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Comment by dotancohen | Hacker News Reader

  > 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?

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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.

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)

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

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.

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