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Question #43616

Difficulty: 8/10 | 🟢 Free

The Question

A stationary point radioactive source contains $N_0$ nuclei with a decay constant $\lambda$ . Each nuclear decay instantaneously and isotropically emits a single gamma photon of energy $E_0$ . A detector of area $A$ is placed at a distance $d$ from the source ( $d \gg \sqrt{A}$ ), oriented such that its surface is perpendicular to the incident photons. Assume the detector has $100\%$ efficiency for incident photons and the medium between the source and detector is a perfect vacuum. Consider a time interval $\Delta t = \frac{4\pi d^2}{N_0 \lambda A}$ , and assume $\lambda \Delta t \ll 1$ so that the source activity remains constant. What is the probability that the detector registers exactly one gamma photon during the interval $\Delta t$ ?

Options

A.
$1$
B.
$e^{-1}$
C.
$1 - e^{-1}$
D.
$\frac{1}{2}$

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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

Instantaneous Nature|Statistical Nature of Decay|Activity Units and Transitions

Why This Question Is Tricky

Common Mistakes

This question tests the distinction between classical time-averaged intensity and the discrete, statistical nature of quantum emission.

Students often fall into the trap of treating gamma radiation as a continuous energy flow.

By calculating the classical energy received ( $E_{rec} = I A \Delta t$ ) and finding it exactly equals the energy of one photon ( $E_0$ ), they incorrectly conclude that exactly one photon will be detected with 100% certainty (Option A).

However, radioactive decay is a probabilistic process governed by Poisson statistics.

The calculated value is merely the expectation value ( $\mu = 1$ ), meaning the actual probability of detecting exactly one photon is $e^{-1}$ .

Quick Facts About This Question

Question Information

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