How to approximate electromagnet-magnet interaction

Question

I need a simple model of a small electromagnet interacting with a small magnet. The core of the problem is "If the magnet is located in this position, what about the attraction of the electromagnet over the magnet?"

What about the dipole-dipole interaction? If suitable, the overall magnetic moment could be given by $\vec{m}= \vec{m_{coil}} + \vec{m_{core}}$, where $\vec{m_{coil}}= Ni\vec{S}$ (i = current, N = number of winding, $|\vec{S}|$ = area of a single loop). Is it correct?

If the dipole - dipole approximation is not suitable, what else?

I cannot figure out anything, I'm looking at datasheets of electromagnets but I cannot find anything useful. The only things that are reported consist in: holding force, length and diameter. However, I guess that the latter two quantities are related to the case dimensions, hence no hints about the internal structure. No help from the former quantity, whose formula depends on the structure of the electromagnet....

Answer 1

If you are interested in the static problem, you could use the following steps:

1. Use Biot-Savart's law to calculate the magnetic field $\vec B(x,y,z)$. Use a grid to obtain the field at different positions.
2. The energy of the "small" magnet is given by $E = -\vec \mu \cdot \vec B$.
3. Take the "derivative" (gradient) to obtain the force $\vec F$.

For certain configurations there are exact formulas. Hence, you should probably cross-check your numerics using these special configurations.