I've found the following equation in this source: $$B_\text{N,S} = \frac{\mu m}{4\pi r_\text{N,S}^2}$$ Where $r$ is the distance of "one pole"? I do not get how this can be well defined. The effective field is then something like $$ B_{\text{eff}} = B_\text{N} - B_\text{S} .$$ Where $\text{S}$ is the south pole and $\text{N}$ the north pole.
I thought that the equation magnetic dipole would be the starting point for the derivation of the above formula - but I dont see how this formula can be transformed: $$\vec{B} = \frac{\mu_0}{4\pi}\left(\frac{3\vec{r}(\vec{m}\vec{r})}{r^5} - \frac{\vec{m}}{r^3}\right)$$ The magnitude $B$ is $$B = \frac{\mu_0}{4\pi}\left(\frac{3m\cos(\alpha)}{r^3} - \frac{m}{r^3}\right).$$ Are $B_\text{eff}$ and $B$ equivalent?
