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Question #43677

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The Question

Consider a hypothetical heavy nucleus of mass number $A$ and atomic number $Z$ , modeled classically as a rigid solid sphere of uniform mass density and uniform charge density. The nucleus rotates about its diametric axis with a constant angular velocity $\omega$ . Given that the nuclear radius follows the empirical relation $R = R_0 A^{1/3}$ (where $R_0$ is a constant) and the mass of a single nucleon is $m_p$ , which of the following expressions correctly represents the magnitude of the magnetic dipole moment $\mu$ of this rotating nucleus?

Options

A.
$\frac{1}{5} Z A^{2/3} e R_0^2 \omega$
B.
$\frac{1}{2} Z A^{2/3} e R_0^2 \omega$
C.
$\frac{1}{5} Z A^{5/3} e R_0^2 \omega$
D.
$\frac{1}{2} Z A^{5/3} e R_0^2 \omega$

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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

Nuclear Density Invariance|Magnetic Dipole Moment of Revolving Electron|Vector Cross-Product

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to transition from point-particle quantum models (like the Bohr model) to classical rigid-body electrodynamics for extended charge distributions.

A common trap is applying the point-particle magnetic moment formula ( $\mu = \frac{1}{2} Q R^2 \omega$ ) directly to the nucleus, ignoring its spatial extent and treating it as a revolving ring of charge.

By enforcing the constant nuclear density constraint ( $R = R_0 A^{1/3}$ ), the problem mandates the use of the solid sphere moment of inertia ( $I = \frac{2}{5}MR^2$ ).

Students must recognize that the gyromagnetic ratio ( $\gamma = Q/2M$ ) holds for any uniform charge-to-mass distribution, allowing them to bypass complex volume integration by directly scaling the angular momentum.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Nuclei - Class 12
Concept	Nuclear Density Invariance
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free