# How is the force between two magnets calculated?

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#### flofelis

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##### How is the force between two magnets calculated?
« on: 15/12/2015 01:37:04 »
Apologies, too long a question to fit into the subject  =P

Is the force between two diametrically magnetized cylindrical permanent magnets (PMs) the same as that of two axially magnetized cylindrical PMs?

I am interested in finding the force and torque exerted onto a PM by another PM within the vicinity.

So I am just wondering if I could use the same force/torque equations (e.g. for the force as shown below-sorry for the messy typeset) between two axially magentised PMs for the diametrically magnetized ones as well:

F=$$\frac{pi*mu_0*M^2*R^4}{4}$$($$\frac{1}{x^2}$$+$$\frac{1}{(x+2h)^2}$$-$$\frac{2}{(x+h)^2}$$)

Looking forward to hearing from anyone with ideas and advice =) Thank you very much!

« Last Edit: 15/12/2015 08:20:35 by chris »

#### Space Flow

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##### Re: force btw 2 diametrically magnetized magnet & that of 2 axially magnetized same?
« Reply #1 on: 15/12/2015 02:10:03 »
I am certainly no expert and can conclusively tell you nothing on this subject but you invited ideas as well, so I offer this.
A permanent magnet has it's electrons arranged so that each of the electron magnetic moments will be, on average, lined up. Then the material can produce a net total magnetic field.
As the strength of that magnetic field is dependant on the number of atoms available to have their electrons so arranged, I can see no reason for a difference in field strength in the same size material, between axial or diametric alignment.

Not Guarantied correct but it makes sense to me.

Hope it helps.
We are made of Spacetime; with a sprinkling of Stardust.
Matter tells Spacetime how to Flow; Spacetime tells matter where to go

#### flofelis

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##### Re: force btw 2 diametrically magnetized magnet & that of 2 axially magnetized same?
« Reply #2 on: 15/12/2015 03:47:37 »
Thank you, Space Flow =)

I was thinking that the magnetization, M through the diametrically and axially magnetized PMs is different in direction, thus the force on either one of the two PMs might be different with different PM dimensions and positions.

I see that most cases are calculated for axially magnetized PMs. I am working with two diametrically magnetized NdFeB magnets and got pretty confused as to how I should compute the dynamics (force and torque of one PM (primary) when another is closely present).. as well as how far should the NdFeB PMs be apart so I could ignore the effect of the other PM onto the primary PM. Also, I am not sure if the generally available equations for the axially magnetized PMs can be used in my case.

Thanks!

#### vhfpmr

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##### Re: How is the force between two magnets calculated?
« Reply #3 on: 15/12/2015 12:03:39 »
Won't it be the relative positions of the fields that determine the force, rather than the positions of the magnets themselves?

#### wolfekeeper

• Neilep Level Member
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##### Re: How is the force between two magnets calculated?
« Reply #4 on: 15/12/2015 15:50:15 »
In practical terms you solve Maxwell's equations and calculate the force from that. There are software packages that permit you to do that for any given geometry.

#### flofelis

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##### Re: How is the force between two magnets calculated?
« Reply #5 on: 16/12/2015 05:40:39 »

I realised that I should include an sketch of the setup I am working on for better illustration, as per attached =)

So these are the two diametrically magnetized cylindrical NdFeB PMs placed beside one another with a distance, d. Both are rotated by independent external sources (I have solenoids to independently drive the PMs). This will affect each PM in terms of the torque and force.

So I am not sure if the expressions for forces and torque in the case of axially magnetized PMs are not compatible in this case. I do not have a software for this, thus I would need to model it mathematically.. (i have spent weeks trying to find a way to do it but I ran out of idea) but I have been told that it is a very complex problem to solve by hand =(

I kind of visualised it similar to a bar magnet but it is usually rectangular and not cylindrical. So there's this geometric aspect that I have to think about.
Besides, I could only find force analysis on bar magnets (attract/repel) but not torque as bar magnet is rarely expected to rotate the way I need my cylindrical PMs to rotate axially. Pretty perplexing..

Any ideas and comments are appreciated =) thanks again~

#### wolfekeeper

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##### Re: How is the force between two magnets calculated?
« Reply #6 on: 16/12/2015 16:42:35 »
It will be similar to a 2D problem, in other words you can calculate the torque and force between two infinite parallel cylinders, but weakened due to fringing at the ends.

You can solve the 2D problem in FEMM, which is freely available, and then empirically calculate the fiddle factor to deal with the truncated topology.

