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The reason that protons are positive and neutrons have no charge is owed to the fact that quarks, which make up these particles, do not have integral charge. The charge of an up quark is +2/3, and the charge of a down quark is -1/3. When the charges of the quarks are added for the particles, we get the following:

Proton = up quark + up quark + down quark = 2/3 + 2/3 - 1/3 = 4/3 - 1/3 = 3/3 = +1

Neutron = up quark + down quark + down quark = 2/3 - 1/3 - 1/3 = 2/3 - 2/3 = 0 (zero)

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Q: If protons are positive and are made of two up quarks and one down quark why do neutrons have no charge but are made of two down quarks and one up quark?
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