How strong is the electric force between quarks in a proton?

Question

Does the electron force between quarks in a proton still depend on the inverse of distance squared, and are charges still of the order of $1.6 \times 10^{-19} \ \text{C}$?

Answer 1

If I’m not mistaken, the electric field is always linear (and inverse square thereby) below the Schwinger limit. After that limit at an electric field strength around $1.32\times10^{18}$ volts per meter (1.32 volts per attometer), the field becomes nonlinear and deviations from an inverse-square law might apply.

Given that protons are around 880 attometers wide and the charge of a down quark is $\frac{1}{3}q_e$, the electric field produced by a down quark at sub-proton distances is approximately $-6.191\times10^{20}$ volts per meter - much higher than the Schwinger limit. I used the standard $E=k\frac{Q}{R^2}$ formula for that, which is of course the one used for linear electric fields, but it at least illustrates that the field strength is extremely high and that the field is probably nonlinear within the proton itself.

The charges of the quarks don’t care about velocity or distance scale - charge is Lorentz invariant.

Answer 2

Approximately, yes the force still depends on the inverse distance squared and the charges are still of the order of $1.6 \times 10^{-19}$ C. The caveat is that quantum mechanics rules at this scale, which makes the interactions very complicated and the particles do not behave much like point particles.