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Question #42655

Difficulty: 8/10 | 🟢 Free

The Question

Two identical point charges, each of charge $+Q$ , are fixed at positions $(d, 0, 0)$ and $(-d, 0, 0)$ in a Cartesian coordinate system. A small ideal electric dipole of dipole moment $\vec{p} = p \hat{j}$ (where $p > 0$ ) is constrained to move only along the $y$ -axis with its orientation fixed. The dipole is placed near its stable equilibrium position $y_0 > 0$ on the $y$ -axis. If the system is in thermal equilibrium at absolute temperature $T$ , find the root-mean-square (RMS) displacement of the dipole from its equilibrium position. Assume the displacement is small enough that the restoring force is linear. Let $k_B$ be the Boltzmann constant and $\epsilon_0$ be the vacuum permittivity.

Options

A.
$\sqrt{ \frac{9\sqrt{3}\pi\epsilon_0 d^4 k_B T}{8pQ} }$
B.
$\sqrt{ \frac{9\sqrt{2}\pi\epsilon_0 d^4 k_B T}{8pQ} }$
C.
$\sqrt{ \frac{3\sqrt{3}\pi\epsilon_0 d^4 k_B T}{4pQ} }$
D.
$\sqrt{ \frac{27\pi\epsilon_0 d^4 k_B T}{4pQ} }$

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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

Neutral Equilibrium Points|Dipole in Non-Uniform Field|Gravitational Potential Energy (System)|Energy of Vibrational Modes

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to correctly model the interaction of a dipole in a non-uniform electric field and couple it with statistical mechanics.

A common trap is treating the non-uniform field as uniform ( $F=qE$ ) or incorrectly evaluating the restoring force by taking the first derivative of the field at the origin rather than the second derivative of the potential energy at the true equilibrium point (the Hessian).

By requiring the use of the Equipartition Theorem, the problem forces students to rigorously derive the stiffness constant ( $k = \partial^2 U / \partial y^2$ ) of the potential energy landscape, penalizing any algebraic shortcuts or dimensional inconsistencies.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electric Charges and Fields - Class 12
Concept	Neutral Equilibrium Points
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free