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Question #43788

Difficulty: 8/10 | 🟢 Free

The Question

A spherical satellite of radius $R$ is in deep space where the ambient temperature is $0 \text{ K}$ . Inside the satellite, a source contains a sample of $He^+$ ions. Exactly $N$ ions per second undergo an electronic transition from the first excited state to the ground state. All the emitted photon energy is completely absorbed by the satellite's interior. The satellite's outer surface has a latitude-dependent emissivity given by $\epsilon(\theta) = \epsilon_0 \sin^2\theta$ , where $\theta$ is the polar angle ( $0 \le \theta \le \pi$ ) and $\epsilon_0$ is a dimensionless constant. Assuming the satellite is highly thermally conductive so that its surface temperature $T$ is uniform, what is the steady-state temperature $T$ ?

(Let $E_0$ be the ionization energy of a hydrogen atom in its ground state, and $\sigma$ be the Stefan-Boltzmann constant.)

Options

A.
$\left( \frac{9 N E_0}{8 \pi R^2 \sigma \epsilon_0} \right)^{\frac{1}{4}}$
B.
$\left( \frac{3 N E_0}{2 \pi R^2 \sigma \epsilon_0} \right)^{\frac{1}{4}}$
C.
$\left( \frac{9 N E_0}{32 \pi R^2 \sigma \epsilon_0} \right)^{\frac{1}{4}}$
D.
$\left( \frac{3 N E_0}{\pi^2 R^2 \sigma \epsilon_0} \right)^{\frac{1}{4}}$

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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

Energy Level Scaling|Stefan-Boltzmann Power Law|Emissivity and Absorptivity|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to bridge discrete quantum energy scaling with macroscopic continuous thermal radiation.

The primary trap lies in the calculation of the radiated power.

Students often incorrectly assume emissivity can be averaged linearly or forget the geometric weighting factor ( $\sin\theta$ ) in the differential area element $dA = 2\pi R^2 \sin\theta d\theta$ when integrating the Stefan-Boltzmann law over a spherical surface.

Furthermore, a secondary trap exists in the Bohr model scaling: students might forget the $Z^2$ dependence for the $He^+$ ion, leading to an incorrect input power.

The problem demands rigorous application of surface integrals to non-uniform material properties in thermal equilibrium.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Atoms - Class 12
Concept	Energy Level Scaling
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free