Problem 58 - University Physics with Modern Physics

Question

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?

Step-by-Step Solution

Answer

(a) \( T \approx 1.51 \times 10^{-16} \) s; (b) \( I \approx 1.06 \times 10^{-3} \) A; (c) \( \mu \approx 9.35 \times 10^{-24} \) J/T.

Step 1: Calculate the Orbital Circumference

In a circular orbit, the circumference of the orbit is given by the formula: \( C = 2 \pi r \), where \( r = 5.3 \times 10^{-11} \) m. Thus, the circumference is: \( C = 2\pi \times 5.3 \times 10^{-11} = 3.33 \times 10^{-10} \) m.

Step 2: Find Orbital Period of Electron

The orbital period \( T \) is the time it takes for the electron to complete one orbit. It is given by \( T = \frac{C}{v} \), where \( C = 3.33 \times 10^{-10} \) m is the circumference and \( v = 2.2 \times 10^6 \) m/s. Therefore, \( T = \frac{3.33 \times 10^{-10}}{2.2 \times 10^6} \approx 1.51 \times 10^{-16} \) s.

Step 3: Calculate the Current due to Electron Motion

The current \( I \) is the charge divided by the time period, \( I = \frac{e}{T} \), where \( e = 1.6 \times 10^{-19} \) C. From the previous step, \( T = 1.51 \times 10^{-16} \) s. Thus, \( I = \frac{1.6 \times 10^{-19}}{1.51 \times 10^{-16}} \approx 1.06 \times 10^{-3} \) A.

Step 4: Calculate Magnetic Moment of the Electron

The magnetic moment \( \mu \) is given by \( \mu = I \cdot A \), where \( A \) is the area of the loop. The area \( A = \pi r^2 = \pi (5.3 \times 10^{-11})^2 \). Thus, \( A \approx 8.83 \times 10^{-21} \) m². Therefore, \( \mu = 1.06 \times 10^{-3} \times 8.83 \times 10^{-21} \approx 9.35 \times 10^{-24} \) J/T.

Key Concepts

Magnetic Moment

The magnetic moment is a concept used to express the magnetic strength and orientation of a magnet or another object that produces a magnetic field. In the Bohr model of the hydrogen atom, the motion of the electron around the nucleus can be equated to a tiny loop of current, which consequently creates a magnetic field. Hence, the hydrogen atom possesses a magnetic moment due to the electron in motion.
To calculate this magnetic moment, we first determine the current formed by the moving electron. This current is associated with the electron's charge and the time it takes to complete one orbital cycle, known as the orbital period. Using the formula \( \mu = I \cdot A \), where \( I \) is the current and \( A \) is the area of the circular path taken by the electron, we can find the magnitude of the magnetic moment.
The area \( A \) can be obtained using \( A = \pi r^2 \), with \( r \) being the radius of the electron's orbit. Hence, this provides insights into how the structural geometry of the electron's path influences the atom's magnetic properties.

Orbital Period

The orbital period is the time it takes for an electron to complete one full orbit around the nucleus. In the Bohr model, the electron travels in a designated circular orbit, which allows us to precisely calculate the orbital period. This calculation is crucial for understanding how often the "current loop," or revolving electron, can generate effects such as a magnetic field.
Given the speed of the electron and the circumference of its orbit, the orbital period can be calculated using \( T = \frac{C}{v} \), where \( C \) is the orbital circumference and \( v \) is the electron's speed. In our scenario, the orbital period is found to be approximately \( 1.51 \times 10^{-16} \) seconds, indicating an incredibly fast revolutionary motion around the hydrogen nucleus.

Current Loop

A moving charge can be thought of as an electrical current. In the context of the Bohr hydrogen atom, the electron's motion in its orbit is equivalent to a small current loop. This is because a circulating charge generates a magnetic effect akin to a current flowing through a wire.
This concept allows us to use formulas for calculating current as \( I = \frac{e}{T} \), where \( e \) is the charge of the electron and \( T \) is the orbital period. In this context, the electron behaves like a tiny electric current producing magnetic fields, contributing to the atom's overall magnetic characteristics.
This simplification shows how atomic-scale phenomena can be analyzed using basic electromagnetic concepts, highlighting the interconnectedness of electron motion, magnetic fields, and current.

Electron Motion

In the Bohr model of the hydrogen atom, the electron revolves around the nucleus in a circular path. This motion is significant as it lays the foundation for several other derived properties, such as the atom's magnetic moment and the induced current.
The motion of the electron can be characterized by its speed and path radius. For instance, the speed \( 2.2 \times 10^6 \) m/s and radius \( 5.3 \times 10^{-11} \) m are used to describe its path and compute further quantities like the orbital period and current. This reinforces the notion of the electron's dual role as both a particle in motion and a contributor to the quantum properties of the atom.
Understanding electron motion is critical as it underpins many atomic interactions and characteristics. It's a fundamental aspect of how atoms, such as hydrogen, exhibit their unique properties in the physical world.