Summary

Following the good old tradition to build something between the years I decided to make a little test setup to explore permanent magnet-coil interaction.
The first picture shows the setup.
The 3d printed contraption consists of a lever that contains on one side of the fulcrum one or more stacked strong permanent magnets which interact with a coil, as well as a container for testweights. On the other side is a counterweight, so that the lever is horizontally balanced. If a current is then applied to the coil the magnets get attracted to the coil and weight can be added to the container. The weight at which the lever tips over is equivalent to the maximum vertical force between magnets and coil.
Since the distance between fulcrum and testweight is twice that of fulcrum and magnet the testweight has to be doubled to give the actual holding force.

Two coils have been made. One is embedded in a material with permeability mu_r of 1 the other one in a material that contains about 25% iron powder by volume. The magnets are 18mm in diameter and 3mm in height.
What do you think the amount of holding force over applied current looks like for both coils? Do you think it's enough to hold a chocolate bar, which is equivalent to 1N (where I live).

The second picture give the empirical data.
For the second coil there is quite a significant cogging force that has been reduced from the measured data in order to give better comparison between the curves. I kind of expected the curves for both coils to be about the same, for the same reason that iron filled filament as magnetic core material doesn't improve much. On the other hand I didn't expect the height of the magnet has such an influence.
I think there is no analytic formula to approximate the force (correct me if I'm wrong) so I went out and tried to simulate the problem. It is an intrinsic three dimensional problem but in order to keep it simple I made a two dimensional simplification. If you look at the problem sideways it seems that it could be possible to get an approximate result with a planar setup as depicted. The program that I used is a really good two dimensional, free simulation program by David Meeker called FEMM. The lua script for the simulation is attached.
Unfortunately the simulation results are always too high by a factor of 2, I expected them rather to be lower than the real values because the front and back parts of the coil are missing in this simplification.