Show that the units of \(\epsilon_{0}\) may be written as \(\mathrm{F} / \mathrm{m}\).

Coulomb's Law is a foundational principle in electrostatics. It quantifies the electrostatic force between two point charges and specifies how the force changes with the distance between them. Expressing the law, we have the formula \( F = k \frac{{q1 \cdot q2}}{{r^2}} \), where \( F \) represents the force, \( q1 \) and \( q2 \) are the magnitudes of the charges, \( r \) is the distance between the centers of the two charges, and \( k \) is the Coulomb’s constant. Understanding this law is essential as it also leads us to determine the nature and magnitude of electrostatic forces that charged particles exert on each other.

One important aspect of Coulomb's Law is the inverse-square relation—highlighting that the force diminishes with the square of the distance. As the distance \( r \) doubles, the force \( F \) decreases to a quarter of its original value, which is an essential characteristic for calculations involving electrostatic interactions. This relationship is pivotal across various applications, playing a crucial role in understanding and designing electrical circuits, sensors, and even in the computation of intermolecular forces in chemistry and physics.

Electrostatic force is the force of attraction or repulsion that acts between particles that are charged. This force is central to the study of electromagnetism and plays a critical role in understanding how charged particles interact within an electric field. According to Coulomb's Law, this force can either pull charged particles towards each other—when they have opposite charges—or push them away when the charges are of the same type.

Working with the electrostatic force often involves not only computing its magnitude but understanding its direction as well. The force carries a vector quality, which means it has both magnitude and direction. This vector nature determines how objects will move when subjected to an electric field, leading to a variety of phenomena in the natural and scientific world—from the behavior of electrons in an atom to the principles that govern electric currents in conductors. Moreover, its understanding is essential in grasping concepts related to electrical potential energy, electric field intensity, and capacitors in circuits.

The electric constant, also known as the permittivity of free space \( \epsilon_0 \), is a measure of how an electric field affects and is affected by a medium—free space in this case. It is a fundamental constant in physics that forms an integral part of electromagnetism's underpinnings. The permittivity of free space \( \epsilon_0 \) is used as a reference point to define the relative permittivity or dielectric constant of materials, which tells us how a material modifies the electric field within it.

The Quantified Electric Constant

When dealing with the math involving electric fields and forces, the constant \( \epsilon_0 \) takes a quantifiable form. The unit of \( \epsilon_0 \) in the International System of Units (SI) is farads per meter (F/m), providing a standardized way to apply this constant in equations and calculations—like Coulomb's Law. Understanding and being able to manipulate the units of \( \epsilon_0 \) is crucial for students tackling electrostatics, especially when it comes to understanding how electric fields propagate through different media and how materials can influence the behavior of electric fields.