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Question #42196

Difficulty: 8/10 | 🟢 Free

The Question

A refrigerator is placed inside an evacuated chamber whose walls are maintained at a constant temperature $T_0$ . The refrigerator maintains its cold interior at a temperature $T_c$ and operates with a constant Coefficient of Performance $\beta$ . Heat leaks into the cold interior at a constant rate $P$ . The refrigerator rejects heat to the surroundings exclusively via radiation from its outer fins, which have a total surface area $A$ and emissivity $\epsilon$ . Assuming steady-state operation, what is the temperature $T$ of the radiator fins? (Given $\sigma$ is the Stefan-Boltzmann constant).

Options

A.
$\left[ T_0^4 + \frac{P(\beta + 1)}{\sigma \epsilon A \beta} \right]^{\frac{1}{4}}$
B.
$\left[ T_0^4 + \frac{P}{\sigma \epsilon A} \right]^{\frac{1}{4}}$
C.
$\left[ T_0^4 + \frac{P \beta}{\sigma \epsilon A} \right]^{\frac{1}{4}}$
D.
$\left[ \frac{P(\beta + 1)}{\sigma \epsilon A \beta} \right]^{\frac{1}{4}}$

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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

Refrigerator Coefficient of Performance|Stefan-Boltzmann Power Law|First Law Application

Why This Question Is Tricky

Common Mistakes

This question tests the application of the First Law of Thermodynamics to a refrigeration cycle coupled with the Stefan-Boltzmann law.

The primary trap is neglecting the work done by the compressor.

Students often incorrectly assume that the heat rejected by the radiator fins equals the heat extracted from the cold reservoir ( $\dot{Q}_h = \dot{Q}_c = P$ ), leading to the second option.

They fail to recognize that the compressor adds mechanical work to the system, which acts as an additional thermal load that must also be dissipated as heat ( $\dot{Q}_h = \dot{Q}_c + \dot{W}$ ).

A secondary trap involves ignoring the ambient temperature $T_0$ in the net radiation exchange, which would lead to the fourth option.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Thermodynamics - Class 11 Part 2
Concept	Refrigerator Coefficient of Performance
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free