In the Bohr model of the hydrogen atom (see Section 39.3),...

In the Bohr model of the hydrogen atom (see Section 39.3), in the lowest energy state the electron orbits the proton at a speed of 2.2 $\times$ 10$^6$ m/s in a circular orbit of radius 5.3 $\times$ 10$^{-11}$ 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 is a seminal model of the hydrogen atom where electrons move in fixed, circular orbits with quantized angular momentum. It combines classical circular motion with early quantum ideas, leading to discrete permitted orbits with specific radii and energies. This simplified model helps us understand atomic spectra and the fundamental interactions within atoms.

Circular Motion and Orbital Period

In classical mechanics, an object moving in a circular orbit has an orbital period defined as the time it takes to complete one full revolution. The orbital period can be calculated by dividing the circumference of the orbit (2? times the radius) by the speed of the object. This fundamental concept is used to determine how long it takes the electron to circle the nucleus in the Bohr model.

Current Loop

A moving charge in a circular orbit can be considered equivalent to a current loop. In this analogy, the electron's motion creates a current, which is defined as the charge passing a point per unit time (I = charge/orbital period). This concept is critical in linking the microscopic movement of charges to macroscopic electromagnetic effects.

Magnetic Moment

The magnetic moment of a current loop is a measure of its tendency to align with a magnetic field and is calculated as the product of the current and the area of the loop. In the context of the Bohr atom, the magnetic moment arises from the orbiting electron and is important for understanding the magnetic properties of atoms, such as their interactions with external magnetic fields.

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