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The electrostatic potential energy of Z protons uniformly distributed within a spherical nucleus of radius R is expressed as E = (3/5) * Z(Z−1)e² / (4πε₀R). The masses of neutron, ¹H, ¹⁵₇N, and ¹⁵₈O are 1.008665 u, 1.007825 u, 15.000109 u, and 15.003065 u, respectively. If the radii of ¹⁵₇N and ¹⁵₈O nuclei are identical, and given 1 u = 931.5 MeV/c², e²/(4πε₀) = 1.44 MeV fm, and 1 fm = 10⁻¹⁵ m, calculate the radius of either nucleus, assuming the binding energy difference between ¹⁵₇N and ¹⁵₈O arises solely from electrostatic energy.

1. 2.85 fm
2. 3.03 fm
3. 3.42 fm
4. 3.80 fm

Correct answer: 3.42 fm

Solution

The binding energy difference between ¹⁵₇N and ¹⁵₈O is attributed to the electrostatic potential energy difference. Using the given formula and data, the radius of the nucleus is calculated to be 3.42 fm.

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