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Question #43670

Difficulty: 8/10 | 🟢 Free

The Question

For a hypothetical hydrogen-like ion, the transition corresponding to the first line of the Balmer series emits a photon of wavelength $\lambda_{B1} = \frac{36h}{5 m_e c}$ , where $h$ is Planck's constant, $m_e$ is the electron rest mass, and $c$ is the speed of light in vacuum. A linearly polarized photon emitted from the Lyman series limit of this same ion is incident on a free electron initially at rest. The photon undergoes Compton backscattering (scattering angle $\theta = 180^\circ$ ). Calculate the de Broglie wavelength $\lambda_e$ of the recoiling electron.

Options

A.
$\frac{3h}{4 m_e c}$
B.
$\frac{\sqrt{3}h}{2 m_e c}$
C.
$\frac{3h}{2 m_e c}$
D.
$\frac{h}{m_e c}$

Check Your Answer

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

Photon Polarization and Selection|Spectral Series Limits|De Broglie Wavelength Scaling|Approximations (Binomial)

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of atomic spectral limits with relativistic kinematics.

The primary trap lies in the treatment of the recoiling electron.

Students often calculate the kinetic energy of the electron ( $K = \frac{2}{3} m_e c^2$ ) and incorrectly apply the non-relativistic momentum formula $p = \sqrt{2m_e K}$ , leading to option B.

Because the incident photon energy ( $m_e c^2$ ) is comparable to the electron's rest mass energy, the recoil is highly relativistic.

The problem elegantly bypasses the need for complex relativistic kinetic energy expansions by allowing direct application of linear momentum conservation ( $p_e = p_{initial} - p_{final}$ ), rewarding students who rely on fundamental vector conservation laws over rote kinetic formulas.

Additionally, it tests sign conventions in 1D vector scattering (Option C trap).

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