electromagnetism - What are the limits of applicability of Coulomb's Law?

Coulomb's law is formally parallel to Newton's Law of Universal Gravitation, which is known to give way to General Relativity for very large masses. Does Coulomb's Law have any similar limits of applicability? What physics takes over then? Does the law hold for very large and very small charges?

Answer

On the upper end, Coulomb's law has not been observed to break for any large collection of charge that can be put together. In principle, if you tried to put more and more charge together then there would be a lot of energy stored in the field, and if the mass equivalent of this energy density got too high, there would be general relativistic effects to consider. In practice, unless you're dealing with a charged black hole, the charge distribution will tear itself apart many orders of magnitude before that.

There is no corresponding breakdown as such for "very small" charges, on the other hand, because charge is quantized: no free charge smaller than the electron charge, $e=1.6\times10^{-19}\textrm{ C}$, has ever been observed.

Coulomb's law also breaks down if charges are moving, and particularly if they're moving fast (comparably to the speed of light) or there are charge movements in an otherwise neutral conductor. This is fixed by extending the electrostatic case into the full electromagnetic theory, as developed by Maxwell, which is fully compatible with special relativity.

In the domain of the small, the electrostatic force remains unaltered for standard quantum mechanics. It does change for the relativistic case, in which case you should use quantum electrodynamics (QED), which describes a bunch of nonclassical phenomena that occur when charged elementary particles go fast.

There is, however, one very interesting application of QED to stationary charges, and it happens in the short distance limit: as you get in closer, the electron looks like it has more charge, and the force goes up faster than $1/r^2$. This is called charge screening and results from a cloud of virtual particle pairs that momentarily pop into and out of existence.