Question

(a) What is the measured component of the orbital magnetic dipole moment of an electron with${\mathit{m}}_{\mathbf{l}}\mathbf{=}\mathbf{1}\mathbf{?}$(b) What is the measured component of the orbital magnetic dipole moment of an electron with${\mathit{m}}_{\mathbf{l}}\mathbf{=}\mathbf{-}\mathbf{2}\mathbf{?}$

Short answer

(a) The orbital magnetic dipole moment of an electron with$m_l=1$ is$-9.3\times {10}^{-24}\mathrm{J}/\mathrm{T}$

(b) The orbital magnetic dipole moment of an electron with$m_l=-2$ is$1.9\times {10}^{-23}\mathrm{J}/\mathrm{T}$

Step-by-step solution

Listing the given quantities

$m_l=1$

$m_l=-2$

Understanding the concepts of orbital angular momentum

An electron in an atom has an orbital angular momentum, and for this, we can

associate an orbital magnetic dipole moment. Orbital angular momentum is quantized. By substituting the value of$m_l$, we can find the corresponding component of the orbital magnetic dipole moment.

Formula:

${{\mu }_{orb}}_z=-{\mu }_{B}m$

(a) Calculations of the orbital magnetic dipole moment of an electron with ml=1

${{\mu }_{orb}}_z=-{\mu }_B{m}_l$

For $m_l=1$

role="math" localid="1663153694077" $\begin{array}{rcl}{{\mu }_{orb}}_z& =& -{\mu }_B{m}_l\\ & =& -{\mu }_B\\ & =& -9.3\times {10}^{-24}\mathrm{J}/\mathrm{T}\end{array}$

The orbital magnetic dipole moment of an electron with $m_l=1$is$-9.3\times {10}^{-24}\mathrm{J}/\mathrm{T}$

(b) Calculations of theorbital magnetic dipole moment of an electron with ml=-2

${{\mu }_{orb}}_z=-{\mu }_B{m}_l$

For $m_l=-2$

$\begin{array}{rcl}{{\mu }_{orb}}_z& =& -{\mu }_B{m}_l\\ & =& +2{\mu }_B\\ & =& 2\times 9.27\times {10}^{-24}\mathrm{J}/\mathrm{T}\\ & =& 1.9\times {10}^{-23}\text{J/T}\end{array}$

The orbital magnetic dipole moment of an electron with $m_l=-2$is$1.9\times {10}^{-23}\mathrm{J}/\mathrm{T}$