Electromagnet force on object at some distance

[Machinia]

I've searched high and low and come up with a lot of "maybes" and "could be's" and "try this" to a problem which I feel should have at least a few example calculations floating around the internet. The questions is trivial but it seems like the solution is fairly complicated. I'm not looking for an exact solution as one probably doesn't exist, but at least an approximation would be nice.

Say I have an electromagnet and some distance away from it along its axis is a highly magnetically permeable material like iron. What is the force exerted on that object by the electromagnet?

 

[scottdave]

This may not be what you are looking for, but maybe a start. I found this as one of the results for solenoid force.
http://www.daycounter.com/Calculators/Magnets/Solenoid-Force-Calculator.phtml

 

[Machinia]

Thanks for the link but I did come across that while I was searching. The problem with that formula is it's for a closed magnetic circuit meaning the magnetic flux stays within a core material except for a small airgap. With my problem, the flux is only inside the electromagnet's core, and leaves the north pole and travels through the air back to the south pole. My example electromagnet below. The object it's applying force to would be some distance from either pole.

 

[scottdave]

OK, I misunderstood your meaning. Applications, such as yours, most likely have a wide range of variables to consider.
One thing is for sure: the magnetic field density (B) is inversely proportional to the square of the distance; see Biot-Savart Law: http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/Biosav.html#c1
The attractive force is proportional to the B-field. So you may be able to make some findings from a known piece of metal, at closer distances, then make some predictions about behavior at further distances.

 

[Machinia]

Alright so first case, let's say best case scenario. Tho object I'm lifting is the same material and diameter as the core, and is a fraction of the length of the core. In which case the force exerted on it at 0 distance would be:
$$F = \frac{B^2 A}{2 \mu_0}$$

This is the formula Wikipedia provides for the force on the core of a solenoid.

If the force is proportional to the inverse squared of the distance, then the graph would look something like:

I just need to turn that curve into a usable formula.Edit:
Thanks to you I have been finding more useful information. It looks like the field strength for a monopole drops at at a squared rate, but for a dipole like an electromagnet or magnet, it drops off at a cubed rate.