 2 p \alpha}$

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

Work Done in Stretching|Dipole Potential Angular Dependence|Conservative Field Path Independence

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to couple electrostatic potential energy with mechanical strain energy in a non-uniform field.

The primary trap is assuming the dipole's position remains fixed at $x=L$ during rotation, leading to the incorrect answer $-p\alpha L^2$ .

A secondary trap involves calculating the new equilibrium position but failing to include the positive work done to stretch the spring in the system's total energy balance.

It rigorously evaluates the understanding that $W_{ext} = \Delta U_{total}$ and that the electric force itself is position-dependent, creating a feedback loop with the spring's restoring force.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Mechanical Properties of Solids - Class 11 Part 2
Concept	Work Done in Stretching
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free