91Ó°ÊÓ

Coulomb's Law is a fundamental principle which describes how electric charges interact with each other. It states that the electric force between two charged objects is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between their centers. This important law can be expressed with the formula: \[ F = \frac{k \cdot |Q_1 \cdot Q_2|}{r^2} \] where:

• \( F \) is the electric force between the charges
• \( k \) is Coulomb's constant, approximately equal to \( 8.99 \times 10^9 \text{ Nm}^2/\text{C}^2 \)
• \( Q_1 \) and \( Q_2 \) are the magnitudes of the charges
• \( r \) is the distance between the charges

In the context of the given problem, Coulomb's Law provides the foundational understanding for how the electric field is computed from a point charge. Understanding this principle helps us relate the physical setup (distances and fields) to the mathematical expressions in the step-by-step solution.

A point charge is simply an idealized model where the electric charge is considered to be located at a single point in space. This concept is crucial when calculating electric fields because it simplifies our calculations and allows us to use the symmetry in the problem. In our exercise, the problem uses a point charge to create different electric fields at different distances. The electric field (\( E \)) from a point charge is mathematically defined as: \[ E = \frac{kQ}{r^2} \] where:

• \( k \) is Coulomb's constant, a crucial element for calculations
• \( Q \) represents the amount of charge
• \( r \) indicates the distance from the charge

Understanding the point charge helps us effectively use this formula to evaluate how the electric field strength changes with different distances, as seen in the problem scenario.

In the given exercise, one key objective is to determine the ratio of two distances (\( r_2/r_1 \)) at which the electric field magnitudes are specified. The concept of having distance ratios is essential when dealing with problems related to electric fields, especially when there's a point charge. To solve for the distance ratio:

• Remember that the electric field strength varies inversely with the square of the distance. This is captured in the equation \( E = \frac{kQ}{r^2} \).
• By carefully comparing the fields at two distances, \( r_1 \) and \( r_2 \), we derive that the field magnitude relates to the distance squared as shown in \( \frac{E_2}{E_1} = \frac{r_1^2}{r_2^2} \).

By using this relationship, we solve for the distance ratio as: \[ \left( \frac{r_2}{r_1} \right)^2 = \frac{E_1}{E_2} \] The result is that solving this equation gives us a deeper understanding of how electric fields decrease as we move further away from a point charge, thanks to the inverse square law.