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Coulomb's Law describes the electric force between two point charges. These are simply charged particles located at specific points in space. The law states that the force (\( F \)) between two charges is proportional to the absolute product of the charges and inversely proportional to the square of the distance between them.
The mathematical expression for Coulomb's Law is given by:

• \[F = \frac{k \cdot |Q \cdot q|}{r^2}\]where \( k \) is Coulomb's constant \( (8.99 \times 10^9 \, \text{Nm}^2/\text{C}^2) \), \( Q \) and \( q \) are the magnitudes of the point charges, and \( r \) is the distance separating the charges.

Coulomb's Law helps us calculate the magnitude of the force, which can be attractive or repulsive: like charges repel and unlike charges attract. This fundamental principle is crucial in understanding how charged particles interact in an electric field.
By applying Coulomb's Law in conjunction with the concept of electric potential energy, we can better analyze how energy transforms as the positions of these charges change.

In physics, a point charge is an idealization where the charge is assumed to be located at a single point in space. This simplifies calculations by ignoring any physical dimensions the charge may have.
Point charges are used in physics to model objects that are charged but are small enough that their size doesn’t significantly affect the calculations of their interactions with other charges.

• Point charges allow the use of simple equations like those derived from Coulomb's Law to predict behaviors of electric forces and energies.

A real-world application is where we consider tiny charged particles, such as electrons and protons, as point charges due to their relatively minute size compared to the distances over which we calculate forces and energies.
This concept is key in calculations regarding electric potential energy and electric fields because it allows us to assume the charge is acting from one specific location without accounting for any distribution around it.

The principle of conservation of energy states that energy in a closed system remains constant; it can neither be created nor destroyed, but can only change forms. In electric systems involving point charges, this principle is crucial to understand the transition between different types of energy.
In our scenario, as a point charge moves within an electric field, its electric potential energy changes. According to the conservation of energy:

• The decrease in electric potential energy converts to an increase in kinetic energy, resulting in the motion of the charge.
• The sum of potential energy and kinetic energy at any point is constant.

Mathematically, this can be expressed as:\[U_i + K_i = U_f + K_f\]where \( U \) represents electric potential energy, and \( K \) is kinetic energy. Thus:

• \( \Delta U = U_f - U_i \), and the change equals \( \Delta K = K_f - K_i \).

In the step-by-step solution, this principle was employed to find the speed of a moving point charge after it was released from rest, by equating the potential energy lost to the kinetic energy gained.