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The relative strength of gravity and electromagnetic forces is obvious — stand on a sheet of paper, and even with the whole of Earth pulling, your motion is stopped by the electric fields inside that sheet of paper.

Great.

This is often phrased as "gravity is the weakest of all forces" or some variant thereof. This seems equivalent to saying "the mass-to-charge ratio of fundamental particles is such that charge dominates".

My problem with phrasing it as "gravity is the weakest" is that different particles have different mass-to-charge ratios: according to this chart on Wikipedia, mass-to-charge ratios vary by over 6 orders of magnitude, even if you exclude massive neutral particles like the neutrinos where gravity is infinitely stronger.

With such a wide range of mass-to-charge ratios, why is the question usually phrased as being about the strength of forces? Why invoke extra dimensions for gravity to leak into (for example) when one also needs to explain an extra factor of -5.588×10^6 between top quarks and electrons? (I'm assuming that infinities would get explained away differently, but perhaps not?)

(Hope this isn't a duplicate, my searching mainly showed a lot of "why is gravity weak?" type questions, which isn't what I'm curious about — I want to know why phrasing it like that is seen as the best way of thinking about the problem).

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Comparing forces for particles brings us to the quantum mechanical framework, the underlying framework of nature.

Forces, quantum mechanically are represented as exchanges of particles in Feynman diagrams, and the probability of the interaction happening follows mathematically from that.

For example here is the Feynman diagram for same charge repulsion:

e-e-scatter

A physics student at graduate level, from this diagram follows a recipe that leads to a computable integral, and the calculation will show the repulsion of like charges.

The strength of the interaction enters at the two vertices of the diagram and is given by the coupling constant.

Here are the coupling constants of the rest of the interactions.

em

weak

strong

gravity

It is in this sense that gravity is compared as the weakest of all forces.

The diagram of the two electrons could represent the exchange of a graviton, but when compared to the electromagnetic exchange of a photon , the probability of the gravitational interaction is tiny : in the integral the coupling constants enter multiplicatively and the calculational recipe raises them to the fourth power .

This is the framework where all known forces are compared. Of course gravity is not yet quantized rigorously, only effective theories exist, but it is within this theory that the statement of gravity being weakest is clearly evident.

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The reason one refers to gravity as "weak," is because, it is. Even if different particles have mass-to-charge ratios of 6 orders of magnitude, the ratio of the charge constant (C) to the gravitational constant (G) is 20 orders of magnitude!

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    $\begingroup$ These constants have different dimensions, so comparing them seems pretty meaningless. In SI units the ratio between Coulombs constant $k_e$ and the gravitational constant $G$ is $k_e/G \approx 1.3 \times 10^{20} \, \mathrm{kg}^2/\mathrm{C}^2$. But in Planck units $G = k_e = 1$ so the ratio is $k_e/G = 1$. $\endgroup$ – Olof Dec 16 '15 at 8:57
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There is a smallest possible charge an electromagnetic field exerts a force. There is no elementary mass we could compare the gravitational force. But if you think of subatomic particles (which are somehow the smallest units), like electrons and protons, the electromagnetic force exceeds the gravitational force by far (even at very different masses). The weak and strong nuclear force only act on subatomic particles, so that's the probably reason you compare all forces on this subatomic scale.

You can continue with different mass-to-charge ratios, where the electromagnetic forces are still dominating over a large range (charged atoms, molecules, ...). When you reach macroscopic objects, you attain a balance between the forces (e.g. the balloon stick to the ceiling due to a static charge) and end up at larger object where the gravitational force is dominating (no electrostatic charge will keep you at a ceiling).

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You might argue that it is comparatively weak by virtue of being utterly negligible on the quantum scale, unlike the other three forces. If, say, EM was apparent at that scale, but in some strange world neither the strong nor weak nuclear forces operated (like gravity), then we might describe EM as a 'strong' force, rather than gravity as a 'weak' one.

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