Hey everyone! Today, we're diving into a classic physics problem: calculating the electrostatic force between charged particles. We've got a scenario where we need to find the force exerted on one particle by another, and we'll break down the steps to get there. Physics can be intimidating, but together we'll make short work of this problem.
Problem Setup
Here's the problem we're tackling:
- Particle 1: Has a charge of -28.1 × 10⁻⁶ Coulombs (-28.1 μC).
- Particle 3: Located 0.600 meters to the right of Particle 1, with a charge of -47.9 × 10⁻⁶ Coulombs (-47.9 μC).
- Goal: Determine the force on Particle 3 due to Particle 1.
Let's get this done! We need to calculate the magnitude and direction of the electrostatic force using Coulomb's Law. This law is the key to understanding the interactions between charged objects.
Understanding Coulomb's Law
Before we jump into the calculations, let's quickly recap Coulomb's Law. This fundamental law describes the electrostatic force between two point charges. The formula looks like this:
F = k * (|q1 * q2|) / r²
Where:
- F is the electrostatic force between the charges (in Newtons).
- k is Coulomb's constant, approximately 8.9875 × 10⁹ N⋅m²/C².
- q1 and q2 are the magnitudes of the charges (in Coulombs).
- r is the distance between the charges (in meters).
- The vertical bars || denote the absolute value, meaning we only consider the magnitude of the charges, not their sign, for the force magnitude calculation.
The law tells us a few important things:
- Force and Charge: The force is directly proportional to the product of the charges. Bigger charges mean a bigger force.
- Force and Distance: The force is inversely proportional to the square of the distance. As the distance increases, the force decreases rapidly.
- Attractive or Repulsive: The sign of the charges determines whether the force is attractive (opposite signs) or repulsive (same signs). Since both particles in our problem are negatively charged, we can anticipate a repulsive force pushing Particle 3 away from Particle 1.
Diving Deeper into Coulomb's Law
To truly grasp Coulomb's Law, let's break down each component and see how it impacts the electrostatic force. The electrostatic force, governed by Coulomb's Law, is a fundamental interaction that dictates how charged particles interact. This force can either attract or repel particles, depending on their charges. At the heart of the law is the constant k, known as Coulomb's constant, which has a value of approximately 8.9875 × 10⁹ N⋅m²/C². This constant is pivotal in determining the magnitude of the electrostatic force in different systems of units.
The magnitude of the charges, denoted as q1 and q2, plays a crucial role in Coulomb's Law. The force is directly proportional to the product of these charges. In simpler terms, the larger the charges, the stronger the electrostatic force between them. For instance, if you double one of the charges, you double the force, and if you double both charges, the force quadruples. This direct proportionality highlights the importance of charge magnitude in determining the force's strength. Distance, represented as r in Coulomb's Law, is another critical factor. The force is inversely proportional to the square of the distance between the charges. This inverse square relationship means that even small changes in distance can significantly affect the force. For example, if you double the distance, the force decreases to one-quarter of its original value. This rapid decrease in force with increasing distance is a key characteristic of electrostatic interactions.
Determining whether the force is attractive or repulsive depends on the signs of the charges. Like charges (both positive or both negative) repel each other, while opposite charges (one positive and one negative) attract. This behavior is a cornerstone of electromagnetic interactions and explains many phenomena, from the behavior of atoms to the functioning of electronic devices. In our specific problem, both Particle 1 and Particle 3 have negative charges, indicating that the force between them will be repulsive. This means that Particle 3 will experience a force pushing it away from Particle 1. Understanding these nuances of Coulomb's Law is essential for accurately calculating electrostatic forces and predicting the behavior of charged particles. By carefully considering the magnitude of the charges, the distance between them, and their signs, we can determine both the strength and direction of the electrostatic force. Now, let's put this knowledge into action and calculate the force between Particle 1 and Particle 3.
Applying Coulomb's Law to Our Problem
Alright, let's plug the values from our problem into Coulomb's Law and see what we get!
