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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?

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Whats a pure magnetic dipole? – Man May 28 '13 at 21:51
@AmanAbhishek Either an infinitely small and infinitely powerful bar magnet, or what you approximate with various configurations of spinning and resting electricly or magnetically charged spheres. – Art M May 28 '13 at 21:59
up vote 3 down vote accepted

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.

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As you approach a physical dipole, there is another term you need to consider. The field inside a physical dipole (e.g. inside a magnetized material), can be calculated using the expression (2/3)*mu_0*M, where mu_0 is the permeability of free space, and M is the magnetization per volume. See Griffiths among others for clarification. Turns out this term gives rise to atomic hyperfine structure! – burgerking Dec 13 '13 at 16:31

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.

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Respectfully disagree that there are no "pure" dipoles. There are no magnetic monopoles, but magnetic dipoles are in no way forbidden, and exist. How else would you account for particle spin? – burgerking Dec 13 '13 at 16:26

It's infinity, since the denominator goes to zero and, in the limit, from every direction, the numerator does not.

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