What is the smallest and biggest distance in which Coulomb's law is valid?
Please provide a reference to a scientific journal or book. Just saying that this law is valid from this range to that range is not enough. Which experiments did result to these ranges?
Answer
The validity of Coulomb's Law over large distances is equivalent to bounding the mass of the photon. In quantum field theory, where one derives Coulomb's law, if the photon had a mass $m$, then the Coulomb potential gets replaced by the Yukawa potential (in natural units where $\hbar=c=1$ and Gaussian units): $$ \frac{e^{-mr}}{4\pi r}\ , $$ where $m$ is the mass of the photon. There are experimental bounds for the mass which can be viewed as an upper bound on the validity of Coulomb's law. Here is a fairly recent paper on this topic titled Upper bounds on the photon mass.
As @count-iblis (correctly) points out, you can think of the Compton wavelength of an electron (about $2\times 10^{-12}$m) as a lower bound for the validity of Coulomb's law. However, quantum field theory predicts the nature of the corrections. Given that it is the only way to "derive" Coulomb's law, I would say that it is a matter of taste as to whether Coloumb's law is actually breaking down at this length scale.
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