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

Your question holds true for any physical phenomena that we have mathematical relations for. The trouble with asking “is it 100% accurate” versus “or a really good approximation and don’t know any better” is that there is no experimental difference. Experiments reveal physical reality up to a measurement uncertainty, and the best that we can claim is that a certain model is consistent with measurements and other results that are derived from that assumption. This is the closest we can get to “100% accurate”. Any deviations from that are buried within uncertainty, assuming the experimental measurements are free of artifacts. Asking “what if it’s slightly different” is then the objective of further research.

Now if I were to reinterpret your question less philosophically and more practically, where we assert that 1/r^2 scaling exactly true, we could also talk about how unlikely it is to fail. Electromagnetism is so fundamental to all subfields of modern physics that no field is completely free of it. Maxwell’s equations are not only directly tested, but also indirectly tested when we use those in experiments and models. For example, the Coulomb interaction is incredibly important in atomic and solid state physics. Its functional dependence is incredibly sensitive due to small distances (i.e. big forces bc small r) and the many-body nature of these systems (i.e. summing over ensembles). Using this 1/r^2 form, we derive electronic orbitals, use those to build our quantum theories, and are used for analysis in pretty much all of condensed matter physics.