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Followup to "Why do electron and proton have the same but opposite electric charge?".

It is argued that even a tiny residual charge would result in huge amounts of electricity in bulk matter, everything would be different, etc. I do not find that a convincing answer: suppose $n$ protons plus $n+1$ electrons are neutral. Why wouldn't we also expect there to be $n$ protons to every $n+1$ electrons? That is, there is no bulk matter problem if every $n$'th atom is a negative ion (for hydrogen).

Now, what empirical lower bound can we give for $n$ in that kind of scenario?

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In $\beta$ decay a neutron turns into a proton, an electron and an electro antineutrino. So if the proton and electron charge were not the same either the neutron must originally carried a net charge or the antineutrino must carry a charge.

For the neutrino current limits are reported by the particle data group as less than 10$^{-15}$ of the electron charge. (I'm a bit surprised this limit isn't tighter given how weakly neutrinos interact - oh well).

For the neutron the particle data group report an even tighter limit of less than 10$^{-21}$ of the electron charge.

The particle data group report on the proton gives a figure for $|q_p + q_e|/e$ of 10$^{-21}$.

So basically experiment shows the charge difference is less than 10$^{-21}e$.

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I haven't traced the reference, but the 10^21 limit is said to be based on the "neutrality of matter", which sounds suspiciously like the argument I mentioned. The 10^15 limit OTOH seems to be based on a variety of neutrino experiments, hmm. –  Retarded Potential Mar 1 '13 at 16:34
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suppose n protons plus n+1 electrons are neutral. Why wouldn't we also expect there to be n protons to every n+1 electrons? That is, there is no bulk matter problem if every n'th atom is a negative ion (for hydrogen).

Maybe in a science fiction world, though I doubt the mathematics would hold up to the stress. It would need a new type of solid state that has not been observed: one that is neutral in bulk but when seen microscopically is charged. It would have been observed in microcircuit technologies, to say the least.

In our reality there exist more than charges that we have studied with great accuracy. Atoms, as given by the periodic table of elements, and they have been studied for almost two hundred years. The whole structure depends on having an equal number of protons to the electrons of the atom, and it is not a hypothesis, it is supported by experimental numbers. Thus matter as we know it has equal numbers of protons to the electrons. Note "number", not charge. Now suppose each electron charge to be different from the proton charge by a delta(q), small enough not to bother the electromagnetic solutions for the atoms and be consistent with spectroscopic data. Nevertheless, since one mole of molecules contains something like 10^23 atoms, this tiny charge would add up to enormous charge in bulk. That is the argument, and it is an irrefutable proof in our reality. It is called proof by reduction to the absurd.

Now if you argue, why are there not excess free electrons around, the answer is : because we have not measured them.

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Well didn't I mention this very argument and explain why it's not convincing? Suppose in fact q/delta(q) = Avogadro's number / 100, and suppose in each mole there are 100 excess ions. Can we rule this out? –  Retarded Potential Mar 1 '13 at 16:33
    
have a look at John's answer above for numbers. Below the error anything goes, as will always be the case , If I say take the ocean as an example which has orders of magnitude of avogadro's number, or the ground, which is 0 potential. Would we not have measured a charge ? –  anna v Mar 1 '13 at 16:35
    
I have looked at his answer. You seem stuck on the bulk matter thing, I guess I have not been totally clear: no, we would not measure a charge. Whatever charge difference there is would, on a large scale, be cancelled by the presence of excess ions. If there is a charge difference between protons and electrons then we can also expect there to be slightly more of one or the other. They wouldn't have to be "free", they would happily form ions, like just a bit extra $H_3O^+$ concentration in the ocean. –  Retarded Potential Mar 1 '13 at 16:44
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