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Question #43433

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The Question

An electron (mass $m$ , charge $-e$ ) orbits a stationary nucleus of atomic number $Z$ in a circular orbit corresponding to the principal quantum number $n$ . The motion is governed by the Coulombic central force and satisfies Bohr's quantization condition. Let $\vec{\mu}$ be the orbital magnetic dipole moment of the electron, $a_0$ be the Bohr radius, and $\mu_0$ be the vacuum permeability. Assume all quantities are in SI units. Which of the following expressions correctly represents the magnetic field $\vec{B}$ at the center of the nucleus produced by this orbiting electron?

Options

A.
$\frac{\mu_0 Z^3}{2\pi a_0^3 n^6} \vec{\mu}$
B.
$-\frac{\mu_0 Z^3}{2\pi a_0^3 n^6} \vec{\mu}$
C.
$\frac{\mu_0 Z^3}{4\pi a_0^3 n^6} \vec{\mu}$
D.
$-\frac{\mu_0 Z^3}{4\pi a_0^3 n^6} \vec{\mu}$

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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

Magnetic Dipole Moment of Revolving Electron|Quantization of Angular Momentum|Coulombic Central Force

Why This Question Is Tricky

Common Mistakes

This question tests the student's understanding of the vector relationship between magnetic field, angular momentum, and magnetic dipole moment for a negatively charged particle.

A common trap is assuming that because the electron has a negative charge, the magnetic field it produces at the center must be antiparallel to its magnetic dipole moment, leading to an incorrect negative sign.

However, both $\vec{B}$ and $\vec{\mu}$ are defined relative to the conventional current, making them strictly parallel regardless of the charge carrier's sign.

Additionally, students often misapply the Bohr radius formula or forget the factor of $1/2$ in the magnetic moment of a revolving charge ( $\mu = \frac{1}{2}evr$ ), which incorrectly leads to a $4\pi$ factor in the denominator.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Magnetism and Matter - Class 12
Concept	Magnetic Dipole Moment of Revolving Electron
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free