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Question #42184

Difficulty: 8/10 | 🟢 Free

The Question

A capacitor of capacitance $C$ with an initial charge $Q_0$ is connected in series with an inductor of inductance $L$ , a switch, and a resistor of resistance $R$ . The resistor is placed inside a rigid, thermally insulated cylinder containing $n$ moles of a monoatomic ideal gas. The switch is closed at $t=0$ . The resistance $R$ is chosen such that the electrical circuit is critically damped. What is the rate of change of the temperature of the gas, $\frac{dT}{dt}$ , at the exact instant the current in the circuit reaches its maximum magnitude? (Here $R_{gas}$ is the universal gas constant and $e$ is the base of the natural logarithm).

Options

A.
$\frac{4 Q_0^2}{3 n R_{gas} e^2 C \sqrt{LC}}$
B.
$\frac{2 Q_0^2}{3 n R_{gas} e^2 C \sqrt{LC}}$
C.
$\frac{4 Q_0^2}{5 n R_{gas} e^2 C \sqrt{LC}}$
D.
$\frac{2 Q_0^2}{3 n R_{gas} e C \sqrt{LC}}$

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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

First Law Application|LC Circuit Oscillations|Differential Equations

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of transient circuit analysis and the First Law of Thermodynamics.

The primary trap lies in the assumption of energy conservation within the LC components, ignoring that the critically damped condition necessitates a specific resistance $R = 2\sqrt{L/C}$ which continuously dissipates energy.

Students often fail to correctly derive the time-domain current for a critically damped circuit, mistakenly applying underdamped sinusoidal formulas, or they incorrectly equate the total initial energy to the instantaneous power.

Furthermore, it rigorously tests the application of $dU/dt = dQ/dt$ by linking the instantaneous Ohmic heating power $i^2R$ to the rate of change of internal energy of a monoatomic gas $n C_v dT/dt$ .

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Thermodynamics - Class 11 Part 2
Concept	First Law Application
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free