91Ó°ÊÓ

Coulomb's Law is fundamental in understanding how charged particles interact with one another. It quantifies the electrostatic force between two charges. The law states that the magnitude of the force (\( F \)) between two point charges (\( q_i \) and \( q_j \)) is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance (\( r \)) between their centers. This relationship is expressed by the formula:

\[F_{ij} = \frac{k |q_i \, q_j|}{r^2}\] Where \( k \) is the electrostatic constant, approximately \( 8.99 \times 10^9 \, \text{N m}^2 / \text{C}^2 \). This formula illustrates a few key points:

• Proportionality to Charge: The force increases if the magnitude of either charge increases.
• Inversely Related to Distance: A greater distance between charges reduces the force significantly.

By applying Coulomb's Law, we can derive the electrostatic forces exerted between multiple charge pairs step by step, as done in the provided exercise.

Charge interaction is a concept that describes how charged objects exert forces on each other due to their charges. These forces can either be attractive or repulsive, depending on the nature of the charges:

• Like Charges Repel: Two charges with the same sign (both positive or both negative) will repel each other.
• Opposite Charges Attract: A positive and a negative charge will attract each other.

In the exercise, four particles form a square with specific charges assigned to each corner. The interaction between these charged particles results in a net electric force acting on each particle. This net force is the vector sum of the individual electrostatic forces exerted by each of the other charges.

In overcoming these interactions, understanding charge relationships allows us to balance forces, as required to solve the given problem: finding the condition when the net force on charge \( q_1 \) is zero.

Vector decomposition is crucial when managing forces acting at angles. In the provided scenario, there are forces acting diagonally, particularly the force due to charge \( q_4 \), which requires decomposition into components along the x and y axes.

For any vector \( F \) at an angle \( \theta \), the components can be calculated as:

• Horizontal Component: \[ F_x = F \cdot \cos(\theta) \]
• Vertical Component: \[ F_y = F \cdot \sin(\theta) \]

In the exercise, the diagonal force \( F_{14} \) needed to be split into x and y components using \( 45^\circ \) for both components since a square geometry was involved:

• \( F_{14x} = \frac{\sqrt{2}kQ(2Q)}{(\sqrt{2}a)^2} \cdot \frac{1}{\sqrt{2}} = \frac{kQ^2}{a^2} \)
• \( F_{14y} = \frac{\sqrt{2}kQ(2Q)}{(\sqrt{2}a)^2} \cdot \frac{1}{\sqrt{2}} = \frac{kQ^2}{a^2} \)

Decomposing forces is essential in simplifying problems, making it easier to apply conditions of balance, specifically in calculating the forces required to ensure the net force equals zero.