Are these short cylinders or long cylinders. If they're long compared to the diameter then FEMM will give you a reasonably accurate answer.

#### evan_au

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##### Re: How is the force between two magnets calculated?
« Reply #7 on: 16/12/2015 21:07:39 »
Are you trying to transmit torque through a container wall (eg for a stirrer driven by an external motor, without the need for a hole through the container wall)?

In this case, it might be better to have 2 strong magnets, separated by a distance of a few cm, facing end-on, rather than parallel. The greater the distance between the "transmitter" magnets, the greater the torque you can transmit without blocking the torque entirely.

This would give the advantage of steady torque. It seems that in the arrangement shown, the torque might vary at different angles of rotation?

#### flofelis

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##### Re: How is the force between two magnets calculated?
« Reply #8 on: 17/12/2015 00:10:16 »
Thank you so much, wolfekeeper and evan_au =)

wolfekeeper, the face of the two cylindrical PMs will be randomly positioned with x,y,z distances relative to the coordinate frame of the primary PM. I should be able to take that into account as magnetic field vectors, but I have yet been able to model this mathematically while considering the rotation (which in this case, I assume will also affect the net magnetic field at different angles)
Are these short cylinders or long cylinders. If they're long compared to the diameter then FEMM will give you a reasonably accurate answer.
These diametrically magnetized cylindrical PMs are of 1cm diameter x 1cm thickness. Sounds like the geomtry of the PMs is the tricky part? I wonder if there are any assumptions I can make to still get a valid result?

evan_au,
Are you trying to transmit torque through a container wall (eg for a stirrer driven by an external motor, without the need for a hole through the container wall)?
yes =) about the same concept. I am transmitting torque on the PMs across the barrier so the PMs would have contactless rotation. I am wondering if anyone has used this method before in any applications. It would be a nice reference.

Since the PMs will be randomly positioned, I would have no guarantee how the actual arrangement will be like. So my model would have to cater for the difference between the two in all the x,y and z axes. The transmitter magnets though, would most probably be pretty close to one another, unfortunately =( as my workspace is pretty limited (to about 30x30cm slightly spherical surface). Any idea how I can find out the optimal distance between the magnets for the highest torque transmitted?

Thanks again! =) =) =)

#### wolfekeeper

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##### Re: How is the force between two magnets calculated?
« Reply #9 on: 17/12/2015 00:18:42 »
In the picture shown, why are the magnets rotating in the same direction. They should be rotating in opposite directions should they not?

#### flofelis

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##### Re: How is the force between two magnets calculated?
« Reply #10 on: 17/12/2015 00:30:24 »
Wolfekeeper, the PMs magnets are driven by external magnetic sources which rotate these PMs independently. Thus, the direction of rotation on each PM can vary differently. Apologies, I didn't include that in my illustration. The two PMs are not coupled with each other and only the magnetic fields from both sides are interacting. I hope I am not misunderstanding this myself =)

#### wolfekeeper

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##### Re: How is the force between two magnets calculated?
« Reply #11 on: 17/12/2015 00:32:29 »
With the geometry shown, with a magnetic field coupling their rotation, they will naturally rotate in opposite directions. If they are instead face-to-face they will rotate in the same direction.

edit:

Quote
Any idea how I can find out the optimal distance between the magnets for the highest torque transmitted?
Would you be shocked if I said it was As Close As Possible?

Also if you stick both the magnets on arms, so they're as far from the rotation axis as possible, maximum torque goes up.
« Last Edit: 17/12/2015 00:51:26 by wolfekeeper »

#### flofelis

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##### Re: How is the force between two magnets calculated?
« Reply #12 on: 17/12/2015 00:56:05 »
Wolfekeeper, yea..totally agree..I guess my diagram is a little misleading. sorry about that.. I am not able to quickly illustrate it in 3D but it is something like the diagram that I have quickly updated (as attached). The internal PMs shouldn't affect or be coupled with each other, and have to be strictly driven only by the external driven ones.

So with the internal magnets driven independently, the torque and forces onto one will be affected by the other as well. So I would need to find out the net force and torque onto one of that driven magnet considering the affect of the other. thanks!
« Last Edit: 17/12/2015 00:58:31 by flofelis »

#### wolfekeeper

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##### Re: How is the force between two magnets calculated?
« Reply #13 on: 17/12/2015 15:26:24 »
In the type of ranges showed in the picture, you can approximate each magnet with a magnetic moment, and get a fairly accurate estimate of the torque. This will probably fail though at closer range.