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Question #42323

Difficulty: 8/10 | 🟢 Free

The Question

A parallel plate capacitor has rectangular plates of length $L$ , width $w$ , and separation $d$ . It is given a charge $Q$ and then isolated. A dielectric slab of mass $m$ , dielectric constant $K$ , and dimensions $L \times w \times d$ is attached to a relaxed spring of spring constant $k$ . The slab is partially inserted into the capacitor by a distance $x$ parallel to its length. The system reaches an equilibrium state where the slab is inserted by a distance $x_0$ ( $0 < x_0 < L$ ). If the slab is slightly displaced from this equilibrium position, what is the angular frequency $\omega$ of the resulting small oscillations? Assume friction and edge effects are negligible.

Options

A.
$\sqrt{\frac{k}{m} \left( \frac{L + 3(K-1)x_0}{L + (K-1)x_0} \right)}$
B.
$\sqrt{\frac{k}{m} \left( \frac{L + (K-1)x_0}{L + 3(K-1)x_0} \right)}$
C.
$\sqrt{\frac{k}{m} \left( \frac{L - (K-1)x_0}{L + (K-1)x_0} \right)}$
D.
$\sqrt{\frac{k}{m}}$

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

Spring Potential Energy|Energy Density Locality|Non-Uniform Dielectric Filling

Why This Question Is Tricky

Common Mistakes

This question tests the critical distinction between constant voltage and constant charge scenarios in dielectrics.

Students often memorize that the electrostatic force on a dielectric slab is constant, which is only true when connected to a battery (constant voltage).

For an isolated capacitor (constant charge), the force depends on the insertion distance $x$ , creating a non-linear restoring force.

If students incorrectly assume a constant force, they will conclude $k_{eff} = k$ and choose the trap option.

The problem forces them to apply the Principle of Virtual Work ( $F = -dU/dx$ ) and evaluate the second derivative of the local field energy to find the true state-dependent effective spring constant.

Quick Facts About This Question

Question Information

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