[NOTES/EM-07017] Magnetic Moment of a System of Point Particles

It will be shown that for point charges it is easy to see that the magnetic moment is related to the
angular momentum.\\

Recall that the magnetic moment for a current distribution is given by
\begin{equation}
\vec m = \frac{1}{2} \int \vec r \times \vec J \, d^3 r
\end{equation}
Note that for a point particle the current density is
\begin{equation}
\vec{j}(\vec{r})=q_k \vec{v}_k\delta(\vec{r}-\vec{r}_k).
\end{equation}

Here the following notation has been used for \(k^\text{th}\) charged
particle.\\
\(q_k=\)charge; \(M_k=\)mass; \(\vec{v}_k=\)velocity;
\(\vec{p}_k =\) momentum; \(\vec{\ell}_k\) angular momentum.

Therefore, the magnetic moment for system of point particles is given by
\begin{equation}
\vec{m} = \sum_k \frac{q_k}{2M_k}\big\{
\vec{r}_k\times \vec{p}_k\big\} = \sum_k
\frac{q_k}{2M_k}\vec{\ell}_k.
\end{equation}