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Question #43884

Difficulty: 8/10 | 🟢 Free

The Question

A rigid superconducting square coil of mass $m$ , side length $a$ , and self-inductance $L$ is pivoted freely about its top horizontal edge. The coil is placed in a uniform vertical magnetic field $B$ directed downwards. When the coil hangs vertically ( $\theta = 0$ ), its plane is vertical. Initially, the coil is held at a small angle $\theta_0$ with the vertical, and the current in the coil is zero. It is then released from rest. Find the maximum kinetic energy of the coil during its subsequent motion. Assume $\theta_0$ is small enough that $\sin\theta \approx \theta$ and $\cos\theta \approx 1$ , and let $g$ be the acceleration due to gravity.

Options

A.
$\frac{m^2 g^2 a L \theta_0^2}{4 (m g L + 2 B^2 a^3)}$
B.
$\frac{(m g a L + 2 B^2 a^4) \theta_0^2}{4 L}$
C.
$\frac{1}{4} m g a \theta_0^2$
D.
$\frac{m^2 g^2 a L \theta_0^2}{2 (m g L + B^2 a^3)}$

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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

Series LCR Impedance Vector|Transient Current in RL Circuit|Physical Pendulum Inertia

Why This Question Is Tricky

Common Mistakes

The core trap lies in the misapplication of Faraday's Law for superconducting circuits.

Students often write the induced EMF as $\mathcal{E} = B a^2 \dot{\theta}$ and equate it to $L \frac{di}{dt}$ , incorrectly integrating to $L i = B a^2 \theta$ by assuming the integration constant is zero.

This ignores the initial condition where the coil is released from $\theta_0$ with zero current.

Consequently, they formulate the magnetic potential energy as $\frac{1}{2} \frac{B^2 a^4}{L} \theta^2$ , which shifts the equilibrium position to $\theta=0$ and leads to an incorrect maximum kinetic energy (Option B).

The problem rigorously tests the understanding that in a zero-resistance circuit, it is the total flux that is strictly conserved from the moment of release, creating a shifted harmonic oscillator.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Alternating Current - Class 12
Concept	Series LCR Impedance Vector
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free