91Ó°ÊÓ

Coulomb's Law is a foundational principle in electromagnetism that describes the electric force between two point charges. Formulated by Charles-Augustin de Coulomb in the 18th century, this law states that the force (\( F \)) between two point charges is directly proportional to the product of the magnitudes of the charges (\( q_1 \) and \( q_2 \)) and inversely proportional to the square of the distance (\( r \) that separates them. Mathematically, this can be expressed as: \( F = k \frac{\|q_1q_2\|}{r^2} \) where \( k \) is Coulomb's constant (approximately \( 8.99 \times 10^9 N m^2/C^2 \)).

In the context of the textbook exercise, Coulomb's Law is used to calculate the magnitude of the electric fields that the two point charges on the x-axis generate at a point on the y-axis. Since the charges are equal and positioned symmetrically relative to the y-axis, their fields will be equal in magnitude but will point in opposing directions along the x-axis. Therefore, the electric force they exert on another charge placed on the y-axis can be calculated using Coulomb's Law as well.

The electric force is a vector quantity that represents the interaction between charged objects. According to Coulomb's Law, the electric force exerted by one point charge on another is both attractive and repulsive, depending on whether the charges are of opposite or like signs, respectively.

In our application, after determining the electric fields due to the two charged points using Coulomb's Law, we calculate the electric force on a third charge. The charge, in this case a negative \( -3.00-\mu C \) charge, experiences a force when placed in the electric field created by the other charges. To find this force, we employ the formula \( F = qE \), where \( q \) is the charge on which the force is acting, and \( E \) is the electric field strength at that point. Since the charge here is negative, the direction of the force will be opposite the direction of the electric field, a nuance that's crucial to understanding electric force interactions in electric fields.

The vector sum of electric fields is a critical concept that allows us to determine the resultant electric field at a point due to multiple charges. When more than one charge is present, each charge produces its own electric field. To find the total electric field at a given point, we sum up the vector fields produced by all individual charges.

The sum is not a simple arithmetic addition; since we are dealing with vectors, both magnitude and direction must be considered. In our exercise, the charges are placed symmetrically on the x-axis, and their electric fields at a point on the y-axis have equal magnitude but opposite directions. As a result, their horizontal components cancel each other, and we are left with vertical components that add up. By calculating and summing these components, we obtain the total electric field at the point of interest, considering the angles and signs to ensure all vector components are accurately accounted for, leading to a correct calculation of the overall field.