The electrostatic energy of \( Z \) protons uniformly distributed t...

Описание: The electrostatic energy of \( Z \) protons uniformly distributed throughout a spherical nucleus of radius \( R \) is given by
P
\[
E=\frac{3}{5} \frac{Z(Z-1) e^{2}}{4 \pi \varepsilon_{0} R}
\]
W
The measured masses of the neutron, \( { }_{1}^{1} \mathrm{H},{ }_{7}^{15} \mathrm{~N} \), and \( { }_{8}^{15} \mathrm{O} \) are \( 1.008665 \mathrm{u}, 1.007825 \mathrm{u}, 15.000109 \mathrm{u} \) and \( 15.003065 \mathrm{u} \), respectively. Given that the radii of both the \( { }_{7}^{15} \mathrm{~N} \) and \( { }_{8}^{15} \mathrm{O} \) nuclei are same, \( 1 \mathrm{u}=931.5 \mathrm{MeV} / c^{2} \) ( \( c \) is the speed of light) and \( e^{2} /\left(4 \pi \varepsilon_{0}\right)=1.44 \mathrm{MeV} \) fm. Assuming that the difference between the binding energies of \( { }_{7}^{15} \mathrm{~N} \) and \( { }_{8}^{15} \mathrm{O} \) is purely due to the electrostatic energy, the radius of either of the nuclei is \( \left(1 \mathrm{fm}=10^{-15} \mathrm{~m}\right) \)
(1) \( 2.85 \mathrm{fm} \)
(2) \( 3.03 \mathrm{fm} \)
(3) \( 3.42 \mathrm{fm} \)
(4) \( 3.80 \mathrm{fm} \)
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