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Question #42425

Difficulty: 8/10 | 🟢 Free

The Question

A small electric dipole of dipole moment $p$ and moment of inertia $I$ is placed in a non-uniform electric field $\vec{E} = E_0 \left( \frac{x^4}{4L^4} - \frac{x^2}{2L^2} \right) \hat{i}$ , where $E_0$ and $L$ are positive constants. The dipole is free to move along the $x$ -axis and rotate in the $xy$ -plane. If the dipole is slightly rotated from its fully stable equilibrium configuration, what is the angular frequency of its small rotational oscillations?

Options

A.
$\frac{1}{2}\sqrt{\frac{p E_0}{I}}$
B.
$\sqrt{\frac{p E_0}{I}}$
C.
$\sqrt{\frac{2p E_0}{I}}$
D.
0

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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

Free Body Diagram Isolation|Dipole in Non-Uniform Field|Neutral Equilibrium Points

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to rigorously define stable equilibrium in a coupled translational-rotational system using the Hessian of the potential energy.

A common trap is assuming the dipole will always align with $\theta=0$ and finding the force equilibrium at $x=0$ .

However, at $x=0$ , the electric field vanishes, leading to a neutral equilibrium for rotation ( $U_{\theta\theta}=0$ ).

The true stable equilibrium occurs at $x=\pm L$ , where the local electric field points in the $-\hat{i}$ direction, forcing the stable orientation to be $\theta=\pi$ .

Students who fail to check the second derivative of the potential energy with respect to both position and angle will incorrectly conclude the frequency is zero or undefined.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Laws of Motion - Class 11 Part 1
Concept	Free Body Diagram Isolation
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free