I interpret electric dipole moments and magnetic dipole moments as intrinsic properties of certain materials, but in Griffiths it's literally picked out of expressing the potential and magnetic vector potential using an expansion of Legendre polynomials, where the $n=1$ term picks out the dipole term. For the magnetic dipole part, for instance:

$$ \mathbf{A}_{\text{dip}}(\mathbf{r}) = \frac{\mu_0 I}{4\pi r^2}\oint r'\cos\theta' d\mathbf{l}' = \frac{\mu_0 I}{4\pi r^2}\oint(\hat{\mathbf{r}}\cdot\mathbf{r}') d\mathbf{l}'. \tag{5.81} $$ This integral can be rewritten in a more illuminating way if we invoke Eq. $1.108$ with $\mathbf{c}=\hat{\mathbf{r}}$: $$ \oint(\hat{\mathbf{r}}\cdot\mathbf{r}') d\mathbf{l}' = -\hat{\mathbf{r}}\times\int d\mathbf{a}'. \tag{5.82} $$

I don't know what Eq. $1.108$ does to prove $(5.82)$, namely because I can't actually find it in the textbook, but it seems to be the way to link my conception of a magnetic moment to its apparent appearance in the dipole term for the magnetic multipole expansion. Even if I understood how Eq. $1.108$ could represent this though, I don't think the intuition would be clear for me yet. The math clearly says that magnetic dipole moments have something to do with the magnetic vector potential, but I can't understand it behind this apparent expansion. Any light to be shed would help.


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