It seems like it should be a simple equation, until I realized that the core isn't magnetized until it is induced, then there is a dipole moment, and then as it moves the core of the solenoid gradually changes from air to the core material. This should be a differential I believe. I'm having a hard time finding any information on it.
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2 Answers
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The origin of the force is indeed complicated and linked to side effects. But if the core plunges deep and comes out of the solenoid widely, the force can be found through an energy balance.
The magnetic excitation in the solenoid is $H = nI$ inside and 0 outside.(n is the number of spire by unit length) At the top of the magnetic core, the magnetic field is zero and at the bottom: in the magnetic core $B=\mu nI$ and outside the core, $B=\mu_0 nI$
When the magnetic core of section $ s $ moves from $ dx $ upwards, we replace on the volume $ sdx $ the field $\mu nI$ by $\mu_0 nI$.
In this volume, the magnetic energy $\left(B^2/2\mu\right)sdx=sdx\mu{(nI)}^2/2$ is replaced by $sdx\mu_0{(nI)}^2/2$ and the variation of magnetic energy is: $dE_m=-sdx(\mu{-\mu_0)(nI)}^2/2$
When the current is imposed, we know that the magnetic force is: $F_x=+\frac{dE_m}{dx}=-s(\mu{-\mu_0)(nI)}^2/2$ : attractive force. Hope it can help. (and sorry for my poor english)
answered Jul 10, 2021 at 17:10
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$\begingroup$ Thanks, I'm just having a hard time visualizing some of what you're saying. I think I need a visual representation. $\endgroup$ Commented Jul 10, 2021 at 18:32
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$\begingroup$ I have added an image made quickly. Hope it can help ? $\endgroup$ Commented Jul 10, 2021 at 19:44
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$\begingroup$ Thank you very much, this seems like its exactly what I'm looking for. $\endgroup$ Commented Jul 10, 2021 at 23:54
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$\begingroup$ Just checking to make sure, I'm assuming 's' is supposed to be the cross sectional area? $\endgroup$ Commented Jul 12, 2021 at 21:45
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$\begingroup$ Yes, s is the cross section area of the plunger. Note that the magnetic field is 0 at the top of the plunger, outside the solenoid. $\endgroup$ Commented Jul 13, 2021 at 6:20
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Assuming that the core is (or becomes) uniformly magnetized (perhaps to saturation), then the force acting on any small segment of its volume will depend on its dipole moment / unit volume and the gradient of the field from the solenoid. The gradient will be most significant just outside of the end of the solenoid, where it changes in both magnitude and direction from point to point. This does not lend itself to a simple calculation. One might build tables (or formulas) based on measurements.
