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Question #43369

Difficulty: 8/10 | 🟢 Free

The Question

A uniform cylindrical metallic wire of initial resistance $R_0$ at temperature $T_0$ is rigidly fixed between two immovable supports. The material of the wire has a coefficient of linear thermal expansion $\alpha$ , a temperature coefficient of resistivity $\alpha_r$ , and a Poisson's ratio $\nu$ . If the temperature of the wire is increased by a small amount $\Delta T$ (such that $\alpha \Delta T \ll 1$ ), which of the following expressions correctly represents the fractional change in its electrical resistance, $\frac{\Delta R}{R_0}$ ? Assume the wire does not buckle and the supports do not yield.

Options

A.
$(\alpha_r - 2\alpha(1+\nu))\Delta T$
B.
$(\alpha_r - \alpha)\Delta T$
C.
$(\alpha_r + 2\alpha\nu)\Delta T$
D.
$(\alpha_r - 2\alpha)\Delta T$

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

Temperature Coefficient (Alpha)|Thermal Stress Calculation|Resistivity vs Resistance

Why This Question Is Tricky

Common Mistakes

This question tests the coupling of thermal expansion, linear elasticity, and electrical resistance.

The primary trap is the 'hidden variable' of transverse elastic strain.

Students often incorrectly apply the unconstrained resistance change formula or assume that because the length is fixed, the area only expands thermally (yielding an area strain of $2\alpha\Delta T$ ).

They fail to recognize that the rigid supports induce a compressive longitudinal thermal stress, which in turn causes an additional transverse expansion via the Poisson effect, significantly altering the cross-sectional area and the final resistance.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Current Electricity - Class 12
Concept	Temperature Coefficient (Alpha)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free