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Question #43023

Difficulty: 8/10 | 🟢 Free

The Question

A flexible, straight wire of length $L$ , cross-sectional area $A$ , Young's modulus $Y$ , and coefficient of linear thermal expansion $\alpha$ is rigidly fixed at its two ends. It is positioned exactly midway between two infinite, rigid parallel wires. The distance between the middle wire and each outer wire is $d$ . All three wires lie in the same horizontal plane. The middle wire carries a steady current $I_1$ , and the two outer wires each carry a steady current $I_2$ in the same direction as $I_1$ . Initially, at temperature $T_0$ , the middle wire is perfectly straight and has zero tension. The temperature of the middle wire is then uniformly lowered. Neglecting gravity and the magnetic field of the Earth, what is the minimum temperature drop $\Delta T$ required to keep the middle wire in stable equilibrium in its straight configuration?

Options

A.
$\frac{\mu_0 I_1 I_2 L^2}{\pi^3 d^2 Y A \alpha}$
B.
$\frac{\mu_0 I_1 I_2 L^2}{4 \pi d^2 Y A \alpha}$
C.
$\frac{\mu_0 I_1 I_2 L^2}{2 \pi^3 d^2 Y A \alpha}$
D.
$\frac{\mu_0 I_1 I_2 L^2}{\pi^2 d^2 Y A \alpha}$

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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

Force Between Parallel Conductors|Thermal Stress in Constrained Rods|Calculus (Integration)

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of thermal mechanics and electromagnetism, specifically focusing on the stability of distributed systems.

The primary trap lies in treating the magnetic force as a lumped point load or using a linear 'V-shape' approximation for the deflected wire, which incorrectly estimates the restoring force and leads to option B.

Students must recognize that the magnetic force acts as a distributed lateral load proportional to the local deflection, coupling with the longitudinal thermal tension to form a beam-column buckling problem governed by the differential equation $T y'' + c y = 0$ .

Failing to apply the correct sinusoidal mode shape ( $kL = \pi$ ) leads to incorrect numerical factors like $4\pi$ instead of $\pi^3$ .

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Moving Charges and Magnetism - Class 12
Concept	Force Between Parallel Conductors
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free