Q35P - Fundamentals Of Physics

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 ${\mathit{m}}_{\mathbf{l}}\mathbf{=}\mathbf{-}\mathbf{2}\mathbf{?}$

Step-by-Step Solution

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 1: Listing the given quantities

$m_l=1$

$m_l=-2$

Step 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:

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

Step 3: (a) Calculations of the orbital magnetic dipole moment of an electron with m l = 1

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

For $m_l=1$

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

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

Step 4: (b) Calculations of theorbital magnetic dipole moment of an electron with m l = - 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 $1.9\times {10}^{-23}\mathrm{J}/\mathrm{T}$