Before You Begin

Question #43044

Difficulty: 8/10 | 🟢 Free

The Question

A toroidal solenoid consists of $N$ turns of wire wound uniformly around a non-magnetic, isotropic solid core. The core has an initial major radius $R$ and an initial cross-sectional area $A$ (where $\sqrt{A} \ll R$ ). The material of the core has a coefficient of linear thermal expansion $\alpha$ . A constant direct current $I$ is maintained in the wire. The temperature of the core is uniformly increased at a constant rate $\frac{dT}{dt} = \beta$ . Assuming the wire perfectly accommodates the core's expansion without changing the total number of turns $N$ , what is the magnitude of the induced electromotive force (EMF) in the solenoid? (Assume $\alpha \Delta T \ll 1$ )

Options

A.
$\frac{\mu_0 N^2 I A \alpha \beta}{2\pi R}$
B.
$\frac{\mu_0 N^2 I A \alpha \beta}{\pi R}$
C.
$\frac{3\mu_0 N^2 I A \alpha \beta}{2\pi R}$
D.
$0$

Check Your Answer

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

Toroidal Solenoid Boundary Conditions|Thermal Expansion of Area and Volume|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to couple electromagnetic induction with the thermodynamic properties of materials, specifically avoiding the trap of uniform geometric scaling.

A common misconception is to assume that since the toroid expands isotropically, the ratio of Area to Radius ( $A/R$ ) scales with the volume expansion coefficient ( $3\alpha$ ) or that the turn density $n = N/(2\pi R)$ remains constant, leading to an incorrect flux derivative.

Students often incorrectly apply the area expansion ( $2\alpha$ ) without accounting for the simultaneous expansion of the major radius ( $\alpha$ ) which dilutes the magnetic field $B$ via Ampere's Circuital Law.

By requiring the derivation of self-inductance from first principles, the problem forces the realization that $L \propto A/R$ , leading to an effective expansion coefficient of $2\alpha - \alpha = \alpha$ for the magnetic flux.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Moving Charges and Magnetism - Class 12
Concept	Toroidal Solenoid Boundary Conditions
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free