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I've often heard it said that gravity is much weaker than electromagnetism, and after looking at several questions on SE, I feel that I've got at least a qualitative handle on the concept -- gravity only appears to be stronger macroscopically because positive and negative charges in most everyday matter cancel each other out almost perfectly, leading to low Coulomb forces.

But I've also heard it said that gravity is "thirty orders of magnitude" weaker than EM. How is that figure arrived at, since charge and mass are pretty fundamentally different, at least from what I've learned?

This differs from this question because I'm looking for a quantitative and hopefully more complete understanding of the issue than the answers to that question provided. The closest any of those answers come is vaguely mentioning a coupling constant, without really explaining its nature mathematically.

Qmechanic
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Why-Seven-Six
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    You could, for example, consider a proton and an electron separated by a distance $r$. Calculate the gravitational attraction between them and the electrical force. Now divide those results. – Wood Jan 07 '16 at 03:09
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    But isn't that sort of arbitrary? Why not two protons, or two electrons? Or an electron and a positron? Sure, the direction of the force might change in some of those scenarios (even then, not in the third), but looking only at the magnitude solves that problem -- and the force of gravity would differ massively between them, while the electrostatic force('s magnitude) should remain the same. – Why-Seven-Six Jan 07 '16 at 03:11
  • @Why-Seven-Six I don't know if there's a "standard" way of quantifying it (there probably is). But any of the examples you mentioned would give a ratio greater than 30 orders of magnitude. The electron-proton gives 39. – Wood Jan 07 '16 at 03:18
  • @HDE 226868 -- I saw that question, and asked this one in the hopes of getting a more specific and useful answer. The point of my question is to get a quantitative and hopefully more complete understanding of the issue than the answers to that question provided. The closest any of those answers come is vaguely mentioning a coupling constant, without really explaining its nature mathematically. – Why-Seven-Six Jan 07 '16 at 03:20
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    @Why-Seven-Six On a second thought, using a proton was a bad example, since the down quark in the proton cancels part of the charge of the up quarks. A positron-electron (or electron-electron) makes more sense, and the ratio is $4×10^{42}$. – Wood Jan 07 '16 at 03:27
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    Hi Why-Seven-Six. Despite your pleas otherwise, your question is blatantly a duplicate of What does it mean to say “Gravity is the weakest of the forces”?. If you think the existing answers are inadequate the appropriate way forward is to build up a bit more rep then place a bounty on the previous question. – John Rennie Jan 07 '16 at 07:22
  • Related: https://physics.stackexchange.com/q/755038/226902 – Quillo Mar 13 '23 at 09:05

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