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Question #43331

Difficulty: 8/10 | 🟢 Free

The Question

A series $L-C$ circuit consists of an inductor with inductance $L = 10 \text{ mH}$ and a parallel-plate capacitor. The vacuum capacitance of the capacitor (without any dielectric) is $C_0 = 1.0 \text{ \mu F}$ . The space between the capacitor plates is completely filled with a dispersive liquid dielectric. The relative permittivity (dielectric constant) of this material varies with the angular frequency $\omega$ of the applied AC source according to the relation $k(\omega) = 4 + \frac{\beta}{\omega^2}$ , where $\beta = 36 \times 10^6 \text{ rad}^2\text{s}^{-2}$ . What is the resonant angular frequency $\omega_r$ of this circuit? (Assume resonance occurs when the total reactive impedance of the circuit is zero).

Options

A.
$4000 \text{ rad/s}$
B.
$5000 \text{ rad/s}$
C.
$8000 \text{ rad/s}$
D.
$10000 \text{ rad/s}$

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

Capacitive Reactance Dependency|Medium Dependency|Electric Field Line Geometry

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to adapt standard AC circuit theory to non-linear, frequency-dependent components.

The primary trap is the blind application of the standard resonance formula $\omega = 1/\sqrt{LC}$ .

Students often assume the dielectric constant is a static scalar and might evaluate it at the high-frequency limit ( $k=4$ , yielding $5000 \text{ rad/s}$ ) or ignore the dispersion entirely.

By embedding $\omega$ within the capacitance itself, the impedance operator becomes non-linear.

The correct approach requires returning to the fundamental definition of resonance ( $X_L = X_C$ ) and solving the resulting modified polynomial equation for $\omega$ , demonstrating a rigorous understanding of capacitive reactance.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Alternating Current - Class 12
Concept	Capacitive Reactance Dependency
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free