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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)?

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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),)

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  • $\begingroup$ In this case, how could A describe a solenoid? A is a measure of surface area. As far as I know, that's better used to describe the surface of a piece of ferromagnetic material. $\endgroup$
    – KEY_ABRADE
    Jun 2, 2022 at 21:38
  • $\begingroup$ (A) would be the cross-sectional area of each solenoid. For (F) to the force on a magnetic material, you would need the permeability and dimensions of the material. $\endgroup$
    – R.W. Bird
    Jun 3, 2022 at 14:24
  • $\begingroup$ In other words, if I replaced (A) with the magnetic permeability of the material and its dimensions (for instance, "permability * width * height * depth" instead of "A"), that would more accurately describe the force by a solenoid on a magnetic material? I would upvote you for all this, by the way, but I don't have the reputation to do so. $\endgroup$
    – KEY_ABRADE
    Jun 3, 2022 at 19:40
  • $\begingroup$ Sorry, I neglected to check your references before answering. The 1st answer under your 1st reference calculates the radial force per unit area (from the longitudinal field) on the pseudo current on the surface of a magnetized bar which is inside a solenoid. That is the formula you site, and the net force on the bar in that case would be zero. My answer to that question, relates to the radial component of the field acting on that same current (near the ends of the solenoid), which gives a net longitudinal force. Refer to the Wikipedia article for the formula for that component of the field. $\endgroup$
    – R.W. Bird
    Jun 4, 2022 at 19:36

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