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Question #42710

Difficulty: 8/10 | 🟢 Free

The Question

An ideal LC circuit consists of an inductor of inductance $L$ and a parallel-plate capacitor of capacitance $C_0$ (when empty). The circuit undergoes undamped electrical oscillations with a maximum charge $Q_0$ on the capacitor plates. A dielectric slab of dielectric constant $K$ is then introduced between the plates, moving at a constant, extremely small velocity $v$ until it completely fills the capacitor. The time taken to fill the capacitor is much larger than the time period of the oscillations. Which of the following expressions gives the new maximum charge $Q_f$ on the capacitor after the slab is fully inserted?

Options

A.
$Q_0 K^{1/4}$
B.
$Q_0 K^{1/2}$
C.
$Q_0 K$
D.
$Q_0$

Check Your Answer

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

Non-Uniform Dielectric Filling|LC Circuit Oscillations|Variable Force Integration

Why This Question Is Tricky

Common Mistakes

This question tests the concept of adiabatic invariants in oscillatory systems with time-varying parameters.

The primary trap is the misapplication of energy conservation.

Students often incorrectly assume that because the LC circuit is 'ideal', its total electromagnetic energy remains constant ( $E_i = E_f$ ), leading to the erroneous answer $Q_0 K^{1/2}$ .

They fail to account for the mechanical work done by the external force required to keep the dielectric moving at a constant velocity against the electrostatic pull of the capacitor.

By recognizing that the time-varying capacitance alters the energy dynamically, students must derive or apply the adiabatic invariant $E/\omega = \text{constant}$ (since $\omega \propto 1/\sqrt{C}$ ) to find the correct amplitude scaling.

Quick Facts About This Question

Question Information

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