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Do particles have a strong nuclear charge in the same way as they have electric charge? If so what unit would be used to measure this? Would it be measured in Coulomb for instance?

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The electromagnetic force couples to electric charge, the gravitational force couples to mass charge and the strong nuclear force couples to color charge. Photons are the exchange bosons in the electromagnetic force, gravitons - if they exist- are the exchange bosons in the gravitational force. The strong nuclear force is mediated by the exchange bosons called gluons.

Do particles have a strong nuclear charge in the same way as they have electric charge?

No, not really, since (as below), all particles (that exist as natural particle states) that take part in the strong nuclear force, will always have a zero net color charge, but can have net electric charge.

Would it be measured in Coulomb for instance?

Color charge is not really analogous to electric charge in this sense. Electric charge has one value (which is some number multiplied by the electron charge). But color charge can be one of six possible values (red, blue, green, and antired, antiblue, antigreen). These are quantum numbers we assign to the six different flavor of quarks which combine to form hadrons. While particles like $\pi^-$, K$^-$, protons, $\Delta^-$, $\Delta^{++}$ etc., can have a net electric charge, all particles containing quarks must always have zero, or neutral, net color charge due to quark confinement

For more about this, see quantum chromodynamics which describes the behavior of the strong nuclear force and strong interactions.

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