# Electromagnet force equation clarification

So, from a previous previous question, I know what plane the force from an electromagnet will affect a ferromagnetic material in. I thought that would be enough, but working through some problems using the force of a solenoid (electromagnet) given by $$F = (NI)^2\mu_0A/2g^2$$ I realized I was unsure where to get $g$ from.

I know that I should take $A$ from plane b, but what point should I take $g$ from? What point along the rod should I take the area from? The farthest away possible from the electromagnet, at the back? As close as possible, at the front? The middle of the rod, an equal distance from both ends? Should I calculate the force at all places along the rod, and average them all? As of right now, I am assuming the force is acting on the middle of the rod (the average distance away from the electromagnet.)

Any help appreciated.

Edit: Thinking about this, I believe the real question I should be asking is "Should I be using volume and g^3 in my equation instead of area and g^2? Because the electromagnet would produce a field, acting on anything in it that is ferrous. So the force would be acting on whatever volume is "immersed" if you will, in the field?"

## 1 Answer

The equation you give is for the static situation in which the iron is already fixed near the solenoid (cf this site). In this set-up, see the image below, we have a coil with surface area $A=\pi d^2/4$ with diameter $d$ placed a distance $g$ below a metal plate (it also seems that the plate must have a surface area > $A$). The volume of the field, $V=gA$, is considered in determining the potential of the field, but this factor of $g$ here is canceled by the derivative (and then two more factors of $g$ are divided through from the magnetic field equation): $$U=\frac{B^2}{2\mu_0}gA$$ $$F=\frac{dU}{dg}=\frac{B^2}{2\mu_0}A$$ the magnetic field of a solenoid is $$B=\frac{F_m\mu_0}{g}$$ where $F_m=IN$ is the magneto-motive force. This then gives us your relation, $$F=\frac{N^2I^2\mu_0}{2}\frac{A}{g^2}$$

It seems, however, that you are trying to determine the forces as the rod/plate is being brought near the solenoid. This is highly nontrivial and will very likely require numerical methods as the rod will dynamically alter the magnetic field lines in a way that a simple force equation likely will not satisfy (remember Maxwell's time-dependent equations!).

• Are you sure about that? The force inside a solenoid is homogenous. I have found multiple websites with this calculation. I would question whether your interpretation is correct, on the basis that I cannot think of a use for such a described equation. Since I found the equation on websites aimed at engineers and hobbyists, I can do nothing but assume that this equation is used for the purpose I want to use it for. I can think of numerous situations calling for the force exerted on a ferromagnetic material entering a solenoid, and only one when it has already entered. – CoilKid Sep 21 '14 at 4:00
• Could you please explain how you reached that conclusion in more detail? – CoilKid Sep 21 '14 at 4:01
• Which conclusion are you referring to? The first one (about the eq'n being force inside the solenoid) or the second one (about having to numerically model it instead)? – Kyle Kanos Sep 21 '14 at 13:42
• The one about how the equation only works if the iron is already in the center of the solenoid. As I said, I can think of only one use for such an equation, and that is to calculate something for a project involving levitation. (A no friction bearing perhaps) – CoilKid Sep 21 '14 at 14:51
• @CoilKid: I've done a bit more research into the equation and have corrected my answer. Please see the update. – Kyle Kanos Sep 21 '14 at 15:20