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Question #43658

Difficulty: 8/10 | 🟢 Free

The Question

A cubical container of volume $V$ holds $N$ atoms of an ideal monatomic gas, each of mass $m$ . The gas is initially in thermal equilibrium at temperature $T$ with pressure $P_0$ . At time $t=0$ , every atom simultaneously emits a single photon of energy $E$ in the $+x$ direction. Assume the emission is instantaneous and the atoms undergo elastic collisions with the container walls without colliding with each other. Let $\Delta P_x$ be the change in the average pressure exerted by the gas on the two walls perpendicular to the x-axis, and $\Delta P_y$ be the change in pressure on the walls perpendicular to the y-axis. Which of the following is correct? (Here $c$ is the speed of light and $n = N/V$ )

Options

A.
$\Delta P_x = \frac{n E^2}{m c^2}, \quad \Delta P_y = 0$
B.
$\Delta P_x = \frac{n E^2}{3 m c^2}, \quad \Delta P_y = \frac{n E^2}{3 m c^2}$
C.
$\Delta P_x = \frac{n E^2}{2 m c^2}, \quad \Delta P_y = 0$
D.
$\Delta P_x = \frac{2 n E^2}{3 m c^2}, \quad \Delta P_y = 0$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Recoil of Atom|Pressure-Kinetic Energy Relation|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the fundamental microscopic definition of pressure as the momentum flux tensor, rather than the macroscopic scalar relation $P = \frac{2}{3} \frac{K}{V}$ .

Students often fall into the trap of calculating the total increase in kinetic energy ( $\Delta K = N \frac{p^2}{2m}$ ) and blindly applying the isotropic pressure formula to conclude $\Delta P = \frac{2}{3} \frac{\Delta K}{V} = \frac{n E^2}{3 m c^2}$ for all walls.

However, the photon emission is highly directional (anisotropic), meaning the velocity distribution only shifts along the x-axis.

Consequently, the pressure increase is entirely confined to the x-walls, requiring an integration of the momentum transfer over the shifted velocity distribution.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Atoms - Class 12
Concept	Recoil of Atom
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free