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Question #42629

Difficulty: 8/10 | 🟢 Free

The Question

An ideal primary LC circuit consists of a capacitor of capacitance $C$ and an inductor of self-inductance $L_1$ . It is magnetically coupled to a stationary secondary closed loop having self-inductance $L_2$ and resistance $R$ . The mutual inductance between the inductor $L_1$ and the secondary loop is $M$ . The primary circuit is set into oscillations. Assuming the coupling is weak ( $M \ll \sqrt{L_1 L_2}$ ) such that the oscillation frequency is $\omega \approx \frac{1}{\sqrt{L_1 C}}$ and the primary's effective inductance is approximately $L_1$ , find the damping coefficient $\alpha$ of the primary circuit's current amplitude (which decays as $e^{-\alpha t}$ ).

Options

A.
$\frac{M^2 R}{2 L_1 (R^2 L_1 C + L_2^2)}$
B.
$\frac{M^2}{2 R L_1^2 C}$
C.
$\frac{M^2 R}{2 L_2 (R^2 L_1 C + L_2^2)}$
D.
$\frac{M^2 R}{2 L_1 (R^2 L_2 C + L_1^2)}$

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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

Damped Oscillation Equation|Mutual Inductance Reciprocity|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the student's understanding of coupled oscillations and the physical reality of mutual inductive back-EMF.

The primary trap is ignoring the self-inductance $L_2$ of the secondary loop, which creates a crucial phase shift between the induced EMF and the induced current.

Students who incorrectly assume the secondary current is simply $\varepsilon_2 / R$ (ignoring the inductive reactance $\omega L_2$ ) will arrive at the distractor option B.

By forcing the calculation of the true power dissipation with the phase-shifted damping term, the problem evaluates mastery of Faraday's Law beyond simple constant-resistance models.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Oscillations - Class 11 Part 2
Concept	Damped Oscillation Equation
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free