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Question #42618

Difficulty: 8/10 | 🟢 Free

The Question

A radioactive sample contains $N_0$ active nuclei at time $t=0$ . The sample has a half-life of $T_{1/2}$ . The root mean square (RMS) lifetime of the nuclei, denoted by $t_{\text{rms}}$ , is defined as the square root of the average of the squared lifetimes of all individual nuclei. Which of the following gives the correct expression for $t_{\text{rms}}$ ?

Options

A.
$\frac{\sqrt{2} T_{1/2}}{\ln 2}$
B.
$\frac{T_{1/2}}{\sqrt{2} \ln 2}$
C.
$\frac{2 T_{1/2}}{\ln 2}$
D.
$\frac{T_{1/2}}{\ln 2}$

Check Your Answer

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

Mean vs. RMS Values|Radioactive Decay Law Dynamics|Integration

Why This Question Is Tricky

Common Mistakes

Students often conflate the 'mean lifetime' ( $\tau = 1/\lambda$ ) with the 'RMS lifetime', assuming they are identical or simply related by a factor of $1/\sqrt{2}$ as in AC circuits.

This question tests the fundamental understanding of statistical averages in continuous probability distributions.

By requiring the evaluation of the second moment of the exponential decay distribution ( $\int t^2 \lambda e^{-\lambda t} dt$ ), it forces students to apply integration by parts or Gamma functions, exposing the trap of rote memorization of the standard average life formula.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Oscillations - Class 11 Part 2
Concept	Mean vs. RMS Values
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free