## How much of a charge excess would be needed in the universe to produce detectable non-gravitational motions?

The ratio of the electromagnetic force to the gravitational force between a proton (M) and an electron (m) is just

```             2
K e       G M m
------  =  ------
2          2
R          R

or
2
K e    = G M m

so
2
42        K e
10     = ------
G M m```

approximately. In terms of the densities of these particles in the universe, N and Rho,

```
2  2           2
K e  N  =   G Rho```

so that for a cosmological density of matter of 10 gm/cc you get

```                  -24
N  =  10     protons/cc```

But the cosmological density of matter divided by the mass of a proton is just 10^-5 atoms/cc with is 10^19 times higher than the density needed in charges so that the electrostatic forces contribute as much as the gravitational forces between particles.

So, this means that if in 10^19 atoms in the universe, you have exactly 1 unit of charge ( 1 electronic charge), the net effect is an electrostatic force that equals the gravitational force from these 10^19 atoms! That's how close we have to be to exact neutrality in the universe before non-gravitational forces predominate!

In fact, no non-gravitational forces have ever been confirmed in the dynamics of galaxies or galaxy clusters, so we know that, on average, matter in the cosmos has even less net charge than one part in 10^19.