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Chapter 7: Problem 33

Magnetic Moment of the Hydrogen Atom. In the Bohr model of the hydrogen atom (see Section \(8.5\) ), 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?

Short Answer

Expert verified
(a) \( 1.52 \times 10^{-16} \text{ s} \); (b) \( 1.05 \times 10^{-3} \text{ A} \); (c) \( 9.25 \times 10^{-24} \text{ A} \cdot \text{m}^2 \).

Step by step solution

01

Calculate the Orbital Period

First, we need to determine the time it takes for the electron to complete one full orbit around the nucleus. The formula for the orbital period \( T \) is \( T = \frac{2\pi r}{v} \), where \( r \) is the radius of the orbit and \( v \) is the speed of the electron. Substituting the given values: \( r = 5.3 \times 10^{-11} \) m and \( v = 2.2 \times 10^6 \) m/s, we calculate:\[ T = \frac{2\pi \times 5.3 \times 10^{-11} \text{ m}}{2.2 \times 10^6 \text{ m/s}} = 1.52 \times 10^{-16} \text{ s} \]
02

Determine the Current

Next, we need to find the equivalent current \( I \) for this electron orbiting like a current loop. The formula for current is \( I = \frac{e}{T} \), where \( e \) is the charge of the electron \( 1.6 \times 10^{-19} \text{ C} \). Using the period calculated in Step 1:\[ I = \frac{1.6 \times 10^{-19} \text{ C}}{1.52 \times 10^{-16} \text{ s}} \approx 1.05 \times 10^{-3} \text{ A} \]
03

Calculate the Magnetic Moment

Finally, the magnetic moment \( \mu \) of the atom can be calculated using the formula \( \mu = I \cdot A \), where \( A \) is the area of the orbit. Since the electron's orbit is circular, \( A = \pi r^2 \):\[ A = \pi (5.3 \times 10^{-11} \text{ m})^2 = 8.81 \times 10^{-21} \text{ m}^2 \]Now, substitute the values of \( I \) and \( A \):\[ \mu = 1.05 \times 10^{-3} \text{ A} \times 8.81 \times 10^{-21} \text{ m}^2 = 9.25 \times 10^{-24} \text{ A} \cdot \text{m}^2 \]
04

Conclusion

We have calculated the orbital period, current, and magnetic moment as follows: the orbital period is \( 1.52 \times 10^{-16} \text{ s} \), the current is \( 1.05 \times 10^{-3} \text{ A} \), and the magnetic moment is \( 9.25 \times 10^{-24} \text{ A} \cdot \text{m}^2 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Magnetic Moment
In the Bohr model of the hydrogen atom, the magnetic moment is a fundamental concept, particularly when considering the electron as a current loop. The magnetic moment, denoted as \( \mu \), arises due to the movement of an electron in its circular orbit. It essentially describes the magnetic strength and orientation of the atom's magnetic field. The formula to calculate the magnetic moment is \( \mu = I \cdot A \), where \( I \) is the current and \( A \) is the area of the electron's orbit.
  • Current \( I \) is found using the charge of an electron \( e = 1.6 \times 10^{-19} \text{ C} \) divided by the orbital period \( T \).
  • The area \( A \) is calculated using the formula for the area of a circle, \( A = \pi r^2 \), where \( r \) is the radius of the electron's orbit.
Understanding the magnetic moment is crucial for explaining magnetic properties at an atomic level, which ties into broader magnetic phenomena observed in various materials.
Hydrogen Atom
The Bohr model of the hydrogen atom provides a simple yet powerful way to understand the behavior of electrons around a nucleus. This model portrays the atom as having a central nucleus with electrons orbiting around in defined circular paths, much like planets around a sun. This model is especially useful for discussing elements with just one electron, like hydrogen.
  • The hydrogen atom consists of one proton and one electron.
  • The electron in the lowest energy state orbits the proton in a specific path dictated by quantized angular momentum.
  • This simplified view captures the essence of electron orbits but doesn't account for electron clouds derived from quantum mechanics. However, it gives accurate predictions for hydrogen spectra.
Despite its simplicity, the model is foundational for understanding more complex atomic structures and interactions.
Electron Orbit
In atomic models, an electron orbit refers to the path an electron takes around the nucleus of an atom. The Bohr model describes electrons orbiting in stable orbits without radiating energy, which was a significant departure from earlier electromagnetic views.
  • Orbits are defined by specific radii, with electrons only occupying certain allowed paths.
  • The speed of the electron and the radius of its orbit determine its orbital period, which is the time taken for a complete revolution.
  • Quantization in Bohr鈥檚 model implies that only certain discrete orbits are permitted, explaining the spectral lines observed in hydrogen.
This concept is essential for understanding atomic emissions and absorptions, which are tied to electron transitions between these allowed orbits.
Current Loop
A fascinating aspect of the Bohr model is treating an electron orbit as a current loop. By analogy, when an electron moves in its orbit, it creates a loop of electrical current due to its negative charge and continuous motion. In this view:
  • The moving electron is akin to a tiny current, defined as \( I = \frac{e}{T} \), where \( T \) is the orbital period.
  • Such a current loop generates a magnetic field, analogous to a tiny magnet or coil.
  • This approach provides insights into the magnetic properties of atoms and how they might interact with external magnetic fields.
This analogy enriches our understanding of atomic magnetism, enabling a deeper examination of phenomena such as electron magnetic moments and their role in atomic spectra.

