Field At Magnetic Dipole Suppose I have a pure magnetic dipole $\mathbf{\vec m} = m\hat z$ located at the origin. What is the magnitude of the field $|\vec B|$ as $r\to 0$? In other words, what is $\lim_{r\to 0}\frac{\hat{r}\cdot \vec{p}}{4\pi\varepsilon_0r^2}$? Is it just zero? $\infty$? Do I have to use some sort of quadripole term?
 A: The magnitude of the fields would go to infinity at zero. However, dipoles are an approximation, at large distances, of the fields created by smaller object (e.g. a current loop). If you zoom closer, the B field does not diverge.
A: fffred's answer seems correct to me. As a supplement, just remember that there are no "pure" dipoles because there are no "point" sources of $\mathbf{B}$ (as far as we know), all magnetic effects are due to charge in motion; it was in fact a major breakthrough for physicists to realize this. If you would like to calculate $\mathbf{B}$ at the origin of a "pure" dipole, you would need to consider an infinitesimally small loop and yet figure how to keep the field finite. Also remember the dipole term comes from a multipole expansion in $r$, so when you cut the series at the second term (the first term vanishes, that's one reason why, mathematically, we know there are no monopoles) you are indeed assuming that $r$ is larger than the size of the loop.
A: It's infinity, since the denominator goes to zero and, in the limit, from every direction, the numerator does not.
