Question

What are the measured components of the orbital magnetic dipole moment of an electron with (a) ${\mathit{m}}_{\mathbf{l}}\mathbf{=}\mathbf{3}$ and (b) ${\mathit{m}}_{\mathbf{l}}\mathbf{=}\mathbf{-}\mathbf{4}$?

Short answer

(a) The measured components of the orbital magnetic dipole moment of an electron with $m_l=3$is role="math" localid="1663055992290" ${\mu }_{orb,z}=-2.78\times {10}^{-23}\raisebox{1ex}{$\text{J}$}\!\left/ \!\raisebox{-1ex}{$\text{T}$}\right.$.

(b) The measured components of the orbital magnetic dipole moment of an electron with $m_l=-4$is ${\mu }_{orb,z}=+3.71\times {10}^{-23}\raisebox{1ex}{$\text{J}$}\!\left/ \!\raisebox{-1ex}{$\text{T}$}\right.$.

Step-by-step solution

Listing the given quantities:

The magnetic quantum number is $m_l=3$.

The magnetic quantum number is $m_l=-4$.

Understanding the concepts of magnetic dipole moment:

An electron, along with revolving in its orbit, spins as well along its axis. Thus, it carries angular momentum of its own; this results in the spin magnetic moment of the electron. Its component along the z-axis is given as-

${\mathbf{\mu }}_{\mathrm{orb},z}\mathbf{=}\mathbf{-}{\mathbf{m}}_l\frac{\mathrm{eh}}{4\mathrm{\pi m}}$ ..... (1)

Here,

The charge of an electron, role="math" localid="1663056261439" $\mathrm{e}=1.6\times {10}^{-19}\text{C}$

Plank’s constant, $\mathrm{h}=6.63\times {10}^{-34}\text{J.s}$

The mass of the electron,$\mathrm{m}=9.1\times {10}^{-31}\mathrm{kg}$

(a) Calculations of the measured components of the orbital magnetic dipole moment of an electron with ml=3:

Substitute known values into equation (1), and you get,

$\begin{array}{rcl}{\mu }_{orb,z}& =& -3\times \frac{(1.6\times {10}^{-19}\mathrm{C})\times (6.63\times {10}^{-34}\mathrm{J}.\mathrm{s})}{4\times 3.14\times (9.1\times {10}^{-31}\mathrm{kg})}\\ & =& -2.78\times {10}^{-23}\raisebox{1ex}{$\mathrm{J}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{T}$}\right.\end{array}$

Hence, the measured components of the orbital magnetic dipole moment of an electron with $m_l=3$is ${\mu }_{orb,z}=-2.78\times {10}^{-23}\raisebox{1ex}{$\mathrm{J}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{T}$}\right.$.

(b) Calculations of the measured components of the orbital magnetic dipole moment of an electron with ml=-4:

Putting known values into equation (1), and you have

$\begin{array}{rcl}{\mu }_{orb,z}& =& -(-4)\times \frac{(1.6\times {10}^{-19}\mathrm{C})\times (6.63\times {10}^{-34}\mathrm{J}.\mathrm{s})}{4\times 3.14\times (9.1\times {10}^{-31}\mathrm{kg})}\\ & =& +3.71\times {10}^{-23}\raisebox{1ex}{$\mathrm{J}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{T}$}\right.\end{array}$

Hence, the measured components of the orbital magnetic dipole moment of an electron with $m_l=-4$is .$\begin{array}{rcl}& & +3.71\times {10}^{-23}\raisebox{1ex}{$\mathrm{J}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{T}$}\right.\end{array}$.