The electric potential energy of a uniform sphere of charge...

The electric potential energy of a uniform sphere of charge $q$ and radius $r$ is given by $$U=\frac{3 q^{2}}{20 \pi \varepsilon_{0} r}$$ (a) Does the energy represent a tendency for the sphere to bind to-gether or blow apart? The nuclide $^{239} \mathrm{Pu}$ is spherical with radius 6.64fim. For this nuclide, what are (b) the electric potential energy $U$ ac- cording to the equation, (c) the clectric potential cnergy per proton, and (d) the electric potential energy per nucleon? The binding energy per nucleon is 7.56 $\mathrm{MeV}$ (e) Why is the nuclide bound so well when the answers to $(\mathrm{c})$ and (d) are large and positive?

The electric potential energy of a uniform sphere of charge $q$ and radius $r$ is given by
$$U=\frac{3 q^{2}}{20 \pi \varepsilon_{0} r}$$ (a) Does the energy represent a tendency for the sphere to bind to-gether or blow apart? The nuclide $^{239} \mathrm{Pu}$ is spherical with radius 6.64fim. For this nuclide, what are (b) the electric potential energy $U$ ac-
cording to the equation, (c) the clectric potential cnergy per proton,
and (d) the electric potential energy per nucleon? The binding energy per nucleon is 7.56 $\mathrm{MeV}$ (e) Why is the nuclide bound so well
when the answers to $(\mathrm{c})$ and (d) are large and positive?

Key Concepts

Strong Nuclear Force

The strong nuclear force is a fundamental force that acts over very short ranges, providing an attractive interaction between nucleons (protons and neutrons) in the nucleus. Its strength is sufficient to overcome the Coulomb repulsion between protons, thus playing a decisive role in holding the nucleus together and ensuring its stability.

Coulomb Repulsion

Coulomb repulsion refers to the force between like charges that pushes them apart, as described by Coulomb's law. In systems where all charges are of the same sign, this repulsive interaction is a key factor that tends to destabilize the system, as it creates a natural tendency for the charges to separate.

Nuclear Binding Energy

Nuclear binding energy is the energy required to disassemble a nucleus into its individual protons and neutrons. It is a measure of the stability of a nucleus; higher binding energy per nucleon indicates a more stable nucleus. This concept is crucial in understanding why, despite repulsive forces present among protons, nuclei can still be tightly bound.

Electric Potential Energy

Electric potential energy is the energy stored in a system of charges due to their positions relative to one another. It quantifies how much work is required to assemble the charge configuration from infinity, with a positive value typically indicating that work must be done against the natural repulsive forces between like charges.

Uniformly Charged Sphere

A uniformly charged sphere is an idealized model where the total charge is distributed evenly throughout a spherical volume. This concept simplifies the calculation of electric potential and energy, leading to specific formulas that account for the distribution, such as the one used to determine the potential energy stored in the sphere.

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