Magnetic Moment of the Hydrogen Atom. In the Bohr model of...

Magnetic Moment of the Hydrogen Atom. In the Bohr model of the hydrogen atom (see Section $38.5 ),$ in the lowest energy state the electron orbits the proton at a speed of $2.2 \times$ $10^{6} \mathrm{m} / \mathrm{s}$ in a circular orbit of radius $5.3 \times 10^{-11} \mathrm{m}$ (a) What is the orbital period of the electron? (b) If the orbiting electron is considered to be a current loop, what is the current $I ?$ (c) What is the magnetic moment of the atom due to the motion of the electron?

Key Concepts

Bohr Model

The Bohr model describes the hydrogen atom as an electron moving in fixed, circular orbits around the nucleus with quantized angular momentum. This model provides insight into the discrete energy levels seen in atomic spectra and forms a basis for understanding atomic structure despite its limitations with more complex atoms.

Orbital Period

The orbital period is the time it takes for an object, such as an electron, to complete one full revolution along its circular path. It is calculated using the circumference of the orbit divided by the object's speed, a concept that applies broadly in classical mechanics when analyzing circular motion.

Current Loop in Electrodynamics

A moving charge that circulates in a closed path is analogous to an electric current loop. In this context, the current is determined by the charge flow over time, making it possible to treat the orbital motion of a charged particle as generating a current, which is a fundamental idea in electromagnetism.

Magnetic Moment of a Current Loop

The magnetic moment is a measure of the strength and orientation of a magnetic source produced by a current loop. It is proportional to both the current and the area of the loop, and it plays a key role in determining how magnetic fields interact with the system.

Relation between Angular Momentum and Magnetic Moment

In atomic and quantum physics, there is a fundamental connection between the angular momentum of a charged particle in motion and the magnetic moment it generates. This relationship is particularly significant in atomic models, where the electron's orbital motion gives rise to magnetic properties that influence its interaction with external magnetic fields.

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