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