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Question #42958

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The Question

An AC voltage source $V(t) = V_0 \sin(\omega t)$ is connected in series with a capacitor of capacitance $C$ and an inductor of inductance $L$ having a ferromagnetic core. The ohmic resistance of the entire circuit is negligible. Due to hysteresis, the ferromagnetic core dissipates an energy $W_h = 2\pi \beta I_{\text{rms}}^2$ per cycle, where $\beta$ is a positive constant with appropriate SI units and $I_{\text{rms}}$ is the root-mean-square current. The angular frequency $\omega$ of the source is continuously varied. It is observed that for a specific fixed value of $C$ , the current amplitude in the circuit attains its maximum possible value exactly at the angular frequency $\omega = \omega_0$ . Which of the following expressions correctly gives the required capacitance $C$ ?

Options

A.
$\frac{1}{\omega_0^2 \sqrt{L^2 + \beta^2}}$
B.
$\frac{1}{\omega_0^2 L}$
C.
$\frac{1}{\omega_0^2 (L + \beta)}$
D.
$\frac{1}{\omega_0^2 (L - \beta)}$

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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

Hysteresis Loop Area|Average Power over Cycle|Capacitive Reactance Dependency

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to integrate macroscopic thermodynamic losses (hysteresis) with AC circuit theory.

The primary trap is treating the resonance condition as simply $X_L = X_C$ (yielding $C = 1/(\omega_0^2 L)$ ), which is only valid when the circuit resistance is independent of frequency.

By modeling the hysteresis energy loss per cycle, students must deduce that the effective resistance $R_{\text{eff}}$ scales linearly with frequency ( $\beta \omega$ ).

Consequently, the total impedance $Z$ has a frequency-dependent resistive term that shifts the minimum of $Z(\omega)$ .

Students must construct the full impedance function and differentiate it with respect to $\omega$ to find the true condition for maximum current amplitude, preventing a simple static calculation.

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