Question

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

Short answer

(a) The component of the orbital magnetic dipole moment of an electron with the values $m_l=3$will be role="math" localid="1663006758598" $-2.78\times {10}^{-23}\raisebox{1ex}{$\text{J}$}\!\left/ \!\raisebox{-1ex}{$\text{T}$}\right.$.

(b) Measured component of the orbital magnetic dipole moment of an electron with the values $m_l=-4$will be role="math" localid="1663006815783" $3.71\times {10}^{-23}\raisebox{1ex}{$\text{J}$}\!\left/ \!\raisebox{-1ex}{$\text{T}$}\right.$.

Step-by-step solution

Listing the given quantities:

The orbital magnetic dipole is $m_l=3$.

The orbital magnetic dipole is $m_l=4$.

Understanding the concepts of magnetic dipole moment:

You have the formula for orbital magnetic dipole moment. From the given data, you can get the required value.

Formula:

The orbital magnetic dipole moment equation is,

${\mu }_{orb}=-m_l{\mu }_B$

Here, ${\mu }_B$is the magnetic moment having a value of $9.27\times {10}^{-24}\raisebox{1ex}{$\text{J}$}\!\left/ \!\raisebox{-1ex}{$\text{T}$}\right.$.

(a) Calculations of the orbital magnetic dipole moment:

First take the orbital magnetic dipole as,

$m_l=3$

So, the orbital magnetic dipole moment will be,

$\begin{array}{rcl}{\mu }_{orb}& =& -m_l{\mu }_B\\ & =& -3\times (9.27\times {10}^{-24}\raisebox{1ex}{$\text{J}$}\!\left/ \!\raisebox{-1ex}{$\text{T}$}\right.\\ & =& -2.78\times {10}^{-23}\raisebox{1ex}{$\text{J}$}\!\left/ \!\raisebox{-1ex}{$\text{T}$}\right.\end{array}$

Hence, the component of the orbital magnetic dipole moment of an electron with the values $m_l=3$ will be $-2.78\times {10}^{-23}\raisebox{1ex}{$\text{J}$}\!\left/ \!\raisebox{-1ex}{$\text{T}$}\right.$.

(b) Calculations of the orbital magnetic dipole moment at ml=-4:

Take $m_l=-4$.

Thus, the orbital magnetic dipole moment is,

$\begin{array}{rcl}{\mu }_{orb}& =& -m_l{\mu }_B\\ & =& -(-4)\times \left(9.27\times {10}^{-24}\raisebox{1ex}{$\text{J}$}\!\left/ \!\raisebox{-1ex}{$\text{T}$}\right.\right)\\ & =& 3.71\times {10}^{-23}\raisebox{1ex}{$\text{J}$}\!\left/ \!\raisebox{-1ex}{$\text{T}$}\right.\end{array}$

Hence, the component of the orbital magnetic dipole moment of an electron with the values $m_l=-4$will be $\begin{array}{rcl}& & 3.71\times {10}^{-23}\raisebox{1ex}{$\text{J}$}\!\left/ \!\raisebox{-1ex}{$\text{T}$}\right.\end{array}$.