# Magnetic Force on a Ferromagenetic Material

I am currently working on a project involving solenoids, and I needed a force(Newtons, not a measure of magnetic field strength) equation. What I came up with after some digging around on the internet, is the equation:

$$F = (NI)\mu_0\frac{\text{Area}}{2g^2}$$

Where $F$ is force (in Newtons), $N$ is the number of turns in the coil, $I$ is the current being passed through the coil, $μ_0$ is the magnetic permeability of vacuum, and $g$ is the gap between the coil and the ferromagnetic material. (Area $A$ and $g$ can be any units, as long as you're consistent with the usage)

I don't know in which plane exactly the area $A$ is taken.

Assuming I have a rod, moving lengthwise into a solenoid, which plane would $A$ represent?

Plane a, plane b, or another plane that I did not consider relevant to this problem?

Rod:

Edit: I was looking for the force an electromagnet would exert on a ferromagnetic material moving into the coil. something like this.

Edit: If the equation I was using before does not work, I don't suppose anyone has the correct one?

Edit: After looking at the equation some more, I realized I had written it wrong. It should be:

$$F = (NI)^2\mu_0\frac{\text{Area}}{2g^2}$$

• Simple dimension analysis of the right hand side reveals that something is wrong with this formula. It will not give a force. Finding the right formula for you is difficult because you don't describe what force you're interested in.
– jkej
Commented Aug 21, 2014 at 9:24
• @jkej I am interested in the force (in Newtons) that an electromagnet would exert along its axis on a ferromagnetic rod. I can put that in my edit. Commented Aug 21, 2014 at 13:02

I don't know where you got your formula from, but but I derived it this way:

Field inside the solenoid$=\mu_0ni \hat{z}$ (say)

Since the material is ferromagnetic, there is an induced, bound surface current $K\hat{\phi}$ (and $K=M$, where $M$ is magnetization). The magnetization is uniform, so bound current is zero,$$J_b~=~\nabla \times \left(M\hat{x}\right)~=~0 \,.$$

From the Lorentz force, $F=i \vec{l}\times \vec{B}$:

\begin{align} \Rightarrow F &= A\vec{K}\times\vec{B} \\ &=AK\left(\mu_0ni\right) \left(\hat{\phi}\times\hat{z}\right) \\ &=AM(\mu_0ni)\hat{r} \\ &=A(\mu_0ni)(\chi_m*H)\hat{r} \\ &=A(\mu_0ni)(\chi_m*\frac{\mu_0ni}{\mu_0})\hat{r} \\ &=\chi_mA\mu_0 \left(ni \right)^2\hat{r}\,, \end{align}where $A$ is area.

Here $A$ is the area of the surface that $K$ is flowing on, i.e., the curved surface of the cylinder$=2\pi RL$ where $R$ is radius and $L$ the length of ferromagnet. The force is radially out of the surface of the core, stretching it out as if to fill the coil.

I don't know why the force should depend on the "gap" between the two.

• Why would it not? Magnetism decreases with the cube of the distance if I remember correctly. Just get a bar magnet and a piece of iron and you'll see pretty quickly that the force exerted is distance-dependent. Commented Mar 25, 2017 at 17:36
• BTW, you might find this interesting. As to where I got my formula, I got it here. Commented Mar 25, 2017 at 17:38
• Oh, I see. You were looking at a ferromagnet outside the coil. I thought it was a ferromagnet inside the coil, where the field is uniform. Commented Mar 25, 2017 at 17:42
• On the page where you got your formula from, I think the gap they're talking about is rectangular, so they take the volume as gap length g times cross sectional area A. In this case the volume of the gap is $\pi (R^2-r^2)*L$ . Commented Mar 25, 2017 at 17:58
• Also, the first link only finds the field outside a solenoid, but on the axis, so it would be rather complex to find first the field everywhere and then the bound currents and then the force on the ferromagnet. Commented Mar 25, 2017 at 18:00

The atomic magnetic dipoles in a ferromagnetic material experience a torque that tends to line them up with the crystal axis and another which tends to line them up with any magnetic field. At normal temperatures they can often maintain such an alignment. If the field has a gradient they can also experience a net force. If a ferromagnetic bar is lined up with a current carrying solenoid, the field from the solenoid magnetizes the bar and then sucks it in due to the divergence (or convergence) of the field which occurs just inside and outside of the end of the solenoid. This principle is used a lot in electric current relays and fluid flow valves. (As I recall, the doors on the Boston subway trains in the 50's were opened and closed by rods being sucked into long solenoids.) The actual force is going to depend a lot on the properties of the ferromagnetic material. I would suggest an experimental approach (or buy a preexisting device). You might start with a premagnetized bar, but then there is the risk that the solenoid may modify the magnetization.