# Precision of Coulomb's law

Up to which precision has the coulomb law proven to be true? I.e. if you have two electrons in a vacuum chamber, 5 meters appart, have the third order terms been ruled out? Are there any theoretical limits to measure the precision ( Planck's constant?). Obviously there are practical limitations ( imperfect vacuum, cosmic rays, vacuum fluctuation). Still, does anyone know what was the smallest amount ever correctly predicted by that law?

Edit : Summary

On the high end of the energy spectrum a precision of 10^-16 has been shown ( 42 years ago )

For electron point charges at large distances the law might brake down due to practical reasons.

For moving particles QED gives a correction to the law: http://arxiv.org/abs/1111.2303

-
Going to wikipedia and typing in "Inverse square law" directly leads on to this paper. Also, maybe someone wants to say something on QED implications alla vacuum polarization. – NikolajK May 15 '13 at 13:40
I believe this answer physics.stackexchange.com/a/64375/24124 addresses your question as well. – firtree May 15 '13 at 14:19
related : physics.stackexchange.com/q/62469 – Mostafa May 15 '13 at 14:23
@firtree: I don't think it's the same question, and I don't think the answer addresses this question. This one is about small corrections under normal conditions. The other is about conditions under which it becomes a poor approximation. – Ben Crowell May 15 '13 at 19:25
@BenCrowell Corrections are made to the formula, so it might be tested either with high presicion at normal conditions or with lower presicion at some limit conditions. Latter is often easier, and it is often used to deduce upper limits on formula modifications. For example, Feynman deduced upper limits on photon mass from the cosmic static fields - this is far from the lab scale as well. I know this is model-dependent, but I don't see this is off-topic. – firtree May 15 '13 at 19:38

Quoting from my copy of the 2nd edition of Jackson's book on Classical Electrodynamics, section 1.2:

Assume that the force varies as $1/r^{2+\epsilon}$ and quote a value or limit for $\epsilon$. [...] The original experiment with concentric spheres by Cavendish in 1772 gave an upper limit on $\epsilon$ of $\left| \epsilon \right| \le 0.02$.

followed a bit later by

Williams, Fakker, and Hill [... gave] a limit of $\epsilon \le (2.7 \pm 3.1) \times 10^{-16}$.

That book was first published in 1975, so presumably there has been some progress in the mean time.

-
The 2013 edition of Purcell and Morin still gives a reference to the 1971 Williams paper, so I believe that's still the tightest bound by that technique. – Ben Crowell May 15 '13 at 19:09
here is a link without pay wall : moodle.davidson.edu/moodle2/pluginfile.php/13924/mod_resource/… – Anno2001 May 15 '13 at 20:55
follow-up question: how-the-inverse-square-law-in-electrodynamics-is-related-to-photon-mass physics.stackexchange.com/q/62469 – Anno2001 May 15 '13 at 22:42
while this experiment proves the law to be very precise for high energies, can someone give an estimate of when the law becomes inapplicable for very small charges (e) / large distances? – Anno2001 May 15 '13 at 22:53
summary: there is (currently) no need/hope for any correction like Einstein gave to Newtons law $F = -GM/r^2 -3GMh^2/(c^2r^4)$ , in its most primitive form – Anno2001 May 15 '13 at 23:01

Jinawee and dmckee have already given answers describing the bounds from the spherical capacitor technique.

A different, and more model-dependent, approach is to build and test empirically a theory in which the photon has nonzero mass. There are some theoretical difficulties involved, e.g., local gauge invariance is broken, and it's not trivial to show that you can still have a conserved current. If the mass is nonzero, then the Coulomb's force law would have an exponential decay in it, with a very long range.

The most widely accepted upper limit on the photon mass are from Goldhaber 1971 and Davis 1975. Lakes 1998 is tighter, but I believe more model-dependent. A more controversial and much tighter limit is given by Luo 2003. Davis's limit is $8\times10^{-52}$ kg, corresponding to a range on the order of $10^9$ m.

Goldhaber and Nieto, "Terrestrial and Extraterrestrial Limits on The Photon Mass," Rev. Mod. Phys. 43 (1971) 277–296

Davis, PRL 35 (1975) 1402

R.S. Lakes, "Experimental limits on the photon mass and cosmic magnetic vector potential", Physical Review Letters , 1998, 80, 1826-1829, http://silver.neep.wisc.edu/~lakes/mu.html

Luo et al., “New Experimental Limit on the Photon Rest Mass with a Rotating Torsion Balance”, Phys. Rev. Lett, 90, no. 8, 081801 (2003)

-

I know that the inverse square law has been verified at least 1 part in $10^{16}$.

Feynman Lectures said something about that.

-