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

Generally, you take all the hypotheses, throw away the ones that contradict evidence, and the simplest one is the winner. If all experiments agree that *F ~ r****^(k)****,* with *k* being between 1.9999999 and 2.0000001, then we say that *k*=2 and make further inferences from that. We could also claim that *k*=1.999999942069 exactly and consider the implications of that, but so far the universe seems to be governed by relatively simple rules and the likelihood of an oddly specific *k* seems much lower than a nice round 2.

Or consider the [anomalous magnetic dipole moment](https://en.wikipedia.org/wiki/Anomalous_magnetic_dipole_moment). Dirac’s equation predicts that the electron has a certain quantity called the *g*-factor equal to exactly 2. Initially, experiments seemed to agree with this. But increasingly precise ones gave the quantity that was slightly different from 2. Thus, unfortunately, the simplest hypothesis was wrong and it needed to be complicated a bit. The theory was revised and currently it predicts the value *g* = 2.002319304363286 ± 0.000000000001528 while the experimental value is *g* = 2.00231930436146 ± 0.00000000000056.

Note that it doesn’t simply say *g* is equal to that. The new hypothesis was the result of someone thinking, “what if particles interact in all possible ways at the same time”, and the calculations based on that assumption give the tremendously precise *g,* so perhaps it is in fact true that the particles interact in all possible ways at the same time.