# How can we calculate the force of attraction between a coil and a ferromagnetic object?

As the question states I’m trying to figure out how I can calculate the force of attraction between the magnetic field of a solenoid (coil) and a ferromagnetic object. I suppose we should somehow calculate the force between the magnetic field of the solenoid and the induced field of the ferromagnetic object if I’m not wrong? So far I have not found any formulas online able to do this.

Let’s take an iron rod with a diameter of 12 mm and a length of 4 cm as an example. We are talking about a DC solenoid with lets just say, as simple values: 25 windings of AWG 12 wire and just a single layer coil. There will go a current of 2000A through the coil. The object is placed horizontal to the coil with a distance of 2 cm.

Any help on how to tackle this problem would be appreciated!

• Your coil would melt. – Hearth Oct 28 '19 at 14:28
• ...and partially evaporate – Curd Oct 28 '19 at 17:41

It's a very difficult problem to solve. There are simple solutions only for a few very simple geometries.

The best analytical approach is through work and energy, force = d(energy)/d(distance).

As you pull the iron away from the field, you are doing work, which is equal to the force you apply multiplied by the distance traveled. This energy you supply gets stored in the new magnetic field, and depending on how you drive the coil, some may be delivered to the external circuit (how a generator works).

Among the many unknowns are what the field is at all points, and how the field changes when the armature moves.

The simplest geometry to solve is for the force required to open a closed magnetic circuit, for instance the electric door locks you often see on access control doors, as we know accurately the area of the poles, and the flux in them, and we neglect flux in the air. The force can be calculated as a 'pressure' times the pole area, the pressure proportional to flux squared. IIRC, 1Tesla gives 4 bar pressure, I may be wrong about the 4, but I'm not orders of magnitude wrong (do comment with the correct figure if you know or can be bothered to calculate it). You can work this out from first principles using the work approach above, using the energy stored in the newly created airgap.

You'll need to get out your 3D integration calculus or fire up a FEMM solver for any more complicated geometry. It's probably easiest to simply build and measure. The force will be proportional to current squared, until you hit saturation. Yes, saturation is a further complication!

• With 2000A, saturation sounds like the first complication OP will run into. – Harry Svensson Oct 28 '19 at 14:27
• @HarrySvensson 2000 A through 12 AWG? I think the coil melting would be more of a problem. – Hearth Oct 28 '19 at 14:29
• I was talking about a very short surge current of 2000A, sorry for the confusion. According to the Oderdonk's equation that should be possible, or am I doing something wrong here? – jortpepe Oct 28 '19 at 16:18
• @jortpepe no, you weren't talking about a very short surge current. But it's probably something you wanted to mention in your question but forgot to do. – Harry Svensson Oct 28 '19 at 17:23