91Ó°ÊÓ

Direct variation is a fundamental concept in mathematics, particularly when dealing with relationships between two variables. When we say that a variable varies directly, it means that as one variable increases or decreases, the other does so in the same manner. In the scenario presented in the original exercise, the electrostatic force (\( F \) ) varies directly with the product of the charges (\( q \cdot Q \) ).

The expression \( F \propto q \cdot Q \) encapsulates this relationship, where the symbol "\( \propto \)" indicates that one quantity increases or decreases with another. This direct relationship implies:

• If one of the charges increases, the force increases, provided the other charge and the distance remain constant.


• Conversely, if one charge decreases, the force also decreases, again assuming the other variables remain constant.

Recognizing direct variation in equations is crucial because it helps predict how modifying one quantity will affect another.

Inverse variation describes a situation where one quantity increases while another decreases. In the context of the electrostatic force, this manifests in the relationship between the force (\( F \) ) and the square of the distance (\( r^2 \) ) between the charges.

The expression \( F \propto \frac{1}{r^2} \) communicates that as the distance between the charges increases, the force decreases, assuming the charges remain constant. This is the essence of inverse variation, captured in the formula:

• If the distance (\( r \) ) doubles, the force (\( F \) ) becomes one-fourth of its original value.
• If the distance halves, the force quadruples, assuming the charges do not change.

Understanding inverse variations is important because it shows how distance plays a crucial role in the interaction strength between electrical charges.

The concept of a proportionality constant bridges the gap between a proportional relationship and a definitive equation. In the given electrostatic force equation, the constant of proportionality (\( k \) ) allows us to equate the proportional relationship with a specific value.

In the equation \( F = k \frac{qQ}{r^2} \), \( k \) is the constant that depends on elements such as the units used or the medium containing the charges. This constant is what transforms: \( F \propto \frac{qQ}{r^2} \) into an equality and provides a numerical value that can be used for calculations.

• The value of \( k \) can differ based on context, such as whether the charges are in a vacuum or another medium.
• In vacuum situations, \( k \) is often equated to \( 8.9875 \times 10^9 \) Newton square meters per \( C^2 \), which is known as Coulomb's constant.

By incorporating a proportionality constant, we convert abstract proportionality into practical computation.