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Coulomb's Law is fundamental in understanding electrostatic interactions. It describes the force between two charges in space. According to the law, the electrostatic force \(F\) between two point charges is proportional to the product of their magnitudes \(Q\) and inversely proportional to the square of the distance \(r\) between them. Mathematically, this is represented as:\[F = k \frac{{|Q \, Q'|}}{{r^2}}\]where \(k\) is Coulomb's constant, approximately \(8.99 \times 10^9 \, N \, m^2/C^2\). The force is attractive if the charges are of opposite types (positive and negative) and repulsive if they are of the same type.
Understanding this concept is crucial for calculating potential energy in systems with multiple charges, such as protons and electrons in a rectangle, as outlined in the original exercise.

In a system involving protons and electrons, interactions are central to determining the total energy. A proton, being positively charged, and an electron, being negatively charged, will attract each other. This attraction is described by Coulomb's law and results in a potential energy that is negative, indicating a stable, bound state.
In the exercise, when arranging a proton and electron alternately along the rectangle's side, the system's energy is largely influenced by these negative potential energies. Each proton-electron pair contributes a negative amount to the total potential energy of the system, causing a net stabilization.

Charge configuration in an electric field refers to the particular arrangement of charges. The exercise provides two configurations: horizontal and vertical, determined by how protons and electrons are arranged at opposite corners of a rectangle.

• In the horizontal configuration, each proton-electron pair along the longer side \(a\) contributes to a lower potential energy due to increased attraction, minimizing repulsion by ensuring that like charges (protons and protons or electrons and electrons) are further apart.
• In the vertical setup, like charges are closer on the longer side \(b\), potentially increasing energy due to repulsive forces.

Choosing the ideal configuration depends on balancing these attractions and repulsions to achieve lower or higher potential energy, which relates to the work required to assemble the system.

The work-energy theorem connects the work done by forces on a system to its energy change. In an electrostatic context, the work required to assemble charges in a configuration is equal to the change in their electric potential energy.
In the provided exercise, calculating the work involves determining the total potential energy for different charge configurations and comparing these values:

• The horizontal arrangement yields a certain negative potential energy, showing lesser work required due to more favorable proton-electron interactions.
• The vertical arrangement increases potential energy, thus more work or energy input may be required to bring charges into the configuration, especially if \(a > b\).

For some configurations, depending on the distances \(a\) and \(b\), the work done can be positive, implying energy must be added to assemble the system. Using these concepts helps analyze and predict outcomes in various charge distributions.