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 $\frac{1}{r^2}$ fall-off, this implies a two-dimensional world is not all there is.
Is this argument correct? Does the $\frac{1}{r^2}$ 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.
Answer
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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