Two equal but opposite-sign point charges of \(3 \mu \mathrm{C}\) are held \(x\) meters apart by a spring that provides a repulsive force given by \(F_{s p}=12(0.5-x) \mathrm{N}\). Without any force of attraction, the spring would be fully extended to \(0.5 \mathrm{~m}\). (a) Determine the charge separation. (b) What is the dipole moment?

Imagine two kids playing with magnets: when they hold opposite poles close, they feel a pull, but with the same poles, they push each other away. This is somewhat analogous to the behavior of charged particles, which is described by an important principle in physics known as Coulomb's Law. This law was named after Charles-Augustin de Coulomb, who in the 18th century discovered that the force between two point charges is directly proportional to the product of their charges and inversely proportional to the square of the distance between them.

Coulomb's Law is mathematically expressed as: \[ F = k \frac{q_1 q_2}{r^2} \] where:

• \( F \) is the electrostatic force between charges,
• \( k \) is Coulomb's constant, approximately \(8.99 \times 10^9 N m^2 C^{-2}\),
• \( q_1 \) and \( q_2 \) are the magnitudes of the charges,
• \( r \) is the distance between the charges.

To solve problems using Coulomb’s Law, it often involves equating this electrostatic force to another force in play, like a spring force in our example, to find unknown quantities like the distance between charges or the amount of charge itself. Understanding this law is crucial for predicting how charged particles will interact.

The electrostatic force is the interaction force between static (non-moving) electric charges. It manifests as either an attractive or repulsive force, depending on the nature of the charges involved. Similar charges repel each other, while opposite charges attract, much like the familiar behavior of magnetic poles.

In the context of our textbook exercise, we're looking at a scenario where an electrostatic force competes with a mechanical spring force. This concept highlights a real-world application where balancing forces can maintain a system in equilibrium. Understanding electrostatic force is not only crucial for solving physics problems but also for grasping how fundamental principles play out in everyday technology, such as the functioning of computer chips, the behavior of dust particles in air purifiers, and the bonding in materials.

Think of a bar magnet with its north and south poles. This bar magnet can be represented by an electrical analog known as an electric dipole, which consists of two equal and opposite charges separated by a distance. The dipole moment is a vector quantity that provides a measure of the separation of positive and negative charges within a system. It is mathematically defined by the product of the charge magnitude and the distance between the charges: \[ p = qr \]

The direction of the dipole moment points from the negative to the positive charge. It's a useful concept when dealing with molecules, as many of them have dipole moments due to the uneven distribution of electrons. This affects how they interact with each other and with external fields. In the example provided, we calculate the dipole moment of point charges to understand the physical nature of the system better and predict its behavior when subjected to external electric fields.