The Magnetic Force on an Iron Ring

[Arkthanon]

I decided to simulate this scene in Lord of the Rings for a project in school with the purpose of calculating the "real" mass of the One Ring. I've done the experiments according to this principle sketch but I have some troubles with calculating the magnetic force that the copper coil is exercising on my 3,8 g iron ring. The area of the ring is 5,84 cm² and I measured the strength of the magnetic field to be 33 mT in the position of the ring. How would you calculate the force F_B? I'm really having a problem with this calculation and would very much appreciate an answer. Any simplifications and assumption are very welcome if that is needed. Please do also ask me for more information if that is needed.

 

[mfb]

Calculations of magnetic forces on ferromagnetic things are messy. You need the field gradient in addition to the field, unless you assume the ring was magnetized before. But your ring itself modifies the magnetic field...

Unrelated to the specific question: I don't see how this approach would estimate any mass of anything related to the movie.

 

[Arkthanon]

Calculations of magnetic forces on ferromagnetic things are messy. You need the field gradient in addition to the field, unless you assume the ring was magnetized before. But your ring itself modifies the magnetic field...

Unrelated to the specific question: I don't see how this approach would estimate any mass of anything related to the movie.

With the field gradient, do you mean the inclination of the field? For I can assure you that the ring was stationed at the axis of the magnetic field. Would I get a result if I assumed that the ring was magnetized before? In that case, how would that calculation look like? I'm really desperate for some kind of results for a laboratory discussion...

As an answer for your question: My main idea was to vary the magnetic field until the ring that I was dropping on the wooden plan would behave like the ring in the video. I would then add F_B to the gravitational force mg and then divide the sum by g (assuming that the gravitational acceleration is the same on Middle Earth). I know that this method would give a rough estimate, as it ignores elasticity. My mentor did at least approve the method...

 

[Arkthanon]

Calculations of magnetic forces on ferromagnetic things are messy. You need the field gradient in addition to the field, unless you assume the ring was magnetized before. But your ring itself modifies the magnetic field...

Unrelated to the specific question: I don't see how this approach would estimate any mass of anything related to the movie.

I found this formula (source, page 284)
f_m = m * χ₀ * µ₀ * H * grad(H)
m and µ₀ are already known.
χ₀ is determined by

and

(source)
H is determined by B/µ
Is there somehow possible to calculate grad(H) or at least simplify it by making some assumptions? I'm really in need for some results.

 

[mfb]

With the field gradient, do you mean the inclination of the field?

No, the derivative of the field as function of position.

Would I get a result if I assumed that the ring was magnetized before?

Then you don't need the field, just the field gradient.

As an answer for your question: My main idea was to vary the magnetic field until the ring that I was dropping on the wooden plan would behave like the ring in the video. I would then add FB to the gravitational force mg and then divide the sum by g (assuming that the gravitational acceleration is the same on Middle Earth). I know that this method would give a rough estimate, as it ignores elasticity. My mentor did at least approve the method...

I don't see how that would work. What force would you need where, when?

H is determined by B/µ

How do you find B at the position of the ring? If you can find B everywhere, finding grad(B) is easy. Both things are challenging theoretically, and might need some finite element analysis if you are interested in more than an order of magnitude estimate.

 

[Arkthanon]

What force would you need where, when?

The idea is that the weight of the one ring (m_Ringg) would be simulated by the weight of the Ironring + the force of the magnetic field (m_Ironring + F_B), according to the principle sketch.

How do you find B at the position of the ring?

I measured it with a Hall effect sensor.

I appreciate your help, but I've handed in the assignment now.

 

[mfb]

The idea is that the weight of the one ring (mRingg) would be simulated by the weight of the Ironring + the force of the magnetic field (mIronring + FB), according to the principle sketch.

I don't see what there would be to simulate. The mass does not matter much if everything is proportional to it.
Also, the magnet will generate a force that varies a lot with distance.