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Question #95458

Difficulty: 5/10 | 🟢 Free

The Question

An inductor of inductance $L$ and a resistor of resistance $R$ are connected in series with a battery, establishing a steady current $I_0$ . The battery is suddenly short-circuited, allowing the current to decay. The resistor is perfectly thermally insulated from the surroundings but is in ideal thermal contact with a uniform metal rod of mass $m$ , specific heat capacity $s$ , coefficient of linear expansion $\alpha$ , and Young's modulus $Y$ . The rod is rigidly fixed between two unyielding, immovable walls. Assuming all the heat dissipated by the resistor is entirely absorbed by the rod, what is the compressive thermal stress developed in the rod after a long time?

Options

A.
$\frac{Y \alpha L I_0^2}{2ms}$
B.
$\frac{Y \alpha L I_0^2}{ms}$
C.
$\frac{Y \alpha R I_0^2}{2ms}$
D.
$\frac{Y \alpha L^2 I_0^2}{2msR}$

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

Energy Stored in Inductor|Transient Current in RL Circuit|Thermal Conductivity in Series

Why This Question Is Tricky

Examiner Levers

Dimensional Analysis

Tests if the student verifies units/dimensions before selecting.

Difficulty Boost : +2

Common Mistakes

This question tests the intersection of transient circuits, energy conservation, and solid mechanics.

The primary trap is the 'math bloat' of integrating $I^2R(t)$ over time; students who fail to recognize that the total dissipated heat is simply the inductor's initial stored energy ( $\frac{1}{2}LI_0^2$ ) will waste valuable time.

Furthermore, the question tests the conceptual distinction between thermal expansion and thermal stress.

Students often incorrectly assume a fixed rod undergoes physical expansion or misapply the stress formula by including the rod's length or cross-sectional area, which cancel out in the stress ( $\sigma = F/A$ ) calculation.

Finally, dimensional analysis immediately eliminates options containing $R$ or $L^2/R$ in place of $L$ , rewarding students who verify units.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electromagnetic Induction - Class 12
Concept	Energy Stored in Inductor
Difficulty	5
Cognitive Load	6
Target Solve Time	140 sec
Access	Free