Minimum Energy Configuration for Free-to-move Magnets

Check in the back of the book. Maybe the answer is there.

What book would you suggest? (I've published ten.) My interest is in the proof (if there is one) that they would form a colinear arrangement rather than an arrangement in three dimensions.

Well, apparently you haven't published the right ten--as yet, anyway.

Would this be a new book, or an addition to an existing book.

Well, it would depend on exactly where you placed them, wouldn't it? But even if they formed a straight line, they would still be in a 3D universe (unless you believe in string theory, in which case it would be 11). Why don't you make them 2D, and then ask Homer Simpson?

Clearly the minimum energy formation is to ask others to do your homework for you.

There are other things that should be clarified in this question. What is the minimum energy configuration of what? Are you looking for the minimum energy to configure these three magnets in a static position? Is there any friction involved? How do you make a near point source magnet that exhibits any kind of a magnetic field? Do you understand any quantum mechanics because this is the realm of three discrete near point source anythings? Did you understand the mathematics of Erwin Schroedinger's probability equations of the two body Hydrogen atom that deals the complete electromagnetic effect of electron and nucleus? What is the mass of these magnets and do we take into account the expansion of the universe?

Yes. Apparantly, you understand but can't make use of these concepts.

I'd be a little more careful about who and how you challenge somebody. I've asked for seven different clarifications and you've answered none of them. I wonder why?

Equilateral triangle with unlike poles touching.

Yes, triangular, touching, plus some rotation, and or linear velocity.

Not enough data for a better answer.

{edit} Isn't this just a variation on the 3 body gravitational problem?

I'm thinking more like:

Okay, this is just a 'hand-waving' argument, not at all rigorous, but here goes.

The force between two magnets goes roughly as the inverse cube of the distance between them. In the co-linear condition illustrated below the distance between the centers is r1, whereas in the equilateral condition the distance is r2, where r2 is greater than r1. So more force is needed to separate the magnets in the co-linear condition and move them to a large distance, than in the equilateral condition. The work done is f•ds so the net energy required would also be greater (work = energy in a conservative field). Therefore the (side-by-side) co-linear condition is the least energy condition. [The width:length aspect ratio of the magnets will also affect the relative forces and work required in the different conditions.]

Makes sense, but I think the OP meant collinear. You don't show that in your drawings.

We can make all sorts of suppositions about what the OP meant. I'm still looking for an explanation of a point source bar magnet. Are we talking about the magnetic moment of an electron? You cannot get much more of a point source magnet than an electron. If we consider electrons as magnets then the lowest energy of these three magnets would be a lone, non ionized atom of Lithium with a complete 1S shell and a single electron in the 2S shell. Now these magnets will never be static, but they will be in their lowest energy state in these orbitals. One might be tempted to consider Helium also with the nucleus being the third magnet, but due to inertia the nucleus is not really free to move.

But instead of getting any clarification from the OP, I get the mild abuse of a petulant child.

He didn't originally use the term co-linear, but anyway, I understand your point. I used the term myself in the sense of 3 lines parallel to the same line are co-linear, though that definition is perhaps less standard.

That may be plausible, but I will wave a hand the other way:

1. The fairly close north and south poles in the parallel arrangement could make it easier to separate the magnets. 2. Wouldn't the distance between adjacent attracting/repelling poles be more important than distances to the midpoints of the magnets?

I'm not sure which of these effects would predominate. Even if ultimately wrong, for now I still bet on the triangle. (Too bad I don't have 3 bar magnets and a small spring scale around....)

The F dot ds approach is correct. However, the distance as chosen is not correct. The distance should be the mean path length through air/free-space from one pole "face" to another.

Assume the pole face is circular with radius "r". In the side by side arrangement, the mean path distance is pi*r. In the triangle configuration, the mean path distance is (2/3)*pi*r. The shorter path through air/free-space provides the lower energy state.

I disagree. The question is what is the "minimum energy configuration for free-to-move magnets?" The question does not limit the energy to just the energy stored in the magnetic fields. The energy of the whole will be the same regardless of configuration. As posed, the law of the conservation of energy applies. As the question is posed there is no mechanism for energy to leave this system.

This question is an imprecise question asking for a precise answer. I will not be part of a fool's errand.

Thanks. But what if we were to place the third magnet on top of the first two? (I should really break down and buy three magnets.)

Assuming (big assumption) that the magnets look like the ones in the sketch, I think that once two magnets have achieved minimum energy (side-by-side, opposite poles touching) the 3rd magnet will attach itself to the other 2 as my diagram shows, in the same plane. If it initially tried to attach itself 'on top' the repulsive force from the one and the attractive force from the other would flip it over into the orientation in the diagram.

It really depends a lot on the geometry of the magnets and the strength of the magnetic field.

*If you already have one good strong magnet, use it to magnetize 3 needles and then let us know what results you get.*

How do you propose that the 3 needles are free to move in 3D? If they are to float in water, that would be 2D, would it not?

Errr....ummm... Do I have to think of everything?

Actually, I was thinking he could run the experiment backwards. Start with all 3 magnetized needles side-by-side and see how difficult it would be to separate them. Then try putting all 3 into an equilateral triangle (which I think would be unstable) and see how difficult it would be to separate them. Compare results.

Buy some of those magnetized sphere NIB magnets, grease them up with vaseline, and let them tell you the answer.

You're right, he said bar magnets. I agree with you, it depends on the shape of the magnets. Magnets shaped like coins, magnetized from one face to the other, I would think would rather be stacked. Longer, thinner magnets would rather be in an equilateral triangle, possibly with one overlaying the other two to maximize the amount of metal touching to reduce the free space flux.

This problem on a larger scale is addressed in the theory of spin glass (http://en.wikipedia.org/wiki/Spin_glass).

In addition to the actual aspect ratio, the OP also failed to specify a material. Isn't the "energy" in the triplet also a function of the permeability of the bodies themselves? Am I correct in stating that depending on the strength of the magnets in relation to their length, except at one point, either the field or the distance will be the dominant factor in computing the F in the F dot ds equation?

Assuming the bodies do not deform or fracture upon colliding, I'd agree that the safe bet is the 2 closest of the three will adhere their N-S faces into a collinear object then the 3rd will adhere in an unpredictable location. And while stable, it won't necessarily be the minimum energy condition... or will it at that one crossover point?

Thanks! Of course, three objects are always coplanar! (I kept writing collinear which is silly.) Buckyballs does it!

GA I've got some of those too, and, yes it takes a bit of effort to get from this

to this

Whereas going from the line to the triangle is easy. However the forces required are applied over a much greater distance so I think you'd need to do the analysis to be sure.

If we rearranged these magnets into the maximum energy configuration and mounted them on a shaft, could we make a flux energy motor?

In first place the problem was clearly stated by jayC, shouldn't be further questions about.Then a way to solve: imgine three superthin magnet wires, join the six extremes (only north to south way), thats all the energy you could get without demagnetize so the triangle config is the answer.Technicals matters becoming from sizes,etc,shouldn't be considered.Thats all.-