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Question #42554

Difficulty: 8/10 | 🟢 Free

The Question

A Carnot engine operates between a hot reservoir and a cold reservoir maintained at a constant temperature $T_C$ . The hot reservoir consists of a perfectly absorbing plate of area $A$ . A plane electromagnetic wave with a peak electric field amplitude $E_0$ is normally incident on the plate. The plate is maintained at a steady equivalent radiation temperature $T_H$ , defined by the condition that the incident time-averaged Poynting flux equals the blackbody emissive flux $\sigma T_H^4$ (where $\sigma$ is the Stefan-Boltzmann constant). If the engine absorbs heat from the plate at a rate equal to the total incident electromagnetic power, what is the rate at which heat is rejected to the cold reservoir? (Assume the speed of light is $c$ and vacuum permittivity is $\epsilon_0$ ).

Options

A.
$A T_C \left( \frac{\sigma \epsilon_0^3 c^3 E_0^6}{8} \right)^{\frac{1}{4}}$
B.
$A T_C \left( \sigma \epsilon_0^3 c^3 E_0^6 \right)^{\frac{1}{4}}$
C.
$A T_C \left( \frac{\sigma \epsilon_0^3 c^3 E_0^6}{2} \right)^{\frac{1}{4}}$
D.
$\frac{1}{2} A \epsilon_0 c E_0^2 \left[ 1 - T_C \left( \frac{2 \sigma}{\epsilon_0 c E_0^2} \right)^{\frac{1}{4}} \right]$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Carnot Engine Efficiency|Energy Density Locality|Vector Calculus

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of radiation thermodynamics and classical heat engines.

The primary trap lies in confusing the localized energy density with the directional energy flux (Poynting vector) when defining both the heat intake rate and the equivalent blackbody temperature.

Students often erroneously treat the incident EM field as a static thermal bath, failing to account for the vector-dependent radiation flux required to maintain the steady-state gradient.

Additionally, distractors exploit the omission of the time-averaging factor for harmonic waves and the common misreading of calculating mechanical power output (Option D) instead of the heat rejected to the cold reservoir.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Thermodynamics - Class 11 Part 2
Concept	Carnot Engine Efficiency
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free