91Ó°ÊÓ

Coulomb's Law is the fundamental principle describing the interaction between charged particles. It states that the force between two point charges is directly proportional to the product of the charges and inversely proportional to the square of the distance separating them. Mathematically, it is represented as:
\[ F = \frac{k \, |q_1 q_2|}{r^2} \]where:

• \( F \) is the magnitude of the electrostatic force between the charges,
• \( k \) is Coulomb's constant \( (8.99 \times 10^9 \text{ N m}^2/\text{C}^2) \),
• \( q_1 \) and \( q_2 \) are the magnitudes of the charges,
• \( r \) is the distance separating the charges.

Coulomb's Law not only helps in understanding the quantitative aspects of electric force but also plays a key role in deriving other related concepts like the electric field, demonstrating how charges interact within a field.

A point charge refers to an idealized or hypothetical charge located at a single point in space. It is helpful in simplifying the calculations of electric fields and forces, as it assumes the charge is concentrated at a particular location. In reality, charges are distributed over a region, but treating them as point charges allows for easier analysis of their interactions.
When calculating an electric field due to a point charge, the formula used is:
\[ E = \frac{kQ}{r^2} \]where:

• \( E \) is the electric field strength at a distance \( r \),
• \( k \) is Coulomb's constant,
• \( Q \) is the charge causing the field.

It's important to remember that the direction of the electric field depends on the type of charge:

• A positive point charge creates an outward field direction.
• A negative point charge creates an inward field direction.

These properties make point charges a pivotal concept in conceptualizing and calculating electric fields.

The relationship between distance and the electric field is a fundamental concept in electromagnetism. According to Coulomb's Law, the electric field (\( E \)) generated by a point charge decreases with the square of the distance (\( r \)) from the charge. This inverse square law nature means that as you move further away from a charge, the field's strength diminishes rapidly.
In the context of the given exercise, calculating the electric field at two different distances allows us to compare their relative magnitudes. Using the expressions:

• \( E_1 = \frac{kQ}{r_1^2} \)
• \( E_2 = \frac{kQ}{r_2^2} \)

We can derive the ratio of these distances by dividing the equations:
\[ \frac{E_1}{E_2} = \left(\frac{r_2}{r_1}\right)^2 \]This tells us the relation of distances is proportional to the square root of the field magnitudes' ratio.
Intuitively, this means if you know the electric field at two distances, you can determine how much further one distance is from the other. This comprehension helps in predicting how fields change as you adjust your position relative to a point charge.