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Question #43692

Difficulty: 8/10 | 🟢 Free

The Question

A radioactive source containing $N_0$ nuclei with a decay constant $\lambda$ is placed at the origin. It undergoes $\beta^-$ decay, emitting electrons of charge $-e$ . Through a constant external magnetic confinement mechanism, all emitted electrons are immediately captured into a single circular orbit of radius $R$ centered at the origin. The electrons continue to orbit indefinitely with a constant speed $v$ . A small stationary conducting loop of radius $r$ ( $r \ll R$ ) is placed at the origin, coplanar with the electron orbit. Assuming the external confinement field does not change with time, which of the following expressions represents the magnitude of the induced non-conservative electric field at the rim of the small loop as a function of time $t$ ?

Options

A.
$\frac{\mu_0 e N_0 v \lambda r}{8 \pi R^2} e^{-\lambda t}$
B.
$\frac{\mu_0 e N_0 \lambda^2 r}{4 R} e^{-\lambda t}$
C.
$\frac{\mu_0 e N_0 v \lambda r}{4 R^2} e^{-\lambda t}$
D.
$\frac{\mu_0 e N_0 v \lambda r^2}{4 \pi R^3} e^{-\lambda t}$

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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

Radioactive Decay Law Dynamics|Magnetic Moment of Orbiting Electron|Faraday’s Law Vector Form

Why This Question Is Tricky

Common Mistakes

This question tests the critical distinction between the instantaneous rate of particle emission (decay rate) and the accumulated circulating charge that forms a steady macroscopic current.

A common trap for students is to reflexively equate the current $I(t)$ directly to the decay rate $e|dN/dt| = e\lambda N_0 e^{-\lambda t}$ .

This fundamentally misunderstands the magnetic moment of an orbiting electron, where the current is determined by the total accumulated charge divided by the orbital period.

By forcing the student to derive the time-dependent current from the integrated decay function $N_0(1 - e^{-\lambda t})$ , the problem elegantly links stochastic radioactive decay with deterministic electromagnetic induction.

Taking the derivative of the incorrect flux yields a dimensionally similar but physically flawed expression, heavily penalizing the rote application of formulas without physical reasoning.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Nuclei - Class 12
Concept	Radioactive Decay Law Dynamics
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free