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Question #43150

Difficulty: 8/10 | 🟢 Free

The Question

A bimetallic strip with an effective elastic spring constant $k$ acts as a movable armature for an electromagnet. The electromagnet has $N$ turns, a cross-sectional area $A$ , and carries a constant DC current $I$ . Due to a constant thermal power input, the unconstrained equilibrium position of the strip (the position it would take if $I=0$ ) moves towards the electromagnet at a constant speed $u$ . Let $x$ be the instantaneous air gap between the strip and the electromagnet. Assuming the motion is quasi-static (the strip is always in mechanical equilibrium), what is the rate of change of the magnetic flux linkage $\frac{d\lambda}{dt}$ of the electromagnet at the instant the air gap is $x$ ? (Assume the magnetic flux is entirely confined to the air gap, fringing effects are negligible, and the permeability of free space is $\mu_0$ ).

Options

A.
$\frac{\mu_0 N^2 I A k u x}{k x^3 - \mu_0 N^2 I^2 A}$
B.
$\frac{\mu_0 N^2 I A u}{x^2}$
C.
$\frac{\mu_0 N^2 I A k u x}{k x^3 + \mu_0 N^2 I^2 A}$
D.
$\frac{\mu_0 N^2 I A k u x}{k x^3 - 2 \mu_0 N^2 I^2 A}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Transformer Flux Linkage|Bimetallic Strip Curvature|Power-Velocity Relationship

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to couple thermal, mechanical, and magnetic domains using quasi-static equilibrium and power-velocity relationships.

The primary trap lies in treating the air gap as a static parameter or assuming the strip moves at the unconstrained thermal expansion velocity $u$ .

Students often fail to account for the 'Maxwell Stress' (magnetic pull) which acts as a non-linear negative stiffness.

By ignoring the spatial derivative of the magnetic force, they miss the mechanical feedback loop that accelerates the strip towards the electromagnet, ultimately leading to a snap-down instability.

This misapplication of the Work-Energy theorem without the non-conservative magnetic gradient leads directly to incorrect velocity and flux linkage rates.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Alternating Current - Class 12
Concept	Transformer Flux Linkage
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free