# Is this an accurate equation for describing the magnetic force exerted by a solenoid on a ferromagnetic material?

I saw this question, which contained an equation describing the strength of a solenoid: $$F = (NI)^2\mu_0\frac{\text{A}}{2g^2}$$ where: $$F$$ is the force the solenoid exerts on a ferromagnetic material, $$N$$ is the number of turns in the solenoid (i.e. the number of full loops its wire makes), $$I$$ is the current flowing through the solenoid, $$\mu_0$$ is the magnetic permeability of vacuum - i.e. vacuum permeability, $$A$$ is - as far as I know - the surface area of the ferromagnetic material that the magnetic field of the solenoid is flowing on, and $$g$$ is the distance of the ferromagnetic material from the solenoid.

I have two further questions in regards to this; whether or not the second one can be answered depends on the answer to the first one:

• Is this an equation an accurate representation of the magnetic force F a solenoid with N windings and I current passing through it exerts on a ferromagnetic material of area A at a distance g? The original question did not cite a source, but digging around online and piecing together disparate sources, calculators, and whatnot - such as this solenoid calculator and the Wikipedia pages on the units tesla, henry, and weber - have given me the impression that it's correct. I cannot find any sources proving or disproving this, but plugging this equation into the Google search bar certainly gets me a seemingly-accurate result.

• Provided that this is a valid/accurate equation: if a magnetic core were inserted into the center of the solenoid, would the vacuum permeability (i.e. 𝜇0) be replaced with the magnetic permeability of the material that magnetic core was made out of (for instance, 6.3 x 10^-3 [0.0063] henries for 99.8% pure iron)?

The fact that the (NI) is squared suggests to me that this formula may give a reasonable approximation to the force between two identical current carrying solenoids which lie along the same axis. It may require that (A) be small compared with ($$g^2$$). (I would use a (d) instead of (g),)