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Question #42631

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The Question

A rigid square conducting loop of side length $a$ , mass $m$ , self-inductance $L$ , and a series capacitance $C$ is attached to a horizontal spring of spring constant $k$ . The loop is constrained to oscillate along the $x$ -axis on a frictionless surface. A static magnetic field is applied perpendicular to the plane of the loop, given by $\vec{B} = (B_0 + \alpha x)\hat{k}$ , where $B_0$ and $\alpha$ are positive constants. The total electrical resistance of the loop, including dissipative radiation resistance, is $R$ . An external mechanical force $F(t) = F_0 \cos(\omega t)$ drives the loop along the $x$ -axis. If the driving frequency is tuned to $\omega = \sqrt{\frac{k}{m}} = \frac{1}{\sqrt{LC}}$ , what is the steady-state amplitude $X_0$ of the loop's mechanical oscillation? Assume the loop remains in the $xy$ -plane and its sides remain parallel to the axes.

Options

A.
$\frac{F_0 R}{\omega \alpha^2 a^4}$
B.
$\frac{F_0 L}{\alpha^2 a^4}$
C.
$\frac{F_0 \sqrt{L/C}}{\omega \alpha^2 a^4}$
D.
$\frac{F_0 R^2}{\omega^2 L \alpha^2 a^4}$

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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

Driven Oscillations and Resonance|LC Oscillations Energy Cycle|Phasor Rotation Convention

Why This Question Is Tricky

Common Mistakes

This question tests the coupling of a mechanical resonator to an RLC circuit via a time-varying magnetic flux.

A common trap is assuming that at mechanical resonance, the amplitude approaches infinity due to the absence of explicit mechanical friction.

Students often fail to recognize that the induced back-EMF and subsequent Lorentz force act as an effective viscous damping term $c_{eff} = (\alpha a^2)^2 / R$ .

Furthermore, candidates may incorrectly calculate the phase lag between the mechanical displacement and the induced current.

At electrical resonance, the current is exactly in phase with the induced EMF (and thus velocity), making the magnetic force purely dissipative.

The inclusion of radiation resistance $R$ is critical; paradoxically, a smaller resistance leads to a smaller mechanical amplitude because the induced opposing current becomes larger, perfectly canceling the driving force in the limit of zero resistance.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Oscillations - Class 11 Part 2
Concept	Driven Oscillations and Resonance
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free