Problem 12 - College Physics

Question

Two charges originally separated by a certain distance are moved farther apart until the force between them has decreased by a factor of \(10 .\) (a) Is the new distance (1) less than \(10,\) (2) equal to \(10,\) or (3) greater than 10 times the original distance? Why? (b) If the original distance was \(30 \mathrm{~cm},\) how far apart are the charges?

Step-by-Step Solution

Answer

The new distance is greater than 10 times the original, specifically about 94.87 cm.

Step 1: Understanding Coulomb's Law

Coulomb's Law states that the force between two charges is directly proportional to the product of the magnitudes of the charges, and inversely proportional to the square of the distance between them. Mathematically, this is expressed as \( F = k \frac{q_1 q_2}{r^2} \), where \( F \) is the force, \( q_1 \) and \( q_2 \) are the charges, \( r \) is the distance between the charges, and \( k \) is Coulomb's constant.

Step 2: Analyzing the Problem

We are told that the force decreases by a factor of 10. Therefore, the new force is \( \frac{1}{10} \) of the original force. According to Coulomb's Law, this means: \( F_{\text{new}} = \frac{F_{\text{original}}}{10} = k \frac{q_1 q_2}{r_{\text{new}}^2} \). We want to find how the distance \( r_{\text{new}} \) changes in terms of the original distance \( r_{\text{original}} \).

Step 3: Setting Up the Proportion

Since the new force is \( \frac{1}{10} \) of the original force, we set up the proportion: \( \frac{k q_1 q_2}{r_{\text{new}}^2} = \frac{1}{10} \times \frac{k q_1 q_2}{r_{\text{original}}^2} \). Simplifying, we get: \( r_{\text{new}}^2 = 10 \times r_{\text{original}}^2 \).

Step 4: Solving for New Distance

Simplify the equation \( r_{\text{new}} = \sqrt{10} \times r_{\text{original}} \). Since \( \sqrt{10} \approx 3.16 \), then the new distance is about 3.16 times the original distance.

Step 5: Applying the Original Distance

Given that the original distance is 30 cm, we find: \( r_{\text{new}} = \sqrt{10} \times 30 = 94.87 \text{ cm} \). The new distance is approximately 94.87 cm.

Key Concepts

Electric Forces

Electric forces are the interactions between charged particles, which can either be attractive or repulsive. They depend on the nature of the charges involved: like charges repel each other, while unlike charges attract. This behavior is similar to gravitational forces, but instead of mass, it involves electric charge.

• Coulomb's Law: This law provides a way to calculate the strength of the electric force between two charges. It considers the magnitude of each charge and the distance separating them.
• Dependence on charge and distance: The greater the charge, the stronger the force. Likewise, the closer the charges, the more intense the force becomes.

Understanding electric forces is fundamental when analyzing the interaction of charged materials in various physics problems.

Inverse Square Law

The inverse square law is a principle stating that a specified physical quantity is inversely proportional to the square of the distance from the source of that physical quantity.
The formula used in Coulomb's law, \( F = k \frac{q_1 q_2}{r^2} \), clearly illustrates this concept. Here, "\( r^{2} \)" in the denominator shows that the force decreases rapidly as the distance between two charges increases.

• Practical implications: This means that if the distance between two charges is doubled, the force between them reduces to a quarter of its original strength.
• Application to light and gravity: The inverse square law is not limited to electric forces; it also applies to gravitational forces, light intensity, sound, and even radiation.

Recognizing the inverse square law helps predict how forces or other effects diminish with distance.

Charge Interaction

Charge interaction refers to the way electric charges interact with each other, which can manifest as either attraction or repulsion. The strength and direction of these interactions are governed by the properties of the charges and their separation.

• Types of charge: There are two types of charges: positive and negative. Interaction depends on these charges.
• Force direction: If two charges are opposites, they attract each other. If the same, they repel.

Charge interaction is crucial in understanding how charged objects influence one another and how they behave under electric forces. Consistently, these interactions are quantified through commands set by Coulomb's law.

Physical Constants

Physical constants are fixed values that appear in the laws of nature, ensuring consistency in calculations. In the context of Coulomb's law, the constant involved is Coulomb's constant, \( k \). This constant has a value of approximately \( 8.9875 \times 10^{9} \text{ Nm}^{2}/\text{C}^{2} \).

• Importance of constants: They provide a standard value that simplifies the complex relationship between forces and other physical quantities in equations.
• Coulomb's constant: Specifically used in calculating electric forces, it allows relating charge magnitude and distance to force magnitude.

Physical constants like \( k \) provide the necessary means to link theory with real-world measurements, ensuring precision in predictions and calculations.