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Question #43339

Difficulty: 8/10 | 🟢 Free

The Question

A series RLC circuit consists of an ideal inductor of inductance $L$ , a resistor of resistance $R$ , and a parallel-plate capacitor. The capacitor has plate area $A$ and plate separation $d$ . The space between the plates is filled with two distinct dielectric layers, each of thickness $d/2$ . The first layer ( $0 < x < d/2$ ) is an ideal lossless dielectric with relative permittivity $\epsilon_r = 2$ . The second layer ( $d/2 < x < d$ ) is a lossy dielectric with relative permittivity $\epsilon_r = 2$ and a small, constant conductivity $\sigma$ . Assuming the low-loss approximation ( $\sigma \ll \omega_0 \epsilon_0$ ) holds, what is the quality factor $Q$ of this circuit at its resonant frequency $\omega_0$ ?

Options

A.
$\frac{1}{R + \frac{\sigma L}{4 \epsilon_0}} \sqrt{\frac{L d}{2 \epsilon_0 A}}$
B.
$\frac{1}{R} \sqrt{\frac{L d}{2 \epsilon_0 A}}$
C.
$\frac{1}{R + \frac{\sigma L}{\epsilon_0}} \sqrt{\frac{L d}{2 \epsilon_0 A}}$
D.
$\frac{1}{R + \frac{d}{2 \sigma A}} \sqrt{\frac{L d}{2 \epsilon_0 A}}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Quality Factor (Q-factor)|Non-Uniform Dielectric Filling|Frequency Sensitivity of Z

Why This Question Is Tricky

Common Mistakes

This question tests the student's understanding of dielectric losses and their manifestation in macroscopic circuit parameters.

The primary trap is assuming that the quality factor of an RLC circuit is always determined solely by the explicit ohmic resistor $R$ (Option B).

In reality, a lossy dielectric introduces a frequency-dependent equivalent series resistance $R_c(\omega)$ derived from the Fluctuation-Dissipation theorem (modeled here via parallel conductance).

Students must perform a parallel-to-series impedance transformation for the non-uniform dielectric layers and evaluate this dynamic resistance specifically at the resonant frequency $\omega_0$ .

Failing to account for the dynamic nature of $R_c(\omega)$ or incorrectly adding the parallel resistance directly to the series circuit (Option D) leads to lethal collisions with the correct physics.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Alternating Current - Class 12
Concept	Quality Factor (Q-factor)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free