91Ó°ÊÓ

Coulomb’s law is a foundational concept in understanding the interactions between point charges. Simply put, it states that the force (\f\(F\f\)) between two point charges is directly proportional to the product of their charges (\f\(q_1\f\) and \f\(q_2\f\)) and inversely proportional to the square of the distance (\f\(r\f\)) separating them. The equation can be expressed as: \f[ F = K \frac{|q_1 \times q_2|}{r^2} \f], where \f\(K\f\) is Coulomb's constant, approximately equal to \f\(8.99 \times 10^9 N.m^2/C^2\f\).

This fundamental law helps us calculate the magnitude of electrostatic forces and is pivotal for understanding electric fields around point charges. For example, in the given exercise, we use Coulomb's law to deduce the absolute value of charge \f\(A\f\) by rearranging the formula to solve for \f\(q_A\f\), taking into account the known electric field intensity and distance.

The concept of electric field intensity (\f\(E\f\)) is essential in visualizing and calculating how a charge exerts force on other charges around it without physical contact. It is defined as the force experienced by a unit positive charge placed in the vicinity of another charge. The electric field intensity for a point charge is given by the formula: \f[ E = \frac{K \times |q|}{r^2} \f], where \f\(q\f\) is the charge creating the field, \f\(r\f\) is the distance from the charge to the point in question, and \f\(K\f\) is again Coulomb's constant.

In our exercise, the electric field intensity exerted by charge \f\(A\f\) at a specific point is given, and we extend this concept to calculate the fields at the origin due to both \f\(A\f\) and \f\(B\f\), which helps us determine the total electric field at the origin for various scenarios of charge configurations.

The superposition principle plays a crucial role when dealing with multiple charges. It allows us to calculate the net electric field produced by several charges by considering each individual charge's contribution separately and then algebraically adding these vector quantities. According to this principle, the total electric field (\f\(\textbf{E}_{\text{total}}\f\)) created by multiple point charges is the vector sum of the electric fields (\f\(\textbf{E}_{i}\f\)) produced by each charge independently: \f[ \textbf{E}_{\text{total}} = \textbf{E}_1 + \textbf{E}_2 + ... + \textbf{E}_n \f].

This principle is pivotal in solving our exercise since charge \f\(A\f\) and charge \f\(B\f\) both contribute to the electric field at the origin. Depending on whether the charges are of the same or opposite signs, their fields either add or subtract vectorially to produce the net field.

The process of adding electric fields vectorially is an application of the superposition principle specifically to vector quantities. Remember that electric fields have both magnitude and direction, so when combining the fields from multiple charges, we must consider the vector character of these fields. For point charges placed along a line (like in our exercise), this simplifies to a one-dimensional problem where the fields either add or subtract according to their direction.

However, for cases where charges are not collinear, one might have to employ vector addition, involving breaking down the fields into components, typically using a Cartesian coordinate system, and then summing these components. To solve the given textbook problem, we perform such an addition (or subtraction) based on the sign and alignment of charges, to find the resulting net field at the origin.