Problem 54 - University Physics with Modern Physics

Question

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?

Step-by-Step Solution

Answer

(a) 1.52 × 10⁻¹⁶ s; (b) 1.05 × 10⁻³ A; (c) 9.27 × 10⁻²⁴ J/T.

Step 1: Understanding Orbital Period

The orbital period of the electron is the time it takes for the electron to complete one full orbit around the proton. This is given by the formula \( T = \frac{2\pi r}{v} \), where \( r \) is the radius of the orbit and \( v \) is the speed of the electron. We have \( r = 5.3 \times 10^{-11} \text{ m} \) and \( v = 2.2 \times 10^6 \text{ m/s} \).

Step 2: Calculate Orbital Period

Substitute the values into the formula:\[T = \frac{2\pi \times 5.3 \times 10^{-11} \text{ m}}{2.2 \times 10^6 \text{ m/s}}.\] Calculate to find:\[T \approx 1.52 \times 10^{-16} \text{ seconds}.\]

Step 3: Understanding Current in the Orbit

If the orbiting electron is considered as a loop of current, its current \( I \) is given by \( I = \frac{e}{T} \), where \( e \) is the elementary charge \( 1.6 \times 10^{-19} \text{ C} \) and \( T \) is the period of the electron's orbit. Use the value of \( T \) from the previous step.

Step 4: Calculate the Current

Substitute the values into the current formula:\[I = \frac{1.6 \times 10^{-19} \text{ C}}{1.52 \times 10^{-16} \text{ s}}\]Calculate to find:\[I \approx 1.05 \times 10^{-3} \text{ A}.\]

Step 5: Understanding Magnetic Moment

The magnetic moment \( \mu \) due to the orbit of the electron is given by the formula \( \mu = I \cdot A \), where \( A = \pi r^2 \) is the area of the orbit. Use the previous result for \( I \) and \( r = 5.3 \times 10^{-11} \text{ m} \).

Step 6: Calculate Magnetic Moment

Substitute the values into the magnetic moment formula:\[A = \pi \cdot (5.3 \times 10^{-11})^2\]\[\mu = 1.05 \times 10^{-3} \cdot \pi \cdot (5.3 \times 10^{-11})^2\]Calculate to find:\[\mu \approx 9.27 \times 10^{-24} \text{ J/T}.\]

Key Concepts

Orbital Period of an Electron

The concept of the orbital period in the context of the Bohr model revolves around understanding the time it takes for an electron to complete one full orbit around the nucleus. This analogy is similar to the time Earth takes to revolve around the Sun in its orbit. In the Bohr model, the electron orbits the nucleus at a constant speed, making calculations straightforward. This duration is expressed by the formula:

• \( T = \frac{2\pi r}{v} \)

where:

• \( r \) is the radius of the orbit
• \( v \) is the velocity of the electron

For example, with a radius of \( r = 5.3 \times 10^{-11} \) meters and a speed of \( v = 2.2 \times 10^6 \) meters per second, the calculation becomes simple.
The result is approximately \( 1.52 \times 10^{-16} \) seconds, showcasing the incredibly brief time an electron takes to complete its journey around the nucleus.

Current in Electron Orbit

In the Bohr model, the orbiting electron can be thought of as creating a tiny loop of current due to its constant motion around the nucleus. Although it might sound abstract, this approximation is essential for understanding magnetic properties in atomic structures. The current \( I \) generated by this motion is quantified by:

• \( I = \frac{e}{T} \)

where:

• \( e \) is the elementary charge, valued at \( 1.6 \times 10^{-19} \) Coulombs
• \( T \) is the orbital period just calculated

Using the orbital period from earlier, the current is calculated as approximately \( 1.05 \times 10^{-3} \) Amperes.
By conceptualizing the electron's motion as a current loop, we get a foundational understanding of how electrons contribute to magnetic fields in atomic-scale systems.

Magnetic Moment of an Atom

The magnetic moment within the Bohr model is a key concept in grasping how atoms respond to magnetic fields. This moment arises due to the orbiting electron's motion, similar to how a current loop generates a magnetic field. The magnetic moment \( \mu \) is calculated by the formula:

• \( \mu = I \cdot A \)

where:

• \( I \) is the current from the electron's orbit
• \( A \) is the area of the orbit, \( A = \pi r^2 \)

Utilizing the radius \( r = 5.3 \times 10^{-11} \) meters, this calculation yields a magnetic moment of roughly \( 9.27 \times 10^{-24} \) Joules per Tesla.
This value aids in explaining how each electron within an atom contributes to magnetic phenomena, crucial for understanding magnetism at the atomic level.