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

First, all measurements are limited by our scope of the ability to measure. If I take a scrap-bin of lids and measure it on the scale then within a 2-minute span it will measure between 15.05 and 15.10 pounds before settling on 15.06 pounds. If I take it off the scale for a minute, or jostle it while getting my pen out of my pocket, then it might settle on 15.04 pounds or even 15.13 pounds.

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So, for like 80% of cases the difference between an exponent of 2 and an exponent of 2.00000000000001 will probably not be significant enough to make a difference.

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Second, when the substantially-miniscule decimal would actually make a significant meaningful difference, we can test for those values. Which the results of which would require a decimal value much much much smaller than even my proposal of your example.

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And then there’s the fact that we can apply mathematical transformations, such as Integration and Derivation from Calculus, to calculate different variables with different experiments. For example, if you know how far something traveled and how long it was traveling for… you can determine its speed. If you know how long it took to get to that speed, you can determine its acceleration. If you know its mass, you know how much force it had while it was accelerating and thus what force you needed to get it to move at that speed as quickly as it took to get up-to-speed.

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And the same two points from above apply to those calculations as well… so if for some reason the value is actually 2.00000000000000000000000000000001 instead of 2, then it would be 2.00000000000000000000000000000001 for all the associated equations. But, nope, there are some where it’s no larger than 2.0000000000000000000000000000000000000000001. Which means for the equation you ask about, it must be smaller than 2.0000000000000000000000000000000000000000001