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Question #43763

Difficulty: 8/10 | 🟢 Free

The Question

A spherical blackbody of radius $R$ is maintained at a constant temperature $T$ . It is surrounded by a concentric, perfectly reflecting spherical shell of radius $r$ ( $r > R$ ). The shell is expanding radially outward at a constant velocity $v$ . Assuming the space between the blackbody and the shell is vacuum, determine the total force $F$ exerted by the radiation on the expanding shell. (Given: Stefan-Boltzmann constant $\sigma$ , speed of light $c$ )

Options

A.
$\frac{8\pi R^2 \sigma T^4}{c} \left( \frac{c-v}{c+v} \right)$
B.
$\frac{8\pi R^2 \sigma T^4}{c}$
C.
$\frac{8\pi R^2 \sigma T^4}{c} \left( \frac{c}{c+v} \right)$
D.
$\frac{8\pi R^2 \sigma T^4}{c} \left( \frac{c-v}{c} \right)$

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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

Wave-Particle Duality Constraint|Stefan-Boltzmann Power Law|Derived Units via Fundamental Constants

Why This Question Is Tricky

Common Mistakes

This question tests the microscopic application of radiation pressure on a moving boundary.

Students commonly fall into the trap of treating the Stefan-Boltzmann power output as a static pressure field, calculating $F = \text{Pressure} \times \text{Area} = (2I/c) \times 4\pi r^2 = 2P_0/c$ (Option B).

This completely ignores two critical relativistic effects: the reduction in the photon collision rate due to the boundary's recession (flux transformation), and the reduction in momentum transfer per collision due to the Doppler shift of the reflected photons.

Options C and D represent partial corrections where a student accounts for only one of these two effects, failing to synthesize the wave-particle duality constraint with macroscopic kinematics.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Dual Nature of Radiation and Matter - Class 12
Concept	Wave-Particle Duality Constraint
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free