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Question #43808

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The Question

A perfectly reflecting block of mass $m$ is initially at rest on a rough horizontal surface. The coefficients of static and kinetic friction between the block and the surface are $\mu_s$ and $\mu_k$ , respectively ( $\mu_s > \mu_k$ ). A continuous laser beam of cross-sectional area $A$ and intensity $I$ is directed horizontally such that it is fully intercepted by the block. The laser emits photons corresponding exclusively to the transition from the first excited state to the ground state of a hydrogen atom. It is observed that the block just begins to move when the laser is turned on. The laser is kept on, and the block accelerates. Which of the following expressions gives the total number of photons $N$ that strike the block while it undergoes a displacement $D$ ? (Let $R$ be the Rydberg constant, $h$ be Planck's constant, $c$ be the speed of light, and $g$ be the acceleration due to gravity. Assume non-relativistic motion of the block.)

Options

A.
$\frac{4 \mu_s m}{3 R h} \sqrt{\frac{g D}{2 (\mu_s - \mu_k)}}$
B.
$\frac{4 \mu_s m g D}{3 R h c}$
C.
$\frac{4 (\mu_s - \mu_k) m g D}{3 R h c}$
D.
$\frac{4 \mu_s m}{3 R h} \sqrt{\frac{2 g D}{\mu_s - \mu_k}}$

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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

Photon Polarization and Selection|Emission vs Absorption|Derived Units via Fundamental Constants

Why This Question Is Tricky

Common Mistakes

This problem tests the fundamental distinction between the total energy of an incident light beam and the mechanical work done by radiation pressure.

A common trap for advanced students is to erroneously apply the Work-Energy Theorem by equating the total incident photon energy ( $N E_\gamma$ ) directly to the mechanical work done against friction ( $W = \mu_s m g D$ ) or the net work ( $W_{net} = \Delta K$ ), leading to incorrect options B or C.

In reality, radiation pressure is a momentum-transfer phenomenon, and the incident energy $E_{total} = IAt$ depends on the time the beam is active, which is dictated by the macroscopic kinematics of the block accelerating under the net force.

Furthermore, the problem requires mapping the quantum selection rules of the Lyman series to fundamental constants ( $R, h, c$ ) to find the energy per photon, elegantly bridging quantum emission with Newtonian mechanics.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Atoms - Class 12
Concept	Photon Polarization and Selection
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free