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Three quarks for you!

Electricity and Magnetism Level 5 220 points

Neutrons and protons, which make up the nuclei of atoms, are themselves made up of smaller particles called quarks. A proton is made of two "up" quarks and one "down" quark while a neutron is made of two down quarks and one up quark. Quarks have an interesting property called confinement, in that no single quark can be isolated and observed. We can still measure their properties such as mass in accelerators, but there's a simple way to get an estimate of mass differences between quarks using elementary electromagnetism and the masses of the neutron and proton. How much heavier is the down quark than the up quark in kilograms, if the the mass difference between the neutron and proton is due to the different total quark masses and the different electrostatic energy?

To convert energy into mass and vice versa, use \(E=mc^2\).

Details and assumptions

  • The neutron is heavier than the proton by \(2.306 \times 10^{-30}~kg\).
  • The electrostatic constant is \(K=9 \times 10^9~Nm^2/C^2\).
  • The speed of light is \(3 \times 10^8\) m/s.
  • The down quark has a charge \(q_d=e/3\) and the up quark has a charge \(q_u=-2e/3\), where \(e\) is the charge on an electron, \(e=-1.602 \times 10^{-19}\) Coulombs.
  • Quarks can be treated as point charges and the distance between quarks in a proton or neutron is about \(r=10^{-15}~m\).
  • The real answer is around 2 to 3 \(\times 10^{-30}~kg\), so as you will see we're going to get pretty close.

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