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Question #43666

Difficulty: 8/10 | 🟢 Free

The Question

An atom of mass $m$ , initially in its ground state, falls from rest at a very large distance towards a spherically symmetric planet of mass $M$ . At a distance $R$ from the planet's center, the atom absorbs a photon traveling radially outward. The local frequency of the photon at the point of absorption is $\nu$ . The absorption excites the atom to an internal state with energy $\Delta E$ above the ground state and brings the atom instantaneously to rest. Assuming non-relativistic mechanics for the atom's motion, which of the following expressions gives the required photon frequency $\nu$ ? (Here $h$ is Planck's constant, $c$ is the speed of light, and $G$ is the universal gravitational constant).

Options

A.
$\frac{mc^2}{h} \left( \sqrt{1 + \frac{2\Delta E}{mc^2}} - 1 \right)$
B.
$\frac{\Delta E}{h} + \frac{GMm}{hR}$
C.
$\frac{mc^2}{h} \left( 1 - \sqrt{1 - \frac{2\Delta E}{mc^2}} \right)$
D.
$\frac{\Delta E}{h}$

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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

Emission vs Absorption|Gravitational Potential Reference|Escape Velocity Independence

Why This Question Is Tricky

Common Mistakes

This question tests the simultaneous application of energy and momentum conservation in a photon-particle collision within a gravitational field.

A primary trap is assuming the photon energy must directly overcome the gravitational potential energy (leading to Option B), ignoring that the atom's kinetic energy (gained from falling) actually contributes to the excitation.

Another trap is misapplying the recoil energy sign (Option C) by assuming the atom was initially at rest and recoiled, rather than being brought to rest.

The presence of the planet's mass $M$ and distance $R$ acts as a deliberate distractor, testing the student's confidence in local conservation laws over blindly combining all given variables.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Atoms - Class 12
Concept	Emission vs Absorption
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free