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[NOTES/EM-04017] Magnetic Moment of a System of Point Particles

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The magnetic moment for a point particle is shown to be related to the angular momentum \(\ell\) and is given by
\begin{equation}
\vec{m} =
\frac{q}{2M}\vec{\ell}
\end{equation}


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}

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