Force on a ferrous object inside a non-uniform magnetic field

[EnEE]

Hello everyone!

I want to know how to compute the force applied on ferrous inside a non-uniform magnetic field, to make it easier, let's assume that the field direction is constant, and the magnitude decreases linearly.

Now, I know that ferromagnetic material inside a magnetic field will become magnetized, and once he become magnetized, I can compute the induced currents on the surface and inside the matter(curl of M, and Mxn, where M is the magnetization), and therefore I can use Lorentz law to compute the force.
Is that right to do this?

I saw that usually engineers using programs like FEMM to solve such problems, and I try to figure out how to solve it analytically

simple examples:
http://uzzors2k.4hv.org/index.php?page=magneticlevitation - the force applied should be similar to this:
https://en.wikipedia.org/wiki/Force_between_magnets#Force_between_two_cylindrical_magnets

simple problem:
assume we have an air-core solenoid produces a magnetic field(instead of the iron-core in the example), I want to know what is the minimum number of turns or the minimum current I need to pull a given iron cylinder (given distance from the solenoid, and given mass, and for simplify the problem I can assume that the cylinder is right in the symetric axis of the solenoid)

 

[EnEE]

OK, few thougths...
1.my first thougth about induced currents and lorentz law will not work here
2.If I will use the magnetic potential: μ_m=0.5⋅μ⋅H⋅H
I can compute the force by F=-∇μ_m

H is function of z, so F is also function of z. If I omit ellastic forces on the iron cylinder(can I? maybe tension?)
I will get: F=-∫∇μ_m between z1 and z2(z2-z1 is the length of the cylinder)

what do you think?