GATE 2021 PH – Question 58
The Coulomb contribution to a nucleus’s binding energy is 18.432 MeV. The uniform spherical charge distribution has radius 3 fm. Find the corresponding atomic number, rounded to the nearest integer. Given $e^2/(4\pi\epsilon_0)=1.44$ MeV fm.
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Correct answer: 8 to 8 OR 9 to 9
Explanation
The classical electrostatic self-energy of a uniformly charged sphere is $E_C=(3/5)Z^2[e^2/(4\pi\epsilon_0R)]$. Substitution gives $18.432=0.288Z^2$, hence $Z=8$. A nuclear pair-counting convention replaces $Z^2$ by $Z(Z-1)$; it gives $Z=(1+\sqrt{257})/2=8.516$, rounding to 9. The official key explicitly accepts either 8 or 9, and both isolated alternatives are preserved rather than an interval containing arbitrary intermediate values.