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Question #41988

Difficulty: 8/10 | 🟢 Free

The Question

An ideal $L-C$ circuit consists of a capacitor of capacitance $C$ and a solenoid of initial inductance $L_0$ . The solenoid is tightly wound on a solid cylindrical non-magnetic rod having a coefficient of linear thermal expansion $\alpha$ . The circuit is set into electromagnetic oscillations with an initial peak current $I_0$ . The rod is then heated very slowly (adiabatically with respect to the oscillation period) such that its temperature increases by $\Delta T$ . Assuming the total number of turns of the solenoid remains constant and the rod expands isotropically, what is the new peak current $I$ in the circuit?

Options

A.
$I_0 (1 + \alpha \Delta T)^{-3/4}$
B.
$I_0 (1 + \alpha \Delta T)^{-1/2}$
C.
$I_0 (1 + \alpha \Delta T)^{-1}$
D.
$I_0 (1 + \alpha \Delta T)^{-1/4}$

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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

Series LCR Impedance Vector|L-C Circuit Analogy|SHM Definition via Differential Equation|Effective Spring Constant

Why This Question Is Tricky

Common Mistakes

This question tests the application of adiabatic invariants in a time-varying $L-C$ circuit, mapping it to a non-autonomous SHM differential equation.

A common trap is assuming that total electromagnetic energy is conserved ( $E = \text{constant} \implies I \propto L^{-1/2}$ ) or that magnetic flux is conserved ( $\Phi = \text{constant} \implies I \propto L^{-1}$ ).

However, because the physical expansion of the rod alters the inductance dynamically, work is done against the magnetic pressure, changing the total energy.

The rigorous approach requires recognizing the action $J = E/\omega$ as the true conserved quantity for slowly varying parameters.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Alternating Current - Class 12
Concept	Series LCR Impedance Vector
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free