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Question #43819

Difficulty: 8/10 | 🟢 Free

The Question

A stationary, initially neutral radioactive sample is located at the origin of a vacuum chamber. At $t=0$ , it contains $N_0$ radioactive nuclei and begins to decay with a decay constant $\lambda$ . Each decay event produces a heavy daughter nucleus (which remains at rest at the origin) and emits a light particle of charge $+q$ that travels radially outward at a constant speed $v$ . Consider a fixed spherical Gaussian surface of radius $R$ centered at the origin. What is the magnitude of the displacement current $I_d$ through this Gaussian surface at a time $t > \frac{R}{v}$ ?

Options

A.
$q N_0 \lambda e^{-\lambda t}$
B.
$q N_0 \lambda e^{-\lambda (t - R/v)}$
C.
$q N_0 \lambda \left( e^{-\lambda (t - R/v)} - e^{-\lambda t} \right)$
D.
$q N_0 \lambda \left( 1 - e^{-\lambda (t - R/v)} \right)$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Activity Units and Transitions|Flux and Surface Geometry|Gauss Law Applicability

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to dynamically evaluate Gauss's Law for a time-varying, spatially expanding charge distribution.

A common trap is to assume the displacement current depends only on the instantaneous activity at the origin (yielding Option A), ignoring the finite transit time of the emitted particles.

Another trap is considering only the particles in transit as the source of flux change (Option C).

Students must recognize that the enclosed charge depends on the retarded time $t - R/v$ , or alternatively, apply the Ampere-Maxwell law recognizing that the spherically symmetric total current ( $I_c + I_d$ ) must be zero.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Nuclei - Class 12
Concept	Activity Units and Transitions
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free