User:Glover/Fuckin magnets-how they work

Magnetic dipole-dipole interaction

The magnetic field of a magnetic dipole in vector notation is:

${\displaystyle \mathbf {B} (\mathbf {m} ,\mathbf {r} )={\frac {\mu _{0}}{4\pi r^{3}}}\left(3(\mathbf {m} \cdot {\hat {\mathbf {r} }}){\hat {\mathbf {r} }}-\mathbf {m} \right)+{\frac {2\mu _{0}}{3}}\mathbf {m} \delta ^{3}(\mathbf {r} )}$

where

B is the field

r is the vector from the position of the dipole to the position where the field is being measured

r is the absolute value of r: the distance from the dipole

${\displaystyle {\hat {\mathbf {r} }}=\mathbf {r} /r}$ is the unit vector parallel to r;

m is the (vector) dipole moment

μ₀ is the permeability of free space

δ³ is the three-dimensional delta function.δ³(r) = 0 except at r = (0,0,0), so this term is ignored in multipole expansion.

This is exactly the field of a point dipole, exactly the dipole term in the multipole expansion of an arbitrary field, and approximately the field of any dipole-like configuration at large distances.

Also this.