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Most popular questions from this chapter

An electron moves at \(2.50 \times 10^{6} \mathrm{~m} / \mathrm{s}\) through a region in which there is a magnetic field of unspecified direction and magnitude \(7.40 \times 10^{-2} \mathrm{~T}\). (a) What are the largest and smallest possible magnitudes of the acceleration of the electron due to the magnetic field? (b) If the actual acceleration of the electron is one-fourth of the largest magnitude in part (a), what is the angle between the electron velocity and the magnetic field?

You wish to hit a target from several meters away with a charged coin having a mass of \(4.25 \mathrm{~g}\) and a charge of \(+2500 \mu \mathrm{C}\). The coin is given an initial velocity of \(12.8 \mathrm{~m} / \mathrm{s}\), and a downward, uniform electric field with field strength \(27.5 \mathrm{~N} / \mathrm{C}\) exists throughout the region. If you aim directly at the target and fire the coin horizontally, what magnitude and direction of uniform magnetic field are needed in the region for the coin to hit the target? (take \(\left.g=9.8 \mathrm{~m} / \mathrm{s}^{2}\right)\)

An alpha particle (a He nucleus, containing two protons and two neutrons and having a mass of \(6.64 \times 10^{-27} \mathrm{~kg}\) ) traveling horizontally at \(35.6 \mathrm{~km} / \mathrm{s}\) enters a uniform, vertical, \(1.10-\mathrm{T}\) magnetic field. (a) What is the diameter of the path followed by this alpha particle? (b) What effect does the magnetic field have on the speed of the particle? (c) What are the magnitude and direction of the acceleration of the alpha particle while it is in the magnetic field? (d) Explain why the speed of the particle does not change even though an unbalanced external force acts on it.

In designing a velocity selector that uses uniform perpendicular electric and magnetic fields, you want to select positive ions of charge \(+5 e\) that are traveling perpendicular to the fields at \(8.75 \mathrm{~km} / \mathrm{s}\). The magnetic field available to you has a magnitude of \(0.550 \mathrm{~T}\). (a) What magnitude of electric field do you need? (b) Show how the two fields should be oriented relative to each other and to the velocity of the ions. (c) Will your velocity selector also allow the following ions (having the same velocity as the \(+5 e\) ions) to pass through undeflected: (i) negative ions of charge \(-5 e\), (ii) positive ions of charge different from \(+5 e ?\)

A circular loop of wire with area \(A\) lies in the xy-plane. As viewed along the z-axis looking in the \(-z\)-direction toward the origin, a current \(I\) is circulating clockwise around the loop. The torque produced by an external magnetic field \(\overrightarrow{\boldsymbol{B}}\) is given by \(\vec{\tau}=D(4 \hat{i}-3 \hat{J})\), where \(D\) is a positive constant, and for this orientation of the loop the magnetic potential energy \(U=-\overrightarrow{\boldsymbol{\mu}} \cdot \overrightarrow{\boldsymbol{B}}\) is negative. The magnitude of the magnetic field is \(B_{0}=13 D / I A\). (a) Determine the vector magnetic moment of the current loop. (b) Determine the components \(B_{x}, B_{y}\), and \(B_{z}\) of \(\overrightarrow{\boldsymbol{B}} . a^{x}\)
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