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Question #92611

Difficulty: 8/10 | 🟢 Free

The Question

A series RLC circuit consists of a variable capacitor $C$ , an ohmic resistor of resistance $R$ , and an inductor. The inductor is a long solenoid of radius $a$ , length $l$ , and $n$ turns per unit length, with a self-inductance $L$ . Inside the solenoid, coaxial with it, is a solid cylindrical core of radius $b$ ( $b < a$ ) made of a non-magnetic material with electrical conductivity $\sigma$ . The circuit is driven by an AC voltage source of variable angular frequency $\omega$ . For any chosen $\omega$ , the capacitor $C$ is tuned such that the circuit operates at resonance. Assuming the induced magnetic field due to eddy currents is negligible, what is the maximum possible Quality factor ( $Q_{max}$ ) of the circuit as $\omega$ is varied? (Ignore edge effects and any other losses.)

Options

A.
$\sqrt{\frac{2 L a^2}{R \sigma \mu_0 b^4}}$
B.
$\sqrt{\frac{L a^2}{2 R \sigma \mu_0 b^4}}$
C.
$\sqrt{\frac{4 L a^2}{R \sigma \mu_0 b^4}}$
D.
$\sqrt{\frac{L a^2}{R \sigma \mu_0 b^4}}$

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

Bandwidth in RLC|Quality Factor (Q-factor)|Eddy Currents Dissipation|Vector Cross-Product

Why This Question Is Tricky

Examiner Levers

Visualization Stress

Requires mental rotation or 3D mapping (Optics/Geometry/Vectors).

Difficulty Boost : +4

Common Mistakes

Standard intuition for RLC circuits assumes damping is purely ohmic ( $I^2R$ ), leading students to blindly apply the formula $Q = \frac{1}{R}\sqrt{\frac{L}{C}}$ .

This question introduces a physical trap: a conductive core that dissipates energy via eddy currents.

Because eddy current power loss scales with $\omega^2$ , the effective resistance of the circuit becomes frequency-dependent ( $R_{eff} = R + \alpha \omega^2$ ).

This decouples the Q-factor from the standard formula, forcing students to revert to the fundamental energy definition of Quality factor ( $Q = \omega L / R_{eff}$ ) and perform calculus optimization to find the maximum possible $Q$ .

Failing to recognize the non-ohmic, velocity-dependent nature of the damping leads to incorrect, constant-resistance assumptions.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Alternating Current - Class 12
Concept	Bandwidth in RLC
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free