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Question #42416

Difficulty: 8/10 | 🟢 Free

The Question

A radioactive sample initially contains $N_0$ identical nuclei at $t=0$ , each with a decay constant $\lambda$ . Let $N_D(t)$ be the random variable representing the exact number of nuclei that have decayed by time $t$ . Which of the following expressions correctly represents the statistical variance of $N_D(t)$ ?

Options

A.
$N_0 e^{-\lambda t} (1 - e^{-\lambda t})$
B.
$N_0 (1 - e^{-\lambda t})$
C.
$N_0 e^{-\lambda t}$
D.
$N_0 \lambda t e^{-\lambda t}$

Check Your Answer

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

Instantaneous vs Average|Statistical Nature of Decay|Differential Equations

Why This Question Is Tricky

Common Mistakes

This question targets the lethal cognitive trap where students blindly equate radioactive decay with a Poisson distribution, assuming the variance always equals the mean.

While the decay of a single nucleus in an infinitesimal time $dt$ is a Poisson process with parameter $\lambda dt$ , the macroscopic ensemble of $N_0$ nuclei over a finite time $t$ follows a Binomial distribution.

Students who confuse the continuous macroscopic rate approximation with the exact discrete probability will incorrectly select the mean number of decays $N_0(1 - e^{-\lambda t})$ or the mean remaining nuclei $N_0 e^{-\lambda t}$ as the variance.

This exposes the fundamental difference between the constant transition probability per unit time and the statistical variance of the decaying population.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Motion in a Plane - Class 11 Part 1
Concept	Instantaneous vs Average
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free