The energy required to construct a uniformly charged sphere...

The energy required to construct a uniformly charged sphere of total charge $Q$ and radius $R$ is $U=3 k_{c} Q^{2} / 5 R$, where $k,$ is the Coulomb constant (see Problem 66). Assume a ${ }^{40}$ Ca nucleus contains 20 protons uniformly distributed in a spherical volume. (a) How much energy is required to counter their electrical repulsion according to the above equation? (b) Calculate the binding energy of ${ }^{40} \mathrm{Ca}$. (c) Explain what you can conclude from comparing the result of part (b) with that of part (a).

The energy required to construct a uniformly charged sphere of total charge $Q$ and radius $R$ is $U=3 k_{c} Q^{2} / 5 R$, where $k,$ is the Coulomb constant (see Problem 66). Assume a ${ }^{40}$ Ca nucleus contains 20 protons uniformly distributed in a spherical volume. (a) How much energy is required to counter their electrical repulsion according to the above equation?
(b) Calculate the binding energy of ${ }^{40} \mathrm{Ca}$. (c) Explain what you can conclude from comparing the result of part (b) with that of part (a).

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Key Concepts

Interplay Between Strong Nuclear Force and Coulomb Repulsion

This concept highlights the balance between the short-range, highly attractive strong nuclear force and the long-range electrostatic repulsion among protons. By comparing the Coulomb energy (the energy required to counteract the repulsive forces) with the overall binding energy of the nucleus, one can infer how the strong force not only overcomes the repulsion between protons but also provides the cohesion necessary to maintain nuclear stability.

Coulomb Energy of a Uniformly Charged Sphere

This concept involves calculating the electrostatic energy stored in a sphere where charge is uniformly distributed. The formula U = 3 k_c Q² / (5 R) is used to compute the work required to assemble the charge distribution against the repulsive forces between like charges. In the context of a nucleus, it helps estimate the energy contribution from the repulsion among protons.

Nuclear Binding Energy

Nuclear binding energy is the energy released when a nucleus is assembled from its individual nucleons, or conversely, the energy required to break the nucleus apart into free protons and neutrons. This concept is central to understanding the stability of a nucleus, as it reflects the net effect of the attractive strong nuclear force overcoming the repulsive electrostatic forces among protons.

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