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Chapter 32: Problem 35

What is the measured component of the orbital magnetic dipole moment of an electron with (a) \(m_{\ell}=1\) and (b) \(m_{\ell}=-2 ?\)

Short Answer

Expert verified
(a) \(-\mu_B\), (b) \(2\mu_B\)

Step by step solution

01

Understanding Orbital Magnetic Dipole Moment

The orbital magnetic dipole moment of an electron is associated with the motion of the electron around the nucleus. It depends on the quantum number \( m_{\ell} \), which is the magnetic quantum number.
02

Formula for Orbital Magnetic Dipole Moment

The orbital magnetic dipole moment \( \mu_{orb} \) is given by \( \mu_{orb} = -\mu_B m_{\ell} \), where \( \mu_B \) is the Bohr magneton.
03

Calculate for \( m_{\ell} = 1 \)

For \( m_{\ell} = 1 \), the orbital magnetic dipole moment is calculated as \( \mu_{orb} = -\mu_B \times 1 = -\mu_B \). Thus, the measured component is \(-\mu_B\).
04

Calculate for \( m_{\ell} = -2 \)

For \( m_{\ell} = -2 \), the orbital magnetic dipole moment is calculated as \( \mu_{orb} = -\mu_B \times (-2) = 2\mu_B \). Thus, the measured component is \(2\mu_B\).

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Key Concepts

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

Quantum Number
Quantum numbers are like an address system for electrons in an atom. They tell us where an electron is likely to be found and give us information about the electron's energy and other properties. There are four types of quantum numbers: principal ( ), azimuthal ( ), magnetic ( ), and spin ( ).
  • The principal quantum number ( ) indicates the energy level or shell of an electron. It's like the floor of a building where the electron resides.
  • The azimuthal quantum number ( ) describes the shape of the orbital, essentially the paths the electrons travel. Think of it like different rooms on the same floor.
  • Most important for this discussion is the magnetic quantum number ( ), which determines the orientation of the orbitals in space. It is dependent on the azimuthal quantum number and can have values from 鈥� to + .
  • Lastly, the spin quantum number ( ) indicates the two possible spin states of an electron, usually written as +1/2 or -1/2.
Understanding these helps in determining the various possible states an electron may be in within an atom.
Bohr Magneton
The Bohr magneton is a physical constant and serves as the natural unit of atomic magnetic moments. It represents the magnetic moment that's intrinsic to an electron due to its angular momentum.
  • The formula for the Bohr magneton is \( \mu_B = \frac{e \hbar}{2m_e} \), where \( e \) is the elementary charge, \( \hbar \) (read as "h-bar") is the reduced Planck's constant, and \( m_e \) is the electron mass.
  • Essentially, the Bohr magneton quantifies how much an electron behaves like a tiny magnet due to its orbit around the nucleus.
  • This concept is crucial as it gives us a scale to measure the magnetic properties of electrons in atoms, especially when we consider the orbital magnetic dipole moment.
Thus, knowing the value of the Bohr magneton allows us to calculate the magnetic effects of electrons, such as the results of the calculated orbital magnetic dipole moments.
Magnetic Quantum Number
The magnetic quantum number, denoted as \( m_{\ell} \), is crucial in determining the specific orientation of an orbital within a subshell. It plays a significant role in defining 3D space for electron movement.
  • This number reflects the number of orbitals and their orientation within a given subshell, varying from \(- \ell \) to \(+ \ell \), where \( \ell \) is the azimuthal quantum number.
  • For example, if \( \ell = 1 \), \( m_{\ell} \) can be -1, 0, or +1, indicating three orientations for an orbital found within a p-subshell.
  • It is particularly important because it directly influences the orbital magnetic dipole moment, as seen in the formula \( \mu_{orb} = -\mu_B m_{\ell} \). This means the \( m_{\ell} \) value ascertains the strength and direction of the magnetic moment.
  • In practical terms, knowing the \( m_{\ell} \) lets us determine measurable magnetic properties of an electron in a given atomic state.
Understanding the magnetic quantum number helps provide insights into the quantized nature of atomic magnetic phenomena and is a key factor when analyzing electron behavior in magnetic fields.

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Most popular questions from this chapter

The magnetic flux through each of five faces of a die (singular of "dice") is given by \(\Phi_{B}=\pm N\) Wb, where \(N(=1\) to 5 ) is the number of spots on the face. The flux is positive (outward) for \(N\) even and negative (inward) for \(N\) odd. What is the flux through the sixth face of the die?

A parallel plate capacitor has square plates of edge length \(L=1.0 \mathrm{~m}\). A current of \(2.0 \mathrm{~A}\) charges the capacitor, producing a uniform electric field \(\vec{E}\) between the plates, with \(\vec{E}\) perpendicular to the plates. (a) What is the displacement current \(i_{d}\) through the region between the plates? (b) What is \(d E / d t\) in this region? (c) What is the displacement current encircled by the square dashed path of edge length \(d=0.50 \mathrm{~m} ?\) (d) What is the value of \(\oint \vec{B} \cdot d \vec{s}\) around this square dashed path?

A parallel-plate capacitor with circular plates is being charged. Consider a circular loop centered on the central axis and located between the plates. If the loop radius of \(3.00 \mathrm{~cm}\) is greater than the plate radius, what is the displacement current between the plates when the magnetic field along the loop has magnitude \(2.00 \mu \mathrm{T} ?\)

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 \(34.0^{\circ} ?\)

If an electron in an atom has an orbital angular momentum with \(m=0,\) what are the components \(\left(\right.\) a) \(L_{\text {orb }, z}\) and (b) \(\mu_{\text {orb }, z} ?\) If the atom is in an external magnetic field \(\vec{B}\) that has magnitude \(35 \mathrm{mT}\) and is directed along the \(z\) axis, what are (c) the energy \(U_{\text {orb }}\) associated with \(\vec{\mu}_{\text {orb }}\) and (d) the energy \(U_{\text {spin }}\) associated with \(\vec{\mu}_{s} ?\) If, instead, the electron has \(m=-3,\) what are (e) \(L_{\text {orb }, z}\), (f) \(\mu_{\text {orb }, z},(\mathrm{~g}) U_{\text {orb }},\) and \((\mathrm{h}) U_{\mathrm{spin}} ?\)
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