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The masses of protons and neutrons are different. Suppose a proton is a sphere with a uniformly positive charge distribution. Can the mass difference between protons and neutrons be due to the electrical potential energy of the protons? Justify your answer with a simple calculation.

I do not really know the relationship between potential energy and mass difference. Isn't the difference in mass of protons and neutrons due to their quarks? (the neutron is made of two down quarks and an up quark and the proton of two up quarks and a down quark.)

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The reason for mass difference is because a proton contains 2 down quarks and 1 up quark and a neutron contains 2 up quarks and 1 down quarks(you were right about that).And since an charge of up quark is $$+\frac{2}{3}$$ and the charge of a down quark is $$-\frac{1}{3}$$ they are totally different particles so they'll have different masses (even though they fall under the same sub group). If you look up on the standard model you'd see that the mass of a down quark is almost double of that of an up quark. So yeah the mass difference is because of the deference in mass of quarks. Besides in real life a proton shouldn't have a constant mass... quark and antiquark pairs pop in and out of existence inside a proton so that mass of a proton should be constantly changing(conservation of mass doesn't apply since at the quantum level mass and energy conservation is not obeyed. Instead they obey the law of conservation of mass-energy(the mass equivalent of energy + mass))

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    $\begingroup$ "Besides in real life a proton shouldn't have a constant mass." this is not correct. Bound states of quantum mechanical particles have a definite energy level and the whole branch of lattice QCD has developed in order to calculate these levels for hadrons . see arxiv.org/pdf/1203.4789.pdf $\endgroup$
    – anna v
    Sep 29, 2020 at 12:21
  • $\begingroup$ but quark antiquark pairs pop in and out of existence...shouldn't that increase or decrease the mass at times $\endgroup$ Sep 29, 2020 at 12:38
  • $\begingroup$ those are virtual pairs , i.e. diagrams within integrals. Lattice QCD was worked on to be able to calculate over their infinite multiplicity. $\endgroup$
    – anna v
    Sep 29, 2020 at 13:33
  • $\begingroup$ ohhhhhhhhhh okay $\endgroup$ Sep 29, 2020 at 14:28
  • $\begingroup$ I got the potential energy of this proton. It was almost 1 Mev/c^2 . The mass difference of proton and neutron is almost 1 Mev/c^2 , but ignoring the negative sign. In fact, it is a negative sign in the mass difference that indicates that the mass of a proton is less than that of a neutron. Can this be justified? $\endgroup$
    – Michael
    Oct 1, 2020 at 17:21

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