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Coulomb's Law states that the fall-off of the strength of the electrostatic force is inversely proportional to the distance squared of the charges.

Gauss's law implies that a the total flux through a surface completely enclosing a charge is proportional to the total amount of charge.

If we imagine a two-dimensional world of people who knew Gauss's law, they would imagine a surface completely enclosing a charge as a flat circle around the charge. Integrating the flux, they would find that the electrostatic force should be inversely proportional to the distance of the charges, if Gauss's law were true in a two-dimensional world.

However, if they observed a 1/r^2 fall-off, this implies a two-dimensional world is not all there is.

Is this argument correct? Does the 1/r^2 fall-off imply that there are only three spatial dimensions we live in?

I want to make sure this is right before I tell this to my friends and they laugh at me.

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Yes, but since coulomb's law is only verified for large(compared to the planck-scale) distances, it does not exclude the existence of very small spacial dimensions like they are used in some theories beyond the standard model. – CodesInChaos Nov 5 '10 at 16:45
up vote 27 down vote accepted

Yes, absolutely. In fact, Gauss's law is generally considered to be the fundamental law, and Coulomb's law is simply a consequence of it (and of the Lorentz force law).

You can actually simulate a 2D world by using a line charge instead of a point charge, and taking a cross section perpendicular to the line. In this case, you find that the force (or electric field) is proportional to 1/r, not 1/r^2, so Gauss's law is still perfectly valid.

I believe the same conclusion can be made from experiments performed in graphene sheets and the like, which are even better simulations of a true 2D universe, but I don't know of a specific reference to cite for that.

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Why the downvote? – David Z Nov 3 '10 at 5:42
Gauss law is derived from Couloumb's law, not the other way around. – Robert Smith Nov 3 '10 at 15:54
@Robert: No, sorry, but that's really not true. The derivation can go both ways - look at the introduction of the Wikipedia article for Gauss's law, for example, and notice that the citation to the HRW textbook is for the derivation of Coulomb's law from Gauss's law. I get the sense that your education in physics is somewhat limited (forgive me if I'm wrong; it can be hard to tell online), but if you had studied electromagnetism you would know that Gauss's law, being one of Maxwell's equations, is considered part of the foundation of the subject. Not so with Coulomb's law. – David Z Nov 3 '10 at 19:49
I know that the derivation can go both ways, however, the second derivation (Gauss's law->Coulomb's law) is not as strict and fundamental as the first one. Precisely, quoting from the Wikipedia article: "Strictly speaking, Coulomb's law cannot be derived from Gauss's law alone, since Gauss's law does not give any information regarding the curl of E (see Helmholtz decomposition and Faraday's law). However, Coulomb's law can be proven from Gauss's law if it is assumed, in addition, that the electric field from a point charge is spherically-symmetric (this assumption, like Coulomb's law itself, – Robert Smith Nov 3 '10 at 20:23
Maxwell's Equations (which include Gauss's law) are generally regarded as the fundamental laws of classical electrodynamics. That said, Coulomb's law came first historically. – Tim Goodman Nov 18 '10 at 3:11

I would say yes !

Actually some theories explaining quantum gravity use also this reasoning: gravity is a very weak interaction at a quantum level because it "leaks" into other dimensions, not observable at our scale, but that are present at this scale.

The mathematical tools are different, but if you just think about gauss's law you can imagine one explanation why additional dimensions are present in these theories.

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It's more the other way around, I would say. Gauss's law, together with the fact that we live in a world with 3 spatial dimensions, requires that the force between charges falls off as 1/r^2. But there are perfectly consistent analogues of electrostatics in worlds with 2 or more spatial dimensions, which each have their own ``Coulomb's law" -- with a different falloff of force with distance.

More to the point, it's a lot more obvious that we live in a world with 3 spatial dimensions (look around!) than it is that the force between charges has an inverse-square law. So empirically, as well as theoretically, the number of spatial dimensions is more fundamental than the force law.

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Loosely speaking, (super)string theory considers additional spatial dimensions that are "wrapped up" (have unusual topologies of high curvature, I believe). Now it is of course complete speculation, but if these dimensions do exist, electromagnetism would not spread out much into those dimensions, hence it would appear as if there are only three dimensions still (to a very good approximation).

Saying that, your argument is more or less sound (though far from bulletproof). It certainly suggests we don't live in a 2D world, and that any possible extra dimensions are comparatively very small!

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I think it is correct your appreciation if Gauss law holds in a two dimensional world, then the electrostatic force should be inversely proportional to the distance between charges. However, I'm not at all convinced that Gauss law could be true in a two-dimensional world because $F_{q}=k \displaystyle \frac{qq'(r-r')}{|r-r'|^{3}}$ is a consequence of a 3-dimensional space and since we derive Gauss law from such a force law (to be precise, from its electric field), we can not assume the validity of Gauss law independently from a 3-dimensional space.

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I thought for a while about giving you the benefit of the doubt, but after considering it I have to downvote this because Gauss's law can be derived in other ways that are perfectly valid in other numbers of dimensions and have nothing to do with Coulomb's law. – David Z Nov 3 '10 at 19:56
Please see: (Google Books link). Sorry if a reference the first link I found, however, I don't have much time right now. – Robert Smith Nov 3 '10 at 20:40
Unfortunately Google won't let me see that page, but I can try to look up the book in paper form. – David Z Nov 3 '10 at 22:01
Oh, don't worry. Anyway, it is an uninteresting book. Basically, it says "For large distances, r, the gravitational force goes as $1/r^{2}$. This is Newton's inverse square law of gravity, and it has been thoroughly tested for large distances... The inverse square law is a direct consequence of living in a Universe with three large spatial dimensions. But what about short distances, when r is less than R?". Then goes on to describe the changes in the inverse-square law due to n extra spatial dimensions. – Robert Smith Nov 3 '10 at 22:45

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