Feynman starts the derivation of Gauss law by stating the it depends specifically and directly on the fact that force law is inverse square.

Does it mean that if electric force law varied as $1/r^3$, Gauss law wont be true?

I am not able to think. why would that be the case?


The key point is that Gauss' law works for forces that emanate like lines of force out of the sources, whatever is their symmetry. To say that we have an inverse square force is to assume each source is like a point in 3D space, so that its lines of force emanate in all directions and the force acts like a surface density of those lines. We can make "Gaussian pillboxes" that conform to whatever is the symmetry of our sources, so we don't always get an inverse square force, but Feyman means we do if we regard each of our sources as a point in 3D space, and then superimpose those points to get whatever is our actual source symmetry (like a plane of charge and so on). So if the force from a point source fell off like 1/r^3, that could only act like the surface density of lines of force in 4 spatial dimensions, so then Gauss' law would only work in 4 dimensions.

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