Problem 45 - Physics

Question

In the model of the hydrogen atom due to Niels Bohr, the electron moves around the proton at a speed of \(2.2 \times 10^{6} \mathrm{~m} / \mathrm{s}\) in a circle of radius \(5.3 \times 10^{-11} \mathrm{~m}\). Considering the orbiting electron to be a small current loop, determine the magnetic moment associated with this motion. (Hint: The electron travels around the circle in a time equal to the period of the motion.)

Step-by-Step Solution

Answer

The magnetic moment associated with the electron's motion is approximately \(9.27 \times 10^{-24} \text{ A} \cdot \text{m}^2\), called the Bohr magneton.

Step 1: Understand the Problem

We need to find the magnetic moment of an electron moving in a circular orbit. The electron's motion can be considered as a current loop, where the current is the charge of the electron divide by the period of one complete revolution.

Step 2: Find the Period of Motion

The period, \(T\), is the time it takes for the electron to complete one revolution around the orbit. Using the circumference of the circle, \(2\pi r\), and the speed of the electron \(v\), the period is calculated as:\[T = \frac{2\pi r}{v}\]Given \(r = 5.3 \times 10^{-11} \text{ m}\) and \(v = 2.2 \times 10^{6} \text{ m/s}\), calculate:\[T = \frac{2\pi \times 5.3 \times 10^{-11}}{2.2 \times 10^6}\]

Step 3: Calculate the Current

The current \(I\) is defined as the charge per unit time. The charge is the elementary charge of the electron, \(e = 1.6 \times 10^{-19} \text{ C}\), and the period \(T\) from Step 2:\[I = \frac{e}{T}\]

Step 4: Determine the Magnetic Moment

The magnetic moment \(\mu\) is given by\[\mu = I \times A\]where \(A\) is the area of the circle (\(\pi r^2\)): \[A = \pi (5.3 \times 10^{-11})^2\]Therefore, \[\mu = \frac{e}{T} \times \pi (5.3 \times 10^{-11})^2\]Substitute for \(T\) and \(e\) from the earlier steps to get \(\mu\).

Step 5: Calculate the Magnetic Moment Using Given Values

With the calculated period, plug it into the expression for \(\mu\). Using \(T\), \(r\), and \(e\), compute the numerical value of the magnetic moment:\[\mu = \left(\frac{1.6 \times 10^{-19}}{\frac{2\pi \times 5.3 \times 10^{-11}}{2.2 \times 10^6}}\right) \times \pi (5.3 \times 10^{-11})^2\] Compute this to find the value of the magnetic moment.

Key Concepts

Magnetic Moment

Magnetic moment is a fundamental property related to the magnetic characteristics of a system. In the context of the Bohr model of the atom, it pertains to an electron orbiting a nucleus. This is analogous to a tiny magnet. It reflects the strength and orientation of this "magnet".

When an electron orbits a nucleus, it produces a magnetic field similar to a current loop. The magnetic moment, denoted by \(\mu\), is calculated using the equation:

• \( \mu = I \times A \)

where \(I\) is the current generated by the moving electron, and \(A\) is the area of the orbit. In Bohr's model, because electrons are charged particles moving in circles, they behave like a loop of current.

To give perspective, the SI unit for magnetic moment is the joule per tesla (J/T), often expressed in \Ampere square meters (A·m²). Understanding this concept helps grasp how microscopic particles like electrons contribute to the bulk magnetic properties of materials like iron.

Current Loop

Imagine the path of an electron in orbit around a nucleus as a loop. This loop carries current similar to a circular wire in a circuit. This electron path forms what's known as a current loop. The electron’s motion generates a current because it periodically completes a circuit around the nucleus.

In a current loop, the current \(I\) is given by:

• \( I = \frac{e}{T} \)

Here, \(e\) is the electronic charge (\(1.6 \times 10^{-19} \text{ C}\)), and \(T\) represents the period for one full orbit.

This is similar to what happens in a copper wire bent into a circle carrying electric current. The moving charges in both cases create magnetic fields, underlying the fundamental concept of electromagnetism, which unites electric and magnetic phenomena.

Electron Orbit

In Rutherford-Bohr model, the electron orbit is a path around the nucleus in a circular manner. The electron constantly moves with a specific velocity causing this orbital motion. Its orbit defines the quantized energy level of the hydrogen atom.

The orbit is a circle with a radius \(r = 5.3 \times 10^{-11} \text{ m}\). The electron circles the nucleus at a high speed \(v = 2.2 \times 10^6 \text{ m/s}\). This orbit is like a racetrack that the electron continuously loops around, much like a merry-go-round.

Through this model, we comprehend how electrons remain in stable orbits and don't simply collapse into the nucleus, explaining atomic structure from a classical physics standpoint. Although this model has limitations, it paved the way for modern quantum mechanics.

Period of Motion

The period of motion is the time an electron takes to complete a single revolution around the nucleus. It is a critical aspect in calculating the current, as it directly relates to how often the charge completes the orbit.

To determine the period \(T\), use:

• \( T = \frac{2\pi r}{v} \)

Here, \(2\pi r\) is the circumference of the electron's orbit, and \(v\) is its speed. By calculating \(T\), we understand the rhythm of the electron's journey around its path.

This rhythmic journey plays a crucial role in determining the magnetic moment, as it influences the derived current. Understanding this period helps in predicting how electrons contribute to atomic and physical properties.