How do we surely know that electric force is 1/r²? If it is just because “it works” then why can’t it be 1/r^(2.0000001) or something, how do we know?

Question

Edit 1:
I was assuming the correctness of maxwells equations, how flux remains same in the absence of additional charge. So is it to say that that is 100% correct or just simple it can be a really really good approximation and we dont know?

Edit 2:
The best explanation I got till now: We simply do not know if it is really 1/r^2, but experimentally we have found that this model is very accurate to several decimal points(2.0000…). The 1/r^2 and maxwells equations are same, they are correct or wrong together. And in my opinion, the divergence equation for electric field seems very simple, and we found its high accuracy through experiments, so I have a feeling that it must all be exact 1/r^2. But still, I don’t think we have a way to find out because the equations are themselves the starting point which describes the experiments very well, we don’t have derivation for them and I don’t think we will ever have, and on the other hand, our only way to check is by experiments which won’t give 100% confidence accuracy.

Edit 3:
Another perception is about unit-dimentions. Like ( c^2 ) * e0*u0 = 1, which binds the units of e0 and u0, so electric and magnetic formulas must together be right or wrong. I can’t think or anything more right now, but there can be some other unit-dimension constraints that I am missing which will complete the picture.

In: 40

Answer 1

I haven’t seen anyone share the actual reason yet, but not incorrect statements, just not reasonings.

The reason why it’s true is because we live in a world with (supposedly) three spatial dimensions. If we lived in a world with a different number of spatial dimensions then the inverse square law wouldn’t hold. This means that the energy is spread out over an area instead of some other geometry at a particular distance, and as others have said this mathematically translates to an inverse square law.

As for how we know this? Well I said supposedly for a reason. We only think we are in a 3+1 spacetime. We have very good reason to think this, of course, but it is possible we live in a 3.0000000000002+1 dimensional spacetime which would be incredibly weird and makes no sense to our 3+1 evolved brains. If this were the case then it would be an inverse power of 2.0000000000002 law, which looks like an inverse square law down to very small tolerances. The general rule is 1^-(x-1) where x is the number of dimensions that exist. Additionally, micro dimensions, dimensions coiled up on themselves that we can’t perceive, are one hypothesis to explain the weakness of gravity. There’s a lot going on in dimensional studies.