I want to calculate the force between two magnets. In a previous Phys.SE question, I was told that I needed to use the dipole-dipole interaction, but that equation depends on $m$, the magnetic dipole moment of the magnet.
How do I calculate $m$? I found a source that said $m = p \ell$ where $p$ is the "magnetic pole strength" and $\ell$ is the length of the magnet, but I haven't found anything that says how to compute $p$.
A bar magnet is not a pure magnetic dipole, but we can calculate the dipole moment by comparing the magnetic fields of a bar magnet and a magnetic dipole at large distances where contributions from higher order moments become negligible.
The magnetic dipole field a distance $z$ from a dipole $m$ along the dipole axis is: $$B=\frac{\mu_0 m}{2 \pi z^3}\tag{1}$$
For a cylindrical bar magnet, the magnetic field a distance $z$ from the centre along the symmetry axis is
$$B=\frac{B_r}{2}\left (\frac{z+L/2}{\sqrt{{R}^2+{(z+L/2)}^2}} -\frac{z-L/2}{\sqrt{R^2+(z-L/2)^2}}\right )\tag{2}$$
where $B_r$ is the remanence, and $L$ & $R$ are the length and radius of the magnet. (This is just a slight change of variables from the field given in the answer to "How strong is the magnetic field from a neodymium magnet?".)
At large distances from the magnet
$$\lim_{z\gg L,R} B=\frac{B_r R^2 L}{2z^3} \tag{3}$$
So the magnetic moment of a cylindrical bar magnet is
$$m=B_r\frac{AL}{\mu_0}\tag{4}$$
where $A=\pi R^2$ is the cross-sectional area of the bar magnet.
The remanence is not directly measurable, but can be calculated (using Eq. 2) from the magnetic field strength $B_{pole}$ on the magnet axis at the pole surface (i.e. at$z=L/2$):
$$B_r=2B_{pole}\sqrt{1+\left(\frac{R}{L}\right)^2} \tag{5}$$
which gives
$$m=2B_{pole}\frac{A}{\mu_0}\sqrt{L^2+R^2}\tag{6}$$
It is important to remember that although this may be the dipole moment of a bar magnet, a bar magnet has non-zero higher order moments whose contributions to the field fall off faster than the dipole $1/r^3$. The magnetic field of a bar magnet only matches a dipole field for distances much greater the dimensions of the magnet.
Many cell phones have the capability to measure magnetic field. If you hold the device ONE METER away from the magnet center along the axis of interest, the reading will be numerically equal to the magnetic moment in A-m^2 or at least calculable with the appropriate conversion.
