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Question #43558

Difficulty: 8/10 | 🟢 Free

The Question

A stationary nucleus of mass $M$ is located at a very large distance on the optical axis ( $z$ -axis) of an astronomical telescope. The nucleus undergoes beta decay, emitting an electron with momentum $p_e$ along the positive $z$ -axis and an antineutrino with momentum $p_\nu$ in an arbitrary direction. Immediately after the decay, the recoiling nucleus emits a gamma-ray photon exactly along the positive $z$ -axis in its own rest frame. The telescope is in normal adjustment with an objective lens of focal length $f_o$ and an eyepiece of focal length $f_e$ . Assuming $p_e, p_\nu \ll Mc$ (where $c$ is the speed of light), what is the maximum possible angle $\theta_{\text{max}}$ with respect to the optical axis at which the photon can emerge from the eyepiece?

Options

A.
$\frac{f_o p_\nu}{f_e M c}$
B.
$\frac{f_o (p_e + p_\nu)}{f_e M c}$
C.
$\frac{f_e p_\nu}{f_o M c}$
D.
$\frac{f_o \sqrt{p_e^2 + p_\nu^2}}{f_e M c}$

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

Telescope Normal Adjustment|Recoil of Atom|Photon Polarization and Selection

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of nuclear recoil kinematics, relativistic aberration, and ray optics.

The primary trap lies in misapplying conservation of momentum by assuming the electron's momentum contributes to the transverse recoil, leading students to incorrectly use $p_e + p_\nu$ or $\sqrt{p_e^2 + p_\nu^2}$ .

Furthermore, students often memorize the Doppler shift for frequency but neglect the spatial 'headlight effect' (stellar aberration) where the transverse velocity of the source tilts the photon's propagation vector in the lab frame.

Finally, it requires the correct application of the telescope's angular magnification formula $m = f_o/f_e$ for parallel rays in normal adjustment, linking a microscopic quantum decay event to a macroscopic optical observation.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Ray Optics and Optical Instruments - Class 12
Concept	Telescope Normal Adjustment
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free