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Question #42728

Difficulty: 8/10 | 🟢 Free

The Question

A cylindrical metallic rod of initial length $L_0$ , uniform cross-sectional area $A_0$ , and initial electrical resistance $R_0$ is rigidly fixed between two unyielding massive walls. The material of the rod has a Young's modulus $Y$ , Poisson's ratio $\nu$ , coefficient of linear thermal expansion $\alpha$ , and temperature coefficient of resistivity $\alpha_R$ . The temperature of the rod is uniformly increased by a small amount $\Delta T$ . Assuming the walls do not deform and neglecting any buckling of the rod, which of the following expressions correctly represents the fractional change in the electrical resistance of the rod, $\frac{\Delta R}{R_0}$ , to the first order in $\Delta T$ ?

Options

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

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

Resistivity vs Resistance|Thermal Stress in Constrained Rods|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the interplay between thermal expansion, mechanical stress, and electrical resistance in a constrained system.

The primary trap lies in calculating the change in the cross-sectional area.

Students often either assume free expansion (incorrectly applying $\Delta L/L = \alpha \Delta T$ despite the rigid walls) or they recognize $\Delta L = 0$ but only apply the thermal expansion to the area ( $\Delta A/A = 2\alpha \Delta T$ ).

They fail to account for the Poisson effect: the rigid walls exert a compressive mechanical stress to prevent longitudinal expansion, which in turn causes an additional transverse mechanical expansion ( $\nu \alpha \Delta T$ ).

The total transverse strain is the superposition of both thermal and mechanical effects, fundamentally altering the final resistance.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Current Electricity - Class 12
Concept	Resistivity vs Resistance
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free