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Question #41987

Difficulty: 8/10 | 🟢 Free

The Question

A uniformly charged fixed ring of radius $R$ and total charge $Q$ lies in the $y-z$ plane with its center at the origin. A particle of mass $m$ and charge $q$ (where $qQ > 0$ ) is constrained to move along the $x$ -axis. The particle is attached to a light ideal spring of spring constant $k$ , whose other end is fixed at the origin. If $k = \frac{q Q}{32 \pi \epsilon_0 R^3}$ , the particle is displaced slightly from its stable equilibrium position $x_0 > 0$ . What is the angular frequency $\omega$ of its small oscillations?

Options

A.
$\frac{3}{8} \sqrt{\frac{q Q}{2 \pi \epsilon_0 m R^3}}$
B.
$\sqrt{\frac{7 q Q}{32 \pi \epsilon_0 m R^3}}$
C.
$\frac{1}{4} \sqrt{\frac{q Q}{2 \pi \epsilon_0 m R^3}}$
D.
$\frac{3}{4} \sqrt{\frac{q Q}{2 \pi \epsilon_0 m R^3}}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Electric Field Symmetry|Effective Spring Constant|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to find the true stable equilibrium in a non-linear force field and correctly compute the effective restoring force.

A common trap is assuming the equilibrium remains at the origin ( $x=0$ ) and blindly applying $k_{eff} = |k - dF_e/dx|$ , which yields the incorrect frequency for the unstable equilibrium (Option B).

Students must minimize the total potential energy functional $U(x)$ to find the new global minimum $x_0 = \sqrt{3}R$ and evaluate the second derivative $U''(x_0)$ to find the true effective spring constant.

This penalizes formula-plugging and rewards first-principle energy minimization.

Quick Facts About This Question

Question Information

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