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Question #43067

Difficulty: 8/10 | 🟢 Free

The Question

A series RLC circuit is driven by an AC voltage source $V(t) = V_0 \sin(\omega t)$ . The circuit consists of a capacitor $C$ , an ideal resistor $R_0$ , and an inductor with self-inductance $L$ . The inductor is formed by winding $N$ turns of wire around a toroidal magnetic core of mean radius $r$ , cross-sectional area $A$ , and relative permeability $\mu_r$ . The core material exhibits magnetic hysteresis, and the energy dissipated per unit volume per cycle is given by $w_h = \beta B_m^2$ , where $B_m$ is the peak magnetic field in the core and $\beta$ is a material constant. Assuming the magnetic field is uniform across the core's cross-section and ignoring eddy current losses, which of the following expressions represents the quality factor $Q$ of the circuit at its resonant frequency $\omega_0$ ?

Options

A.
$\frac{\pi \omega_0 L}{\pi R_0 + \omega_0 \beta \mu_0 \mu_r L}$
B.
$\frac{2\pi \omega_0 L}{2\pi R_0 + \omega_0 \beta \mu_0 \mu_r L}$
C.
$\frac{\omega_0 L}{R_0 + 2 \omega_0 \beta \mu_0 \mu_r L}$
D.
$\frac{\omega_0 L}{R_0 + \omega_0 \beta \mu_0 \mu_r L}$

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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

Bandwidth in RLC|Hysteresis Loop Area|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to map non-linear magnetic core losses (hysteresis) to an equivalent linear circuit parameter (effective resistance).

A common trap is confusing the cyclic frequency $f$ with the angular frequency $\omega$ when calculating power ( $P = f W_{cycle}$ vs $P = \omega W_{cycle}$ ), or failing to use the RMS current ( $P = I_{rms}^2 R$ vs $P = I_m^2 R$ ) when defining the equivalent ohmic resistance.

Furthermore, students often treat the inductor's resistance as a static material constant, failing to realize that hysteresis loss creates a frequency-dependent effective resistance that modifies the circuit's Q-factor.

By forcing the integration of the hysteresis energy density over the core volume and equating it to average ohmic power dissipation, the problem distinguishes between true physical understanding of energy conservation and rote memorization of RLC formulas.

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