Question

Two point charges are separated by a distance \(r\) and repel each other with a force \(F\). If their separation is reduced to 0.25 times the original value, what is the magnitude of the force of repulsion between them?

Short answer

The magnitude of the new force is 16 times the original force, \(16F\).

Step-by-step solution

Understand the Problem

We have two point charges that repel each other with force \( F \) at a distance \( r \). We need to calculate the new force if the distance between them is reduced to 0.25 of \( r \).

Apply Coulomb's Law

Coulomb's Law states that the force between two point charges is given by \( F = k \cdot \frac{q_1 q_2}{r^2} \), where \( k \) is Coulomb's constant, \( q_1 \) and \( q_2 \) are the magnitudes of the charges, and \( r \) is the distance between them.

Express the New Distance

The new distance between the charges is 0.25 of the original distance, so the new distance is expressed as \( r' = 0.25r \).

Calculate the New Force

Using Coulomb's Law, the new force \( F' \) can be calculated as follows: \( F' = k \cdot \frac{q_1 q_2}{(0.25r)^2} = k \cdot \frac{q_1 q_2}{0.0625r^2} = \frac{k \cdot q_1 q_2}{r^2} \cdot \frac{1}{0.0625} \). Since \( \frac{k \cdot q_1 q_2}{r^2} = F \), it follows that \( F' = F \cdot \frac{1}{0.0625} = 16F \).

Key concepts

Point Charges

Point charges are objects that have either a positive or a negative charge. They are considered as single, infinitesimally small points in space that produce an electric field around them. This notion allows us to simplify calculations in electrostatics by concentrating on the charges' effects at a distance, rather than their actual sizes or shapes.
Each point charge exerts a force on other nearby charges. The direction of this force can be attractive or repulsive, depending on the nature of the charges. Like charges repel, while opposite charges attract.

• Positive charges repel other positive charges.
• Negative charges repel other negative charges.

The concept of point charges is crucial for understanding how charged objects interact with one another, and it forms the foundation of many electrostatic calculations.

Force of Repulsion

The force of repulsion between two point charges is an essential aspect of their interaction. It occurs when both charges are of the same type, either both positive or both negative. This force pushes them apart, acting along the line joining the charges.
Coulomb's law helps us calculate the force of repulsion using the formula
\[ F = k \cdot \frac{q_1 q_2}{r^2} \]
where \( F \) is the force of repulsion, \( k \) is Coulomb's constant, \( q_1 \) and \( q_2 \) are the magnitudes of the charges, and \( r \) is the distance between the charges.

• The greater the magnitude of the charges, the stronger the force.
• The force decreases as the distance between the charges increases.

Understanding the force of repulsion is vital for solving problems involving electrostatic forces.

Distance in Electrostatics

The distance between two point charges is a crucial variable in determining the electrostatic force. According to Coulomb's Law, this force is inversely proportional to the square of the distance between the charges. This means that even a small change in distance can significantly impact the force experienced by the charges.
When the distance is reduced, the force increases, producing a stronger interaction between the charges. Conversely, increasing the distance decreases the force, weakening their interaction.

• Force increases with decreased distance.
• Force decreases with increased distance.

In scenarios where charges move closer together, understanding how to calculate the resulting force is critical for predicting behavior in electrostatic systems.

Coulomb's Constant

Coulomb's constant \( k \) is a value that appears in Coulomb's Law, representing the proportional relationship between force, charge magnitudes, and the distance separating them. Its approximate value is \( 8.9875 \times 10^9 \) N \( \, m^2/C^2 \). This constant essentially scales the force to ensure it is calculated correctly according to our system of units.
Coulomb's constant reflects the strength of the electrostatic force, balancing it against other variables in the equation. It is necessary for accurate calculations in problems involving electrical interactions.

• A higher value of \( k \) would increase the calculated force.
• A lower value would decrease it.

By understanding the role of Coulomb's constant, students can ensure accurate calculations in exercises that use Coulomb's Law.