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Comment by pmontra | Hacker News Reader

Noob question: if whatever is happening is strong enough to raise one of the ends of the sample, why doesn't raise both? After all gravitation is many order of magnitudes weaker than electromagnetism. Did they calibrate the setup to closely match the gravitational force on the sample? Why not push a little more and make it fly up the the cap of the container?

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Magnetic force scales as 1/r^3, not 1/r^2 like gravity. That's why your standard issue fridge magnet measurably attracts stuff only from a very close distance, but when it does, it easily counters the gravitational attraction of the entire planet¹. This 1/r^3 relationship can be derived easily enough by integrating, but essentially it's because magnets are dipoles and the farther away you are, the smaller the apparent distance between the poles and the "more neutral" the magnet looks like.

Anyway, that's why there's an equilibrium distance where the forces balance. But superconductors also exhibit a very strange phenomenon called flux pinning [1] where a levitating object is held in place by magnetic field lines and you can even turn the whole thing upside down and it still levitates even though the forces don't cancel each other out anymore!

[1] https://en.wikipedia.org/wiki/Flux_pinning

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¹ To be fair, "the entire planet" is also around 6000 km away, calculation-wise, but still!

  > Magnetic force scales as 1/r^3, not 1/r^2 like gravity

The magnetic force and gravity are two of the four fundamental forces, no? The others being the strong and weak nuclear force? At what rate do those two scale at?

If you are really close to one pole of a magnet and far away from the other then the force scales like 1/r^2, it is in the far field under the influence of both poles of the magnet that it scales like 1/r^3. Electrostatics scales like 1/r^2 (because you fet isolated charges). The other two are a complicated story.

If you have time, I would love to hear that complicated story.

I remember once hearing that the rate of those forces' decay indicates (but does not prove) that they decay in more than three dimensions. This was an accomplished chemist talking, so I'm sure that she was being concise but factual.

I might not be able to give a super coherent picture but I will have a go. The force that hold nuclei together is based on the residual strong force, it is a complex system so the models used for it tend to be semi-empirical (yukawa or reid potential). They go from being repulsive at short distances to being attractive with something like an exponential decay as you go further away. The force is very strong at 1 femtometre and pretty much negligible 3 femtometres away. This is only the residual of the strong force though, which is what you can think of as holding quarks together. This force is very strange in that it doesn't get weaker the further the quarks get apart, but if it gets far enough then the energy will create new particles. That's just the strong force, I will leave it to someone more knowledgeable than me to try to explain the weak force.

For a point or ball, the formula is F=A/r^2

For a complicated shape, the formula is (something like[1]) F=A/r^2+B/r^3+C/r^4+... , where r is the distance to the "center" (and some parts are closer and some parts are more far away).

An important property of magnets is that A=0, so the r^2 term dissapears. And in many cases it can be simplified to F=B/r^3 [1, again].

[1] The complete formula considers also the directions, so it's longer https://en.wikipedia.org/wiki/Multipole_expansion

weak and strong forces are mediated by particles that have mass and therefore their force falls off basically exponentially, as exp(-m*r) it's not really the same as decaying in extra dimensions, because in that case the force law would look something like 1/r^(n-1)

Yes, in that model, it's not called the magnetic force but the electromagnetic force (combining both types of interactions into one mechanism.)

This one has a table that summarizes the answer to your question

https://en.wikipedia.org/wiki/Fundamental_interaction

On that page, in the table we 1/r² for both electromagnetism and gravitation in the column "Long-distance behavior (potential)".

Is that a mistake?

No, it's not a mistake.

The claim here was that "magnetic force decayed by 1/r³" which is the result of seeing a dipole ("two sources") from far away. Each pole decays by 1/r² and the net result (the combined effect of both poles on a test particle) is proportional to 1/r³

I would look at these:

https://physics.stackexchange.com/questions/184211/why-does-...

https://physics.stackexchange.com/questions/293310/why-does-...

And understand both electrostatic and magnetic dipoles in analogous terms.

> That's why your standard issue fridge magnet measurably attracts stuff only from a very close distance, but when it does, it easily counters the gravitational attraction of the entire planet

You can still get that effect if both forces are r^2. You can even get it with the same force—consider standing on the moon looking up at the earth. It will just be a more marked change with r^3.

Thank you.

Great question. And also a good answer was shared. Thank you both.

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