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Chapter 32: Problem 75

Suppose that \(\pm 4\) are the limits to the values of \(m_{\ell}\) for an electron in an atom. (a) How many different values of the electron's \(\mu_{\text {orb }, z}\) are possible? (b) What is the greatest magnitude of those possible values? Next, if the atom is in a magnetic field of magnitude \(0.250 \mathrm{~T}\), in the positive direction of the \(z\) axis, what are (c) the maximum energy and (d) the minimum energy associated with those possible values of \(\mu_{\text {orb }, z}\) ?

Short Answer

Expert verified
(a) 9 values, (b) magnitude is \( 4 \mu_B \); (c) max energy is \(-\mu_{\text{orb}, z}B\) for highest \(m_\ell\), (d) min energy is \(-\mu_{\text{orb}, z}B\) for lowest \(m_\ell\).

Step by step solution

01

Identify the range of possible quantum numbers

The magnetic quantum number, typically denoted as \( m_{\ell} \), can take values from \(-\ell\) to \(\ell\). According to the problem, \( m_{\ell} \) ranges from \(-4\) to \(4\).
02

Determine the number of possible values

Since \( m_{\ell} \) can be any integer from \(-4\) to \(4\), we find the total number of possible values by counting these integers. The sequence is \(-4, -3, -2, -1, 0, 1, 2, 3, 4\), which gives us 9 values.
03

Calculate possible values of orbital magnetic moment

The orbital magnetic moment along the \(z\)-axis is given by \( \mu_{\text{orb}, z} = m_{\ell} \mu_B \), where \( \mu_B \) is the Bohr magneton. Therefore, the possible orbital magnetic moments are determined by substituting each \( m_{\ell} \) value into this formula.
04

Find the greatest magnitude

The greatest magnitude of \( \mu_{\text{orb}, z} \) occurs for the largest absolute value of \( m_{\ell} \), which is 4. Thus, the greatest magnitude is \( 4 \mu_B \).
05

Calculate maximum energy

In a magnetic field \( B \) along the \(z\) axis, the energy associated with the magnetic moment is \( E = -\mu_{\text{orb}, z} B \). The maximum energy corresponds to \( m_{\ell} = 4 \), thus \( E_{ ext{max}} = -4 \mu_B \times 0.250 \mathrm{~T} \).
06

Calculate minimum energy

Similarly, the minimum energy corresponds to \( m_{\ell} = -4 \), leading to \( E_{ ext{min}} = -(-4 \mu_B) \times 0.250 \mathrm{~T} = 4 \mu_B \times 0.250 \mathrm{~T} \). This results in the lowest energy level.

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Key Concepts

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

Orbital Magnetic Moment
The orbital magnetic moment is a crucial concept in the realm of quantum physics. It relates to how electrons with angular momentum interact with magnetic fields. Specifically, the orbital magnetic moment along the z-axis can be calculated using the formula: \[ \mu_{\text{orb}, z} = m_{\ell} \mu_B \] where
  • \( m_{\ell} \) is the magnetic quantum number,
  • \( \mu_B \) represents the Bohr magneton.
The magnetic quantum number \( m_{\ell} \) ranges between \( -\ell \) to \( \ell \), and provides insight into the component of the angular momentum in the direction of the applied magnetic field. Because of this, orbital magnetic moments can vary greatly, giving rise to a wide array of energy states and levels for electrons.
Bohr Magneton
The Bohr magneton is the constant that links the angular momenta of electrons with their magnetic moments. It is the standard unit for measuring the magnetic moment of an electron as it quantifies how strong the magnetic dipole of an electron is. Mathematically, it is defined as: \[ \mu_B = \frac{e\hbar}{2m_e} \] where
  • \( e \) is the elementary charge,
  • \( \hbar \) is reduced Planck's constant,
  • \( m_e \) denotes the electron mass.
The Bohr magneton becomes a central constant when discussing the magnetic moment of electrons related to their angular momentum, making it instrumental in the calculations involving orbital magnetic moments.
Quantum Numbers
In quantum mechanics, quantum numbers are the "address" of an electron. They describe the probabilities and characteristics of an electron's position in an atom. Key types of quantum numbers include:
  • Principal Quantum Number (\( n \)): Dictates the energy level and size of the electron's orbit.
  • Orbital Angular Momentum Quantum Number (\( \ell \)): Determines the shape of the electron's cloud.
  • Magnetic Quantum Number (\( m_{\ell} \)): Defines the orientation of the orbital, as seen in the exercise, where it plays a role in calculating the orbital magnetic moment.
  • Spin Quantum Number (\( m_s \)): Defines the intrinsic spin of the electron.
Each quantum number provides a different layer of information that collectively forms a comprehensive understanding of an electron鈥檚 behavior within an atom.
Magnetic Field Energy
When an electron with a magnetic moment is placed in a magnetic field, it interacts by aligning its magnetic dipole moment with the field. This interaction changes the energy levels of the electron. This energy shift can be calculated using the formula: \[ E = -\mu_{\text{orb}, z} B \] where
  • \( E \) is the energy associated with the appearance of the magnetic field,
  • \( \mu_{\text{orb}, z} \) is the orbital magnetic moment component along the field direction,
  • \( B \) denotes the magnetic field strength.
The interaction of the magnetic moment with the field can result in either higher or lower energy states depending on the orientation of the moment with respect to the direction of the field. This underpins many phenomena in magnetic resonance and electron paramagnetic resonance spectroscopy.

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Most popular questions from this chapter

A parallel-plate capacitor with circular plates of radius \(R=16 \mathrm{~mm}\) and gap width \(d=5.0 \mathrm{~mm}\) has a uniform electric field between the plates. Starting at time \(t=0\), the potential difference between the two plates is \(V=(100 \mathrm{~V}) e^{-t / \tau}\), where the time constant \(\tau=12 \mathrm{~ms}\). At radial distance \(r=0.80 R\) from the central axis, what is the magnetic field magnitude (a) as a function of time for \(t \geq 0\) and (b) at time \(t=3 \tau\) ?

Measurements in mines and boreholes indicate that Earth's interior temperature increases with depth at the average rate of \(30 \mathrm{C}^{\circ} / \mathrm{km}\). Assuming a surface temperature of \(10^{\circ} \mathrm{C}\), at what depth does iron cease to be ferromagnetic? (The Curie temperature of iron varies very little with pressure.)

A parallel-plate capacitor with circular plates of radius \(0.10 \mathrm{~m}\) is being discharged. A circular loop of radius \(0.20 \mathrm{~m}\) is concentric with the capacitor and halfway between the plates. The displacement current through the loop is \(2.0 \mathrm{~A}\). At what rate is the electric field between the plates changing?

A magnetic rod with length \(6.00 \mathrm{~cm}\), radius \(3.00 \mathrm{~mm}\), and (uniform) magnetization \(2.70 \times 10^{3} \mathrm{~A} / \mathrm{m}\) can turn about its center like a compass needle. It is placed in a uniform magnetic field \(\vec{B}\) of magnitude \(35.0 \mathrm{mT}\), such that the directions of its dipole moment and \(\vec{B}\) make an angle of \(68.0^{\circ}\). (a) What is the magnitude of the torque on the rod due to \(\vec{B} ?\) (b) What is the change in the orientation energy of the rod if the angle changes to \(34.0^{\circ}\) ?

A magnet in the form of a cylindrical rod has a length of \(5.00 \mathrm{~cm}\) and a diameter of \(1.00 \mathrm{~cm}\). It has a uniform magnetization of \(5.30 \times 10^{3} \mathrm{~A} / \mathrm{m}\). What is its magnetic dipole moment?
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