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Question #42505

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The Question

An ideal $LC$ circuit with inductance $L$ and capacitance $C$ initially stores energy with a maximum charge $Q_0$ on the capacitor. The circuit is placed in a large vacuum chamber whose walls are maintained at a constant temperature $T$ . Due to the emission of electromagnetic radiation, the circuit is not perfectly conservative and loses power at a rate $P = \alpha (\ddot{q})^2$ , where $\alpha$ is a known positive constant and $q$ is the instantaneous charge on the capacitor. Assuming the radiative damping is extremely small ( $\alpha \ll L \sqrt{LC}$ ), which of the following expressions correctly gives the effective radiation resistance $R_{rad}$ of the circuit and the entropy $\Delta S$ generated in the chamber walls during the first complete oscillation cycle?

Options

A.
$R_{rad} = \frac{\alpha}{LC}, \; \Delta S = \frac{\pi \alpha Q_0^2}{T (LC)^{3/2}}$
B.
$R_{rad} = \frac{\alpha}{2LC}, \; \Delta S = \frac{\pi \alpha Q_0^2}{2T (LC)^{3/2}}$
C.
$R_{rad} = \frac{\alpha}{LC}, \; \Delta S = \frac{2\pi \alpha Q_0^2}{T (LC)^{3/2}}$
D.
$R_{rad} = \frac{2\alpha}{LC}, \; \Delta S = \frac{\pi \alpha Q_0^2}{T (LC)^{3/2}}$

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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

Entropy and Reversibility|L-C Circuit Analogy|Source of EM Waves|Vector Analysis

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to bridge electromagnetism and thermodynamics by treating radiative damping as an irreversible entropy-producing mechanism.

The primary trap lies in the functional form of the power loss ( $P \propto \ddot{q}^2$ instead of the standard ohmic $P \propto \dot{q}^2$ ).

Students often incorrectly map the maximum power to $I_{max}^2 R$ or fail to properly time-average the squared acceleration, leading to incorrect factors of 2 or $\pi$ .

Furthermore, it challenges the misconception that LC circuits are strictly conservative, forcing the realization that macroscopic electromagnetic energy dissipation into a thermal reservoir fundamentally constitutes an increase in the universe's entropy, governed by $dS = dQ/T$ .

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Thermodynamics - Class 11 Part 2
Concept	Entropy and Reversibility
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free