# [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

\vec{m} =
\frac{q}{2M}\vec{\ell}

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

\vec m = \frac{1}{2} \int \vec r \times \vec J \, d^3 r

Note that for a point particle the current density is

\vec{j}(\vec{r})=q_k \vec{v}_k\delta(\vec{r}-\vec{r}_k).

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

\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.

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