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Question #81752

Difficulty: 8/10 | 🟢 Free

The Question

A non-linear electronic heating element is placed inside a perfectly insulated, rigid chamber of negligible heat capacity containing $n$ moles of an ideal monoatomic gas. The current $I$ through the element and the voltage $V$ across it are related by $I = \frac{\alpha V^3}{T}$ , where $T$ is the instantaneous absolute temperature of the gas and $\alpha$ is a positive constant with appropriate dimensions. An alternating voltage $V(t) = V_0 \sin(\omega t)$ is applied across the element. The frequency $\omega$ is sufficiently high that the gas temperature remains effectively constant over a single electrical cycle. Assuming the heat capacity of the heating element is negligible and the initial temperature of the gas is $T_0$ , what is the time $t$ required for the gas temperature to reach $2T_0$ ?

Options

A.
$\frac{6 n R T_0^2}{\alpha V_0^4}$
B.
$\frac{9 n R T_0^2}{\alpha V_0^4}$
C.
$\frac{3 n R T_0^2}{\alpha V_0^4}$
D.
$\frac{10 n R T_0^2}{\alpha V_0^4}$

Check Your Answer

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|Dynamic Resistance (AC Resistance)|Energy Band Gap Temperature Dependency

Why This Question Is Tricky

Examiner Levers

Concept Bridge

Cross-chapter synthesis (e.g., using Logarithms in Electrochemistry).

Difficulty Boost : +5

Dimensional Analysis

Tests if the student verifies units/dimensions before selecting.

Difficulty Boost : +2

Common Mistakes

This question tests the misapplication of linear AC circuit formulas to non-linear components.

Students commonly fall into the trap of calculating an 'effective static resistance' at the RMS voltage, $R_{static} = V_{rms}/I(V_{rms})$ , and then applying the standard Ohmic power formula $P = V_{rms}^2 / R_{static}$ .

For a non-linear $I-V$ characteristic ( $I \propto V^3$ ), the instantaneous power scales as $V^4$ , requiring the integration of $\sin^4(\omega t)$ over a cycle (yielding a factor of $3/8$ ).

The Ohmic assumption yields an incorrect factor of $1/4$ , leading to a fundamentally flawed differential equation for the thermodynamic heating process.

Furthermore, it tests the integration of electrodynamics with the First Law of Thermodynamics in a variable-temperature feedback loop.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Thermodynamics - Class 11 Part 2
Concept	First Law Application
Difficulty	8
Cognitive Load	8
Target Solve Time	180 sec
Access	Free