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Question #42987

Difficulty: 8/10 | 🟢 Free

The Question

A self-supporting, single-layer, tightly wound solenoid is made of a uniform conducting wire of circular cross-section. The material of the wire has a coefficient of linear thermal expansion $\alpha$ and a temperature coefficient of resistivity $\alpha_\rho$ . If the temperature of the entire solenoid is increased by a small amount $\Delta T$ , what is the fractional change in the inductive time constant $\left(\frac{\Delta \tau}{\tau}\right)$ of the solenoid? (Assume $\alpha \Delta T \ll 1$ and $\alpha_\rho \Delta T \ll 1$ )

Options

A.
$(2\alpha - \alpha_\rho)\Delta T$
B.
$(\alpha - \alpha_\rho)\Delta T$
C.
$(3\alpha - \alpha_\rho)\Delta T$
D.
$-\alpha_\rho \Delta T$

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

Self-Inductance of Solenoid|Temperature Coefficient (Alpha)|Drift Velocity Mechanism

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to correctly apply isotropic thermal expansion to both macroscopic and microscopic geometric parameters of an electromagnetic component.

A common trap is assuming the resistance $R$ simply scales with the temperature coefficient of resistivity $\alpha_\rho$ as $R' = R(1+\alpha_\rho \Delta T)$ , completely neglecting the geometric expansion of the wire's length and cross-sectional area.

This leads to the incorrect fractional change $(\alpha - \alpha_\rho)\Delta T$ .

Another trap is miscalculating the inductance scaling by assuming only the cross-sectional area of the solenoid expands ( $L \propto r^2$ ), forgetting that the length $l$ also expands, which would incorrectly yield a $2\alpha$ contribution from $L$ and lead to $(3\alpha - \alpha_\rho)\Delta T$ .

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electromagnetic Induction - Class 12
Concept	Self-Inductance of Solenoid
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free