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Question #43818

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The Question

To determine the decay constant $\lambda$ of a radioactive isotope, an experiment is conducted using a detector with $100\%$ efficiency. At $t = 0$ , the detector records $C_0$ counts in a small time interval $\Delta t$ . At a later time $t = t_1$ (where $t_1 \gg \Delta t$ ), the detector records $C_1$ counts in an identical time interval $\Delta t$ . Assume $\Delta t \ll 1/\lambda$ such that the activity remains constant during each measurement interval. If the statistical fluctuations in the recorded counts are the only source of uncertainty, which of the following expressions gives the relative statistical uncertainty $\frac{\sigma_\lambda}{\lambda}$ in the calculated decay constant?

Options

A.
$\frac{\sqrt{C_0 + C_1}}{\sqrt{C_0 C_1} \ln(C_0/C_1)}$
B.
$\frac{\sqrt{C_0} + \sqrt{C_1}}{\sqrt{C_0 C_1} \ln(C_0/C_1)}$
C.
$\frac{C_0 + C_1}{C_0 C_1 \ln(C_0/C_1)}$
D.
$\frac{1}{\sqrt{C_0 + C_1} \ln(C_0/C_1)}$

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

Statistical Nature of Decay|Relative Error Propagation in Powers|Approximations

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to bridge deterministic decay equations with the probabilistic nature of quantum decay (Poisson statistics).

The primary trap is failing to recognize that the uncertainty in a counted number of random events $C$ is $\sqrt{C}$ .

Students often misapply standard linear error propagation (adding absolute or relative errors directly without quadrature, leading to Option B) or assume the variance does not require a square root in the final fractional error (leading to Option C).

By requiring the partial derivatives of a logarithmic function and the quadrature addition of Poisson errors, this problem rigorously filters for a deep understanding of statistical mechanics in experimental physics.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Nuclei - Class 12
Concept	Statistical Nature of Decay
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free