Net magnetization with opposing magnetic fields?

[PhiowPhi]

Given this diagram:

Two wires, having parallel and opposing current flows( Where ## I_1 = I_2 ##), creating opposing and equal magnetic fields near a ferromagnetic object(##F##), since the magnetic field of each wire is not uniform, I'm not really certain how to figure out the magnetization of the material, how would I work it out?
If the fields we're uniform I could say that it's zero, since they cancel out(not sure).

 

[Charles Link]

The problem of ferromagnetic samples in a uniform magnetic field such as inside a current carrying solenoid is already reasonably complex, but can be determined by looking at the (measured) hysteresis curve for the particular ferromagnetic material. The typical hysteresis curve for a ferromagnetic material displays considerable non-linear behavior, and it would be difficult to assess how the sample would respond to the uniform magnetic field without such a graph. The scenario that you have with a very non-uniform applied magnetic field created by the electrical current in two wires is far more complicated, and it would be difficult to make any good predictions on how the system might respond. Perhaps other readers will have a better answer, but this is my best assessment.

 

[PhiowPhi]

From the hysteresis curve

It's quite easy for me to "assume" that the net-magnetization would be zero, if there are two uniformly opposing magnetic fields(of equal strength). However, in this case... i think the right side of the ferromagnet(and it's domains) will magnetize in the direction of the blue field lines, the same would happen for the left side magnetizing in the direction of the red field lines. While the center will be upwards? But that's just a simple assumption.