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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?

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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.

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I think there's an item missing from your list, which is "sparking the vacuum." For example, if we could form an atomic nucleus with $Z>137$ (137 being the inverse of the fine structure constant), then we would get pairs of virtual particles being promoted to real particles. This is not completely inaccessible to experiment, although the relevant nuclear systems would be unbound. But my field theory is weak, and maybe what I'm saying is actually covered under one of the ideas you've already listed. –  Ben Crowell May 13 '13 at 1:59
    
QED corrections to Colombo law can be worked out analytically, at least to the first order in $\alpha $. –  Slaviks May 13 '13 at 2:59
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here is a link for @Slaviks QED corrections arxiv.org/abs/1111.2303 –  Anno2001 May 15 '13 at 23:15

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