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Question #43801

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The Question

In a Michelson interferometer experiment, the light source is a discharge tube containing atomic hydrogen gas (mass of a hydrogen atom is $m$ ) at an absolute temperature $T$ . The interferometer is tuned to observe the radiation corresponding to the Balmer series limit. Due to the thermal motion of the atoms, the spectral line is Doppler broadened. Assuming the effective spectral width $\Delta \lambda$ is defined by the Doppler shift of atoms moving along the line of sight with the most probable speed $v_p$ , the interference fringes disappear when the movable mirror is displaced by a minimum distance $d$ from the zero path difference position. Which of the following expressions correctly represents $d$ ? (Let $R$ be the Rydberg constant, $c$ be the speed of light, and $k$ be the Boltzmann constant. Assume $v_p \ll c$ ).

Options

A.
$\frac{c}{R} \sqrt{\frac{m}{2kT}}$
B.
$\frac{c}{R} \sqrt{\frac{2m}{kT}}$
C.
$\frac{c}{R} \sqrt{\frac{m}{3kT}}$
D.
$\frac{c}{4R} \sqrt{\frac{m}{2kT}}$

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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

Spectral Series Limits|Coherence Requirement|Kinetic Theory (Velocity Distribution)

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of atomic physics, kinetic theory, and wave optics.

A common trap is treating spectral limits as perfectly monochromatic mathematical boundaries, ignoring the physical reality of Doppler broadening in a thermal gas.

Students often misapply the interference condition by forgetting that a mirror displacement $d$ in a Michelson interferometer introduces a path difference of $2d$ , leading to an off-by-a-factor-of-two error (Option B).

Furthermore, confusing the most probable speed with the root-mean-square speed ( $v_{rms} = \sqrt{3kT/m}$ ) leads to Option C, while incorrectly identifying the Balmer limit as the Lyman limit ( $\lambda_0 = 1/R$ ) leads to Option D.

The problem forces the recognition that coherence length is inversely proportional to the velocity spread of the emitters.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Atoms - Class 12
Concept	Spectral Series Limits
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free