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Question #42214

Difficulty: 8/10 | 🟢 Free

The Question

An electromechanical system consists of a solenoid with a non-conducting, compressible magnetic core, connected in series with a resistor of resistance $R$ and a DC voltage source $V$ . For $t < 0$ , a constant mechanical compressive force is applied to the core, and the circuit is in a steady state. At $t = 0$ , the compressive force is suddenly removed. Due to the elastic after-effect, the core's strain recovers gradually, causing the inductance of the solenoid to vary with time as $L(t) = L_1 + L_2 (1 - e^{-t/\tau})$ , where $L_1, L_2$ , and $\tau$ are positive constants. Which of the following represents the rate of change of the magnetic energy stored in the solenoid exactly at $t = 0^+$ ?

Options

A.
$-\frac{V^2 L_2}{2 R^2 \tau}$
B.
$+\frac{V^2 L_2}{2 R^2 \tau}$
C.
$-\frac{V^2 L_2}{R^2 \tau}$
D.
0

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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

Elastic After-Effect|Energy Stored in Inductor|Instantaneous Nature

Why This Question Is Tricky

Common Mistakes

This question tests the coupling of mechanical relaxation (elastic after-effect) with electromagnetic induction.

A common lethal mistake is treating inductance as a constant when applying Faraday's law, leading students to write the EMF as solely -L(di/dt) and incorrectly conclude that di/dt = 0 at t=0^+.

This trap yields a positive rate of energy change (Option B).

By forcing the inclusion of the -i(dL/dt) term (motional/geometric back-EMF), the problem reveals that the mechanical expansion actually extracts energy from the magnetic field faster than the changing geometry can store it, resulting in a net negative rate of change of magnetic energy.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Mechanical Properties of Solids - Class 11 Part 2
Concept	Elastic After-Effect
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free