# Calculate the force acting on a magnet inside a solenoid wrapped in Iron

Sounds like a simple question, doesn't it? But it isn't and it has been frustrating me to no end. First, my setup is fairly straightforward: I am building an electromagnetic actuator and have a magnet setup inside a solonoid. I need to produce a specific number of newtons at the highest possible power efficiency within a constrained space.

# Preamble:

After hours of research, I found several formulas which were important:

Magnetic field strength:

$$\dfrac{CIN}{L} = \vec{B}$$

Magentic Moment:

$$m = Coercivity \cdot{} L \cdot{} A_{magnet}$$

Force:

$$F = L (\vec{B}\cdot{}m)$$

Where $$C$$ is a magnetic constant, $$I$$ is current, $$N$$ is number of turns, and $$L$$ is the length of the solenoid (and the magnet). Coercivity of the magnet I am using is 1074 KA/m.

I plugged these numbers in and found that they were good (or at least reasonable, the units worked out to Newtons and the numbers were ballpark based on experimental data). Still, this was like attempt number 8 and my experimental data has fairly large error bars, so I'm concerned that I just got lucky and am using the wrong method to get the right answer.

# Question:

In the course of my research, I read a claim that an iron core increased the power of an electromagnet by up to 200x, but I can't seem to find a way to actually calculate the impact it would have on the magnetic field strength.

# So far:

I have read about magnetic circuits, and I understand that I can increase the magnetomotive force in the core of the solenoid if I decrease the magnetic resistance in the area surrounding the solenoid. I am not very familiar with magnetic circuits, but I am familiar with electric circuits. Apparently, both air and neodymium magnets have a magnetic reluctance about 3000x that of iron. Since the path through the center of the magnet is equal in length to the path up the side of the magnet, my electrically geared mind is telling me even a thin iron sleeve would double the magnetic field within the solenoid.

Can I just do that? Will that be the result? That the magnetic field is just doubled?

I am on very shaky ground here, so please let me know if you are seeing any rookie mistakes in how I'm dealing with this.

• Hmm, I think the "L" in your equation for force should be the gradient. – hank mabucci Jun 18 at 2:42