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Question #42917

Difficulty: 8/10 | 🟢 Free

The Question

A parallel plate capacitor has rectangular plates of length $L$ and width $w$ (Area $A = Lw$ ) separated by a distance $d$ . It is completely submerged in a liquid dielectric of dielectric constant $K_1$ and connected to a battery that maintains a constant voltage $V$ . A solid massless dielectric slab of dielectric constant $K_2$ ( $K_2 < K_1$ ) and dimensions $L \times w \times d$ is initially placed just outside the capacitor. The slab is connected via a massless, inextensible string to a uniform solid cylinder of mass $M$ and radius $R$ , which is hinged at its central horizontal axis. The string is wrapped around the cylinder and pulls the slab horizontally. The cylinder is given an initial angular velocity $\omega_0$ such that it pulls the slab completely into the capacitor. The cylinder comes to a momentary halt exactly when the slab is fully inserted. Assuming the liquid freely flows out with negligible viscous drag and neglecting gravity, find the initial angular velocity $\omega_0$ of the cylinder.

Options

A.
$\frac{V}{R} \sqrt{ \frac{2 \epsilon_0 A (K_1 - K_2)}{M d} }$
B.
0
C.
$\frac{V}{R} \sqrt{ \frac{\epsilon_0 A (K_1 - K_2)}{M d} }$
D.
$\frac{2V}{R} \sqrt{ \frac{\epsilon_0 A (K_1 - K_2)}{M d} }$

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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

Gauss Law Applicability|Dielectric Slab Displacement|Work-Energy Theorem for Rotation

Why This Question Is Tricky

Common Mistakes

This question tests the student's understanding of the physical origin of the force on a dielectric and its integration with classical mechanics.

A common trap is to incorrectly apply Gauss's Law for infinite plates ( $E = V/d$ ), assuming the electric field is strictly uniform and confined between the plates.

Under this false assumption, the field inside the dielectric does not change as it moves, yielding a net horizontal force of zero and leading to the incorrect answer of zero initial angular velocity (Option B).

The problem forces the recognition that the force on a dielectric is fundamentally a fringe-field effect, which is mathematically captured by the spatial derivative of the electrostatic potential energy.

Furthermore, it tests the correct application of the Work-Energy theorem, requiring students to equate the non-conservative work done against the electrostatic fringe fields to the rotational kinetic energy of the solid cylinder, while avoiding common pitfalls like using the moment of inertia of a ring (Option C) or forgetting the factor of 1/2 in the energy formula (Option D).

Quick Facts About This Question

Question Information

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