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Question #42533

Difficulty: 8/10 | 🟢 Free

The Question

A small spherical blackbody of mass $m$ , specific heat capacity $c$ , and radius $r$ falls from rest through a height $H$ in a vacuum chamber maintained at $0\text{ K}$ . Upon striking a perfectly insulating floor, all its mechanical potential energy is converted into internal thermal energy. A thin converging lens of focal length $f$ is positioned at a distance $\frac{f}{3}$ from the object. A distant sensor measures the irradiance and calculates the apparent total radiated power $P_{app}$ by assuming the observed virtual image is an isotropically radiating source. It then computes the apparent initial rate of cooling as $R_{app} = \frac{P_{app}}{mc}$ . What is the value of $R_{app}$ ? (Assume acceleration due to gravity is $g$ and the Stefan-Boltzmann constant is $\sigma$ )

Options

A.
$\frac{9\pi r^2 \sigma}{mc} \left(\frac{gH}{c}\right)^4$
B.
$\frac{4\pi r^2 \sigma}{mc} \left(\frac{gH}{c}\right)^4$
C.
$\frac{6\pi r^2 \sigma}{mc} \left(\frac{gH}{c}\right)^4$
D.
$\frac{81\pi r^2 \sigma}{4mc} \left(\frac{gH}{c}\right)^4$

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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

Emissivity and Absorptivity|Newton’s Law of Cooling Limits|Virtual Object Trap

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to synthesize mechanics, thermodynamics, and optics while avoiding the 'Virtual Object Trap.' Students often incorrectly assume that an optical system merely changes the apparent distance of the object without altering the inferred total radiant flux.

By confusing irradiance with invariant radiant flux, they fail to account for the solid angle transformation.

The trap relies on recognizing that while radiance (brightness) is conserved, the apparent emitting area scales by the square of the transverse magnification ( $M^2$ ), causing the sensor to overestimate the total isotropic power loss and consequently the cooling rate.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Thermal Properties of Matter - Class 11 Part 2
Concept	Emissivity and Absorptivity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free