Magnetic Force between two solenoids

[Xynon]

Consider two thin and long, single layered identical solenoids with same DC in them, positioned coaxially. I'm trying to calculate the magnetic force on each solenoid, when there are identical magnetic cores inside them.

As seen in below, I have the general approximation to the force between two coaxial magnetic dipoles which are much smaller than the distance between them. And I have the calculation of the total magnetic dipole moment of a thin, long coil with a magnetic core.

By combining them, I found that the force is proportional to the relative permeability of the cores squared and to the permeability of the medium between the solenoids.

This seemed to be somewhat wrong to me because by inserting materials like pure iron or mu-metal, we could increase the force to tens of millions times higher than it was with air cores. I don't think it's possible in practice.

Is there something I'm missing or misunderstanding?

Thanks for your time.

 

[Vailanor]

F = μ_0m_1m_2/4πr^3 (1)m = n*I*A_eff (2)whereF is the force between the two solenoidsμ_0 is the permeability of free spacem_1 and m_2 are the magnetic dipole moments of the two solenoidsn is the number of turns in the solenoidI is the current through the solenoidA_eff is the effective area of the core, i.e. the cross-sectional area of the core multiplied by the relative permeability of the core material.