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Question #42785

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The Question

A parallel plate capacitor with vertical rectangular plates separated by a distance $d$ is connected to a constant DC voltage source $V$ . A dielectric slab of mass $m$ , thickness $d$ , and dielectric constant $K$ is suspended vertically between the plates by an ideal spring of spring constant $k$ . The slab is shaped as an isosceles triangle of base $b$ and height $h$ , and is oriented such that its tip points downwards. The slab is inserted tip-first into the capacitor. Let $x$ be the depth of the tip inside the capacitor. The spring is unstretched when $x = 0$ . Assuming the slab remains partially inside the capacitor and neglecting fringing fields, determine the equilibrium depth $x_{eq}$ and the angular frequency $\omega$ of small vertical oscillations about this equilibrium. (Assume $k > \lambda$ , where $\lambda = \frac{\epsilon_0 (K-1) b V^2}{2 h d}$ )

Options

A.
$x_{eq} = \frac{mg}{k - \lambda}$ , $\omega = \sqrt{\frac{k - \lambda}{m}}$
B.
$x_{eq} = \frac{mg}{k + \lambda}$ , $\omega = \sqrt{\frac{k + \lambda}{m}}$
C.
$x_{eq} = \frac{mg}{k - 2\lambda}$ , $\omega = \sqrt{\frac{k - 2\lambda}{m}}$
D.
$x_{eq} = \frac{mg}{k + 2\lambda}$ , $\omega = \sqrt{\frac{k + 2\lambda}{m}}$

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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

Dielectric Slab Displacement|Equilibrium Stability|Variable Force Integration

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to correctly apply energy conservation and equilibrium stability principles to a variable capacitor system.

The primary trap lies in the calculation of the electrostatic force.

Students often mistakenly use $F = -\frac{dU_E}{dx}$ with $U_E = \frac{1}{2}CV^2$ , forgetting that the battery does work $W_B = V^2 \Delta C$ .

This omission leads to an incorrect sign for the electrostatic force ( $F_e = -\lambda x$ instead of $+\lambda x$ ), resulting in the incorrect equilibrium position and frequency (Option B).

Furthermore, the non-linear area insertion requires careful integration to find $C(x)$ , testing their mathematical formulation of physical systems.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electrostatic Potential and Capacitance - Class 12
Concept	Dielectric Slab Displacement
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free