Why does electromagnetism work on quantum scales and gravity doesn't?

Not entirely related to space/astronomy. But why does Coulomb’s law of electrostatic force still work at the quantum level while it’s counterpart Newton’s gravitational force law doesn’t?

Physics questions are also accepted!

Fundamentally, the reason gravity breaks down at the quantum level is because the strength of the gravitational force is much, much weaker than the electromagnetic force. One of the largest outstanding questions in physics is to try and figure out why gravity is so weak, but so far all we’ve managed to do is very precisely measure how much weaker it is. We have a real-world sense of gravity’s weakness, if you stop and think about it - for all gravity holds us to our chairs, if we set a glass of water down on the desk, we expect the glass to sit on top of the desk, not oscillate back and forth until the glass and desk reach a gravitational balance - that’s the power of the electromagnetic force of the atoms in the desk repelling the atoms in the glass.

Once you start going down to quantum scales, you’re dealing almost entirely with fundamental particles of matter - things like protons and electrons, or quarks if you’re going even smaller. If you want to calculate the gravitational force between any two objects, the two pieces of information you’re going to need are the masses of the two things you care about, and how far apart they are from each other. One can then plug in the masses and distances into this handy equation – F=G(m1 x m2)/r^2, where m1 and m2 are the masses of your objects, r^2 is the distance between them squared, and G is the gravitational constant – and out will pop the force the two objects are exerting on each other.

The electromagnetic force works in a very similar way - even the equation looks similar to the equation for the gravitational force. The two things you need to know about for electromagnetism are the charge and the distance - the mass doesn’t play a role here. Our electric force is then written F=k(q1 x q2)/r^2. q1 and q2 are the charges of the objects we’re interested in, r is still the distance, and k is the Coulomb constant. But between the two, we have a very similar looking equation - multiply together the charge or the mass, divide by the distance squared, and scale by some constant.

Here comes the first major difference. The scaling for the electric force is 8.988 x 10^9 N m^2 / C^2. The gravitational constant is 6.674x10^(-11) N m^2 / kg^2. This is twenty orders of magnitude different in scaling. However, one of them is per square kilogram and the other per square Coulomb, and you can’t really compare things with different units (although it can give us a hint as to the direction this is going), so the best way to compare is by doing a calculation.

Let’s assume that two protons are sitting 1 mm away from each other. What’s more important, the gravitational force, or the electromagnetic force?

To work out the strength of the electromagnetic force, we need the charge of a proton (1.6021765 x 10^(-19) Coulombs), a millimeter (0.001 meters), and the Coulomb constant (8.98755 x 10^9 N (m/C)^2). Plugging all that into our equation gives us a final electromagnetic force of Fc=2.307 x 10^(-22) N. Not very large.

However, the same calculation for gravity gives us the following. Plugging in the mass of a proton (1.672621*10^(-27) kg), a millimeter again being 0.001 meters, and the gravitational constant being 6.67384x10^(-11) N (m/kg)^2, we arrive at a gravitational force of Fg=1.867 * 10^(-58) N.

Forces of 10^-22 and 10^-58 Newtons are both extremely small numbers, but we can see that the gravitational force is 36 orders of magnitude weaker than the electric repulsion force. If we were to push our two protons closer together, both forces would get stronger, but since both of these equations use the distance in the same way, the forces would scale up at the same rate, and gravity won’t be able to catch up.

So the problem here for gravity is a combination of the fact that the gravitational constant is twenty orders of magnitude smaller than Coulomb’s constant, and the fact that the mass of any of these particles is much, much smaller than the charge that they bear. The only way to make this better for gravity is to make the mass of the objects you’re looking at much larger; going further down the quantum scale will only make gravity shrink even further into irrelevance.

Something here unclear, or have your own question? Feel free to ask!