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Question #42682

Difficulty: 8/10 | 🟢 Free

The Question

A parallel plate capacitor is formed by two identical vertical triangular metal plates of base $b$ and height $L$ . The bases of the triangular plates are horizontal and located at the bottom edge, while their apexes point vertically upwards. The separation between the plates is $d$ . A rectangular dielectric slab of mass $M$ , thickness $d$ , width $b$ , and length $L$ (with dielectric constant $K$ ) is inserted vertically upwards from the bottom into the space between the plates. The slab is constrained to move only in the vertical direction. The plates are maintained at a constant potential difference $V$ by an ideal battery. Let $x$ be the vertical depth of the slab inserted into the capacitor (measured from the bottom base). Assume $V$ is large enough such that an equilibrium position $x_0$ exists within $0 < x_0 < L$ . Neglecting fringing fields, which of the following correctly gives the equilibrium insertion depth $x_0$ and the nature of this equilibrium?

Options

A.
$x_0 = L - \frac{2MgdL}{\epsilon_0 b (K-1) V^2}$ , Stable
B.
$x_0 = L - \frac{2MgdL}{\epsilon_0 b (K-1) V^2}$ , Unstable
C.
$x_0 = \frac{2MgdL}{\epsilon_0 b (K-1) V^2}$ , Stable
D.
$x_0 = \frac{2MgdL}{\epsilon_0 b (K-1) V^2}$ , Unstable

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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

Dielectric Polarization Logic|Equilibrium Stability|Symmetry and Virtual Work

Why This Question Is Tricky

Common Mistakes

This question tests the 'Battery Work' fallacy in the context of the Principle of Virtual Work and stability analysis.

Students often incorrectly assume that the electrostatic force is the negative gradient of the stored electrostatic potential energy ( $F_e = -dU/dx$ ), forgetting that the battery does work during the displacement ( $dW_{batt} = 2dU$ ).

For a constant voltage source, the correct relation is $F_e = +dU/dx$ .

If a student uses the incorrect relation, they will derive an effective potential energy $U_{eff} = U_{cap} + U_{grav}$ .

Because the area function $A_{in}$ contains a negative $x^2$ term, $U_{cap}$ is an inverted parabola ( $d^2U_{cap}/dx^2 < 0$ ).

This leads the student to falsely conclude that $d^2U_{eff}/dx^2 < 0$ and the equilibrium is unstable.

The correct effective potential energy is $U_{eff} = -U_{cap} + U_{grav}$ , which yields $d^2U_{eff}/dx^2 > 0$ , correctly identifying the equilibrium as stable.

Quick Facts About This Question

Question Information

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