Magnetic Moment

Once a charged particle has angular momentum, it has a magnetic moment. We first define the magnetic moment as the product of the current strength and the area vector $\vec{A}$, which is perpendicular to the area enclosed by the current: \begin{equation} \vec{\mu} = I\vec{A} \end{equation} If one now considers an electron that orbits around an atomic nucleus with the radius $R$ and the velocity $v$ according to the classic atomic model, then an electric current is generated that can be calculated as follows: \begin{equation} I = \frac{e}{t} = e f \end{equation} Inserting this into the definition of the magnetic moment results from the circular area $A=\pi R^2$ and the relationship $\omega = 2\pi f$ \begin{equation} \vec{\mu} = ef \vec{A} = \frac{1}{2}eR^2\vec{\omega } \end{equation} We want to find now a connection between the magnetic moment and the angular moment. For that purpose, we note down the angular momentum derived from its definition of a circular path \begin{equation} \vec{L} = m_e\left(\vec{R}\times \vec{v}\right) = m_e R^2 \vec{\omega} \end{equation} We can see immediately that both quantities, the angular momentum and magnetic moment, point into the direction $\omega$. We can therefore get the correlation between the magnetic moment and angular momentum, following from both of these equations \begin{equation} \vec{\mu} = \frac{e}{2m_e}\vec{L} \end{equation}