Coulomb's force law states that the force between two charged particles is: \(F=k \frac{Q q}{r^{2}} .\) Use this expression to determine the potential energy function.

Understanding the interactions between charged particles is crucial in physics, and Coulomb's force law is the foundational principle describing these interactions. Imagine two particles, each with a charge; one particle is positively charged with an amount Q, and the other is negatively charged with q. According to this law, these particles exert a force on each other that is directly proportional to the product of their charges and inversely proportional to the square of the distance between them. Mathematically, this relationship is expressed as
\( F = k \frac{Qq}{r^2} \),
where F represents the force, k is the Coulomb's constant, and r is the distance between the particles. When the distance between the charges r decreases, the force of attraction or repulsion between them significantly increases. Conversely, as the particles move farther apart, the force diminishes rapidly. The force is also affected by the medium in which the charges reside, but the relationship's core remains the same in a vacuum. Coulomb's force law is pivotal for explaining phenomena in various fields like electrostatics, electronics, and even chemistry where the concept is used to understand atomic and molecular interactions.

Just as gravitational potential energy gives us a sense of how much work is done against gravity, electric potential energy serves a similar role but in the realm of electrostatic forces. It represents the work done in bringing a charge q from infinity to a certain point within the electric field created by another charge Q. The potential energy, denoted as U, is inherently tied to Coulomb's force law. As the charges do work on or against each other, the system's potential energy changes. The relationship is described by the equation
\( U = -k \frac{Qq}{r} + C \),
where C is the constant of integration, essentially a reference point for measuring potential energy. This equation allows us to understand that potential energy is negative when dealing with attractive forces between oppositely charged particles—it costs energy to separate them. The concept gets more interesting when considering systems with multiple charges, where the entire system's potential energy becomes the sum of all the pairwise potential energies.

One of the powerful mathematical tools in physics is integration, which we use to find quantities that are not directly observable. For instance, if we know the force acting on an object as a function of position, how do we determine the work done by the force? Or, in the case of our current discussion, how do we move from knowing the force between charges to finding the potential energy? This is where integration comes into play.Using calculus, we can integrate the force over a distance to obtain potential energy, as shown in the solution to the exercise. Mathematically, to get from the derivative form F = -dU/dr to an expression for U, we need to integrate the force with respect to r. The minus sign in the derivative indicates that as potential energy increases, the force does work in the opposite direction. In physics, integration helps us extrapolate from small, infinitesimal changes to the whole picture, giving us insight into the overall behavior of physical systems. It's a cornerstone of understanding dynamics in systems ranging from atomic scales to astronomical distances, making it an indispensable part of a physicist's toolkit.