-
Identify the Values:
- q1 = -28.1 × 10⁻⁶ C
- q2 = -47.9 × 10⁻⁶ C
- r = 0.600 m
- k = 8.9875 × 10⁹ N⋅m²/C²
-
Plug into the Formula:
F = (8.9875 × 10⁹ N⋅m²/C²) * (|-28.1 × 10⁻⁶ C * -47.9 × 10⁻⁶ C|) / (0.600 m)²
-
Calculate the Numerator (Charge Product):
|-28.1 × 10⁻⁶ C * -47.9 × 10⁻⁶ C| = 1.34599 × 10⁻⁹ C²
-
Calculate the Denominator (Distance Squared):
(0.600 m)² = 0.36 m²
-
Calculate the Force:
F = (8.9875 × 10⁹ N⋅m²/C²) * (1.34599 × 10⁻⁹ C²) / (0.36 m²) F ≈ 33.66 N
So, the magnitude of the force on Particle 3 due to Particle 1 is approximately 33.66 Newtons.
Step-by-Step Calculation Breakdown
Let's dissect this calculation to ensure we're all on the same page. Understanding each step is key to mastering Coulomb's Law and applying it effectively to various problems. First, we identified the given values: the charges of both particles (q1 and q2), the distance between them (r), and Coulomb's constant (k). These values are the foundation of our calculation. The charges, -28.1 × 10⁻⁶ C and -47.9 × 10⁻⁶ C, represent the amount of electrical charge each particle carries. The distance, 0.600 m, is the separation between the particles, and Coulomb's constant, 8.9875 × 10⁹ N⋅m²/C², is a fundamental constant that scales the electrostatic force.
Next, we plugged these values into Coulomb's Law: F = k * (|q1 * q2|) / r². This formula tells us how the force (F) depends on the charges and the distance. To make the calculation manageable, we broke it down into smaller parts. First, we calculated the product of the absolute values of the charges, |-28.1 × 10⁻⁶ C * -47.9 × 10⁻⁶ C|, which resulted in 1.34599 × 10⁻⁹ C². Taking the absolute value ensures we're dealing with the magnitude of the charges, as the sign will determine the direction of the force later. Then, we calculated the square of the distance, (0.600 m)², which equals 0.36 m². Squaring the distance is a crucial part of Coulomb's Law, highlighting the inverse square relationship between force and distance.
Finally, we put it all together to calculate the force: F = (8.9875 × 10⁹ N⋅m²/C²) * (1.34599 × 10⁻⁹ C²) / (0.36 m²). This calculation involves multiplying Coulomb's constant by the product of the charges and then dividing by the square of the distance. The result, approximately 33.66 N, is the magnitude of the electrostatic force between the particles. This step-by-step breakdown clarifies how each value contributes to the final result, making it easier to apply Coulomb's Law to other scenarios. By understanding the process, you can confidently tackle similar problems and gain a deeper appreciation for electrostatic interactions. Now, let's consider the direction of this force.
Determining the Direction of the Force
We've calculated the magnitude, but force is a vector, meaning it has both magnitude and direction. Remember, both particles have negative charges. Since like charges repel, the force on Particle 3 will be directed away from Particle 1. Given that Particle 3 is to the right of Particle 1, the force on Particle 3 will be directed to the right.
Visualizing the Force Direction
Understanding the direction of the electrostatic force is just as crucial as calculating its magnitude. Let's delve deeper into how we determine the direction based on the charges involved. In our scenario, both Particle 1 and Particle 3 possess negative charges. A fundamental principle of electrostatics is that like charges repel each other. This repulsion is the key to determining the force's direction.
Since both particles are negatively charged, they will push each other away. Imagine Particle 1 exerting a force on Particle 3. Because they repel, this force will act along the line connecting the two particles, pushing Particle 3 away from Particle 1. To visualize this, think of Particle 1 as trying to move Particle 3 further away. Given the spatial arrangement of the particles, where Particle 3 is located to the right of Particle 1, the repulsive force will cause Particle 3 to move further to the right. This means the force vector points in the direction from Particle 1 towards the right, away from itself.
