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Electric potential energy is the energy a charge possesses due to its position in an electric field. In simpler terms, imagine you have a set of charged particles. The electric potential energy is like the energy stored because of how these particles are arranged. If they are very far from each other, the energy is low. But when brought close, especially at a specific distance, the energy can become significantly higher. This is because charges interact with each other. Opposite charges attract while like charges repel each other.
In the context of the exercise, each proton is a charge, and we're interested in the energy needed to position them at a certain distance. When calculating electric potential energy, we use the formula:

• \[ U = \frac{k \cdot q_1 \cdot q_2}{r} \]

where:

• \( k \) is Coulomb's constant,
• \( q_1 \) and \( q_2 \) are the charges, and
• \( r \) is the separation distance between the charges.

In the case of three protons, we have multiple pairwise contributions to consider and sum up. This cumulative effect gives us the total work done to achieve the desired configuration.

Coulomb's constant is a crucial part of understanding electrostatic forces and potential energy between charges. It is a proportionality factor in Coulomb's law, which describes the force of interaction between two point charges. The standard value of Coulomb's constant is:

• \( k = 8.99 \times 10^9 \, \text{N} \cdot \text{m}^2/\text{C}^2 \)

This constant allows us to calculate how strongly two charges will push or pull each other when they are at a given distance. It's derived from fundamental properties of electric fields and relates the force experienced by charges to their distance and magnitudes.
In electrostatics problems like the one we are discussing, Coulomb's constant is essential for calculating electrostatic potential energy. Without it, we wouldn't be able to quantify the amount of work required to assemble a system of charges in space. Coulomb's constant essentially bridges the gap between abstract charge interactions and measurable forces and energies.

In physics, work is a measure of energy transfer. Specifically, it is the energy required to move a force over a distance. In the context of electrostatics, the concept of work relates to moving charges within an electric field, which changes their electric potential energy.
When working with electrostatics, the work involved is quite specific: it's the amount of energy needed to assemble a group of charges from infinity to a certain configuration. It's important to note that work only happens when there is a change involved, such as moving the charges closer together in this scenario. The equation used for work from potential energy is:

• \( W = \Delta U \)

where \( \Delta U \) represents the change in potential energy. In our exercise, we calculated the total potential energy required to bring three protons together, at specific distances, resulting in the total work done in the system. The work here reflects how energy is transformed and stored when moving these protons into a compact formation like that found in atomic nuclei.