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Question #43279

Difficulty: 8/10 | 🟢 Free

The Question

A series RLC circuit is driven by an AC voltage source of frequency $f = 50 \text{ Hz}$ such that the RMS current in the circuit is $I_{rms} = 2 \text{ A}$ . The circuit consists of a capacitor of capacitance $C = \frac{1}{600\pi} \text{ F}$ and a real inductor with a ferromagnetic core. The inductor coil has a DC wire resistance $R_w = 2 \ \Omega$ and an inductance $L = \frac{0.14}{\pi} \text{ H}$ . The ferromagnetic core has a volume $V = 10^{-3} \text{ m}^3$ . At this operating current, the area of the B-H hysteresis loop for the core is $A_h = 480 \text{ J m}^{-3} \text{ cycle}^{-1}$ . What is the power factor of the complete circuit?

Options

A.
$\frac{1}{\sqrt{2}}$
B.
$\frac{1}{\sqrt{17}}$
C.
$\frac{3}{5}$
D.
$\frac{5}{\sqrt{89}}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Hysteresis Loop Area|Quality Factor (Q-factor)|Wattless Current

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to bridge thermodynamics (hysteresis energy loss) with AC electrodynamics (complex impedance).

A common misconception is treating a real inductor purely by its inductance and DC wire resistance, ignoring AC core losses.

Students often fall into the trap of calculating the power factor using only $R_w$ (yielding $1/\sqrt{17}$ ), failing to recognize that the hysteresis loop area represents an energy dissipation per cycle that manifests macroscopically as an effective series resistance ( $R_h$ ).

Furthermore, equating the average power loss to $I_{rms}^2 R_h$ (rather than $I_{peak}^2$ ) is crucial.

By forcing the derivation of $R_{eff} = R_w + R_h$ , the problem evaluates a deep understanding of the physical origins of the quality factor and phase angle in non-ideal components.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Magnetism and Matter - Class 12
Concept	Hysteresis Loop Area
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free