If the charges were opposite (one positive and one negative), the force would be attractive instead of repulsive. In that case, the force on Particle 3 would be directed towards Particle 1, pulling them closer together. However, in our problem, the repulsion between the negative charges dictates a force directed to the right. This understanding of how charge signs influence force direction is essential for solving more complex electrostatic problems. By visualizing the forces as vectors pointing in specific directions, we can better analyze the interactions between multiple charged particles. This visual and conceptual clarity is invaluable in mastering electrostatics. Now, let's summarize our findings and present the final answer.
Final Answer
The force on Particle 3 from Particle 1 is approximately 33.66 Newtons, directed to the right. We found this by applying Coulomb's Law, carefully considering the charges, the distance, and the repulsive nature of like charges. You guys nailed it!
Summarizing the Key Steps
Let's recap the essential steps we took to solve this problem. Understanding the process is just as important as getting the correct numerical answer. Here’s a summary to help you tackle similar problems in the future. First, we began by understanding the problem setup. This involved identifying the charges of the particles, their positions relative to each other, and the goal of finding the force on Particle 3 due to Particle 1. A clear understanding of the problem's givens and objectives is the crucial first step in any physics problem.
Next, we reviewed Coulomb's Law, the fundamental principle governing electrostatic forces. We discussed the formula F = k * (|q1 * q2|) / r², where F is the electrostatic force, k is Coulomb's constant, q1 and q2 are the charges, and r is the distance between the charges. Understanding each component of the formula and its implications is essential for applying Coulomb's Law correctly. We then applied Coulomb's Law by plugging in the given values. This involved substituting the charges, distance, and Coulomb's constant into the formula and performing the calculations step by step. Breaking down the calculation into smaller steps, such as calculating the numerator and denominator separately, helps to minimize errors and makes the process more manageable.
After calculating the magnitude of the force, we determined its direction. Since both particles had negative charges, we knew the force would be repulsive, pushing Particle 3 away from Particle 1. Given that Particle 3 was to the right of Particle 1, the force was directed to the right. Understanding the relationship between the signs of charges and the direction of the force is crucial for a complete solution. Finally, we presented the final answer, stating both the magnitude and direction of the force. The force on Particle 3 from Particle 1 is approximately 33.66 Newtons, directed to the right. This clear and concise presentation of the solution ensures that all aspects of the problem are addressed. By following these steps, you can confidently approach and solve a wide range of electrostatic force problems.
Practice Makes Perfect
To truly master electrostatics, practice is key. Try working through similar problems with different charges and distances. Consider what happens if you change the sign of one of the charges. Keep practicing, and you'll become a pro at calculating electrostatic forces!
Further Practice and Exploration
To truly solidify your understanding of electrostatics and Coulomb's Law, engaging in further practice and exploration is essential. Here are some suggestions to help you become even more proficient in this area of physics. One of the best ways to improve your skills is to work through a variety of similar problems with different parameters. Try varying the charges of the particles, the distances between them, and even the signs of the charges. This will help you develop a deeper intuition for how these factors influence the electrostatic force.
For example, you could try calculating the force between two particles with opposite charges or explore how the force changes as you double or triple the distance between the particles. Another valuable exercise is to consider scenarios with more than two charged particles. In these cases, you'll need to calculate the net force on a particle by considering the vector sum of the forces exerted by all other particles. This introduces the concept of superposition and helps you understand how forces combine in more complex systems.
Additionally, exploring real-world applications of electrostatics can make the concepts more engaging and relevant. Electrostatic forces play a crucial role in many technologies, from laser printers and photocopiers to electrostatic precipitators used in pollution control. Learning about these applications can provide a broader context for your understanding of the physics principles involved. You might also consider delving into related topics, such as electric fields and electric potential. These concepts build upon Coulomb's Law and provide a more comprehensive understanding of electrostatics. Understanding electric fields, for instance, can help you visualize the force exerted by a charge on other charges in its vicinity.
Finally, don't hesitate to consult additional resources, such as textbooks, online tutorials, and physics forums. These resources can provide alternative explanations and approaches to problem-solving, which can be particularly helpful if you're struggling with a specific concept. By consistently practicing, exploring real-world applications, and seeking out additional resources, you can build a strong foundation in electrostatics and confidently tackle even the most challenging problems. Remember, the key to mastering physics is persistent effort and a willingness to explore and understand the concepts deeply.