There are two very important laws in physics, namely the Newton's law of universal gravitation and the Coulomb's law, both of which contain a $r^{-2}$. I am curious about the exponent $2$ here.
There is already a well-answered question The square in the Newton's law of Universal Graviation is Really a Square? about the $2$ in the Newton's law of Universal Graviation. It can be seen that historically there have been some attempts in modifying the the exponent $2$ there. I find the Betrand's Theorem explanation in the referred question quite convincing.
For the Coulomb's law, there are also attempts to modify the exponent $2$ and I think this 1970 PRD paper summarizes the effort to measure the exponent. In the 1970 experiment it is shown that the exponent deviates from $2$ by at most $1.3\times10^{-13}$. I don't know whether there are other more accurate results recently.
My question is, if the exponents in these two laws have been questioned about, are the exponents in other fundamental physical laws ever questioned by people and measured in the experiment (to verify they are integers, for example)? Or it is because in macroscopic world the only important fundamental interactions are gravitation and electromagnetic interaction so verifying the exponents in these two laws would suffice?
As it as been commented that this is like a listing question, I am going to further clarify the questions.
- What is the reason behind the fact that there are many experiments measuring the coefficients in physical laws very accurately but there are quite few regarding the exponents?
- Does it suffice to verify that the exponents are indeed $2$ in the two laws stated above to justify that the exponents used in other physical laws are what they are? Why is verifying the number $2$ in Coulomb's law (for example) so important that people are constantly doing experiments on it?
Sorry if I have some misunderstanding above, and if that is the case please point it out. Any help will be appreciated.
