Two positive point charges are infinitely far apart. Is it possible, using a finite amount of work, to move them until they're a small distance \(d\) apart?

Understanding Coulomb's law is the cornerstone of electrostatics, playing as significant a role as Newton's law of gravitation does in mechanics. At its essence, it describes the force between two point charges. Here's the heart of Coulomb's law:

• The electrostatic force (\f\(F\f\)) between two charges is directly proportional to the product of the charges (\f\(q_1\f\) and \f\(q_2\f\)) and inversely proportional to the square of the distance (\f\(r\f\)) between them.
• The force is attractive if the charges are of opposite sign and repulsive if they have the same sign.

The formula is often written as \f\(F = k \frac{|q_1 \times q_2|}{r^2}\f\), where \f\(k\f\) denotes Coulomb's constant. Coulomb's law provides a means to calculate the force at any point in space due to a static charge, which becomes the foundation for understanding how work is done in moving charges in an electric field.

In the realm of electrostatics, we often simplify complex problems by treating objects carrying a charge as point charges. A point charge is a hypothetical charge located at a single point in space. This simplification allows us to apply Coulomb's law without delving into the spatial distribution of charges on the objects.

While real charges are distributed over finite volumes, treating them as point charges is a practical approximation especially when the distance between the charges is much larger than the size of the charged objects. This approach simplifies the math and provides a clear path to understanding the interactions between charges in space.

The work-energy principle is a critical concept that links force, work, and energy in physics. In electrostatics, when you move a charge in the presence of an electric field, you're doing work against the electrostatic forces acting on it. This work can be defined as:

• The work (\f\(W\f\)) done by a force (\f\(F\f\)) on a point charge over a displacement (\f\(r\f\)) is the integral of force along the path of motion.

In terms of point charges, if you're moving one charge in the influence of another, the work done is the area under the force versus distance curve. This energy is stored as potential energy, which can be released as kinetic energy if the charges are allowed to move freely. The work-energy principle thus relates the work done to change the energy state of a system of charges.

Integration is a mathematical tool that plays a key role in physics, particularly when dealing with variable forces. In the context of the exercise regarding point charges, integration allows us to sum the infinitesimal amounts of work done over a continuous range of distances. When you integrate the Coulomb force over the distance between two point charges moving from infinity to a finite distance apart, you're essentially adding up all the tiny bits of work done at each infinitesimal segment of the path.

Forces that vary with distance, like the Coulomb force, which changes with the square of the distance, make integration essential. This process allows us to calculate the total work done in moving a charge in a non-uniform electric field, leading us to understand that bringing two charges from an infinite distance to a finite separation requires a finite amount of work, thus highlighting the beauty and application of calculus in physics.