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Question #43631

Difficulty: 8/10 | 🟢 Free

The Question

In a precision photoelectric experiment, a photon of energy $E$ interacts with a stationary free atom of mass $M$ and ionization energy $\Phi$ . The interaction occurs over a finite time interval $\tau$ , during which internal electromagnetic forces $\vec{F}(t)$ mediate the momentum transfer between the ejecting electron and the parent ion. Assuming the momentum of the incident photon is negligible, what is the exact magnitude of the stopping potential $V_0$ required to halt the emitted electron of mass $m$ ? (Let $e$ be the elementary charge).

Options

A.
$\frac{E - \Phi}{e}$
B.
$\frac{M}{M+m} \left( \frac{E - \Phi}{e} \right)$
C.
$\frac{m}{M+m} \left( \frac{E - \Phi}{e} \right)$
D.
$\frac{M-m}{M+m} \left( \frac{E - \Phi}{e} \right)$

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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

Stopping Potential Sign|Recoil of Atom|Reduced Mass Effect|Integration

Why This Question Is Tricky

Common Mistakes

Students commonly misapply the standard photoelectric equation $eV_0 = E - \Phi$ , treating the parent atom as infinitely massive and ignoring its recoil energy.

This trap is amplified by the red herring of a time-varying internal force $\vec{F}(t)$ and time interval $\tau$ , which tempts students to overcomplicate the problem with impulse integrals.

By recognizing that the photon's momentum is negligible, the system can be treated as isolated.

Conservation of momentum dictates that the electron and atom recoil with equal and opposite momenta, forcing the available kinetic energy to be distributed inversely proportional to their masses.

The reduced mass factor correctly scales the electron's kinetic energy, demonstrating that the standard formula is an approximation valid only when $M \gg m$ .

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Dual Nature of Radiation and Matter - Class 12
Concept	Stopping Potential Sign
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free