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Question #43695

Difficulty: 8/10 | 🟢 Free

The Question

A toroidal coil of rectangular cross-section has an inner radius $a$ , outer radius $b$ , height $h$ , and a total of $N$ turns. A thin, straight capillary tube passes exactly along the axis of symmetry of the toroid. The tube contains a sample of a radioactive isotope that undergoes alpha decay, emitting $\alpha$ particles (each with charge $+2e$ ). At time $t=0$ , the number of radioactive nuclei is $N_0$ , and the decay constant of the isotope is $\lambda$ . An external mechanism ensures that all emitted $\alpha$ particles are immediately swept along the tube in the same direction, forming a continuous narrow beam. Assume the transit time of the particles through the toroid is negligible, so the beam acts as a long straight wire carrying a spatially uniform, time-varying current. What is the magnitude of the maximum induced electromotive force (EMF) in the toroidal coil? (Assume all quantities are in SI units and $\mu_0$ is the permeability of free space)

Options

A.
$\frac{\mu_0 N h e \lambda^2 N_0}{\pi} \ln\left(\frac{b}{a}\right)$
B.
$\frac{\mu_0 N h e \lambda N_0}{\pi} \ln\left(\frac{b}{a}\right)$
C.
$\frac{\mu_0 N h \lambda^2 N_0}{2\pi} \ln\left(\frac{b}{a}\right)$
D.
$\frac{\mu_0 N h \lambda N_0}{2\pi} \ln\left(\frac{b}{a}\right)$

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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

Statistical Nature of Decay|Mutual Inductance Reciprocity|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of statistical radioactive decay and electromagnetic induction, specifically targeting the misconception of treating decay rate directly as an EMF surrogate.

Students often correctly identify that current is proportional to the decay rate ( $I \propto dN/dt \propto \lambda e^{-\lambda t}$ ), but fail to apply Faraday's Law correctly, which requires the time derivative of the magnetic flux ( $\mathcal{E} = -d\Phi/dt$ ).

Consequently, the induced EMF depends on the second derivative of the nuclei count ( $d^2N/dt^2$ ), introducing an additional factor of $\lambda$ .

Furthermore, the trap punishes students who conflate 'decay events per unit time' with 'charge per unit time' by omitting the specific charge of the $\alpha$ particle ( $+2e$ ).

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