Electric Field Intensity Calculation: A Step-by-Step Guide

Hey guys! Today, we're diving into a cool physics problem: calculating the electric field intensity at the midpoint of a line connecting two charged particles. We'll be using the provided values to solve this problem, which is a common scenario in introductory physics and electrostatics. Understanding how to calculate electric fields is fundamental to comprehending how charged objects interact with each other. This knowledge is not only important for academic purposes but also for understanding the behavior of electronic devices and circuits. So, let's get started! We will explore the concepts, equations, and step-by-step calculations needed to tackle this problem effectively. I'll make sure to break everything down so it's super easy to follow. Remember, understanding this is key to grasping more complex concepts later on.

First, let's make sure we're on the same page about the basics. The electric field is a vector field that describes the force a charged particle would experience if placed at a given point in space. The intensity of this field, or its strength, is determined by the magnitude and sign of the charge creating the field and the distance from the charge to the point of interest. It is important to remember that electric fields are vector quantities, which means they have both magnitude and direction. We'll use this to understand the direction of each electric field and sum them together. The electric field concept is a cornerstone of electromagnetism, describing the influence that electric charges exert on each other and on charged objects around them. We are going to calculate the resulting field by taking into account both of the charges. This will make us understand the concept very well.

Now, let's discuss the actual problem, which involves two charges of different magnitudes and signs. The charges are placed a certain distance apart, and we need to determine the electric field intensity at the midpoint between them. The fact that the charges have opposite signs is important because it tells us that the forces will act in different directions. Keep in mind that the electric field is calculated using Coulomb's law, which states that the force between two charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. The electric field at any point is therefore the sum of the fields contributed by all the charges. This is a vector sum, so we have to consider both the magnitude and direction of each field. This is important to remember. The electric field due to a single point charge is given by the formula E = k * |q| / r^2, where E is the electric field, k is Coulomb's constant, q is the charge, and r is the distance from the charge to the point where the field is being calculated. We can then add those fields together to get our answer! Pretty cool, right? In our case, we will need to account for both charges. The sign of the charge determines the direction of the field. Let's delve deeper into how to perform these calculations.

Setting Up the Problem: Defining Variables and Constants

Alright, before we jump into calculations, let's establish our parameters clearly. This part is super important because it helps prevent mistakes! We have two charges, so let's define them like so: charge 1 (q1) = -40 µC and charge 2 (q2) = 30 µC. The distance between the charges is 40 mm. However, since we are calculating the electric field at the midpoint, we will use half of this distance for our calculations. This setup is crucial for any electrostatic problem. Remember to write everything down, okay?

First up, k, Coulomb's constant, is a fundamental constant in physics. Its value is approximately 8.987 × 10^9 N⋅m²/C². This constant is essential for calculating the force between charged particles. Make sure you memorize this value. We'll also need to convert our units to the standard units for consistency. Let’s convert the distance from millimeters to meters. 40 mm is 0.04 m, and the distance to the midpoint from each charge will then be 0.02 m. Remember, the electric field is a vector quantity, so we'll need to think about the direction of the field at the midpoint due to each charge. These directions depend on the sign of the charge. So, we've got all the pieces of the puzzle and we are prepared to start calculating. Now that we have all of our variables defined and our units correct, we'll proceed to the calculations in the next step. Let's start doing some math, guys!

Step-by-Step Calculation of Electric Field Intensity

Okay, guys, let’s get into the actual calculations. This is where the magic happens! We'll calculate the electric field intensity at the midpoint due to each charge separately and then combine them to get the total electric field. Remember our formula: E = k * |q| / r². We'll apply this formula twice, once for each charge, and then we will combine the results, remembering that the electric field is a vector. For q1 (-40 µC), the distance to the midpoint is 0.02 m. So, the electric field (E1) at the midpoint due to q1 is E1 = (8.987 × 10^9 N⋅m²/C²) * (40 × 10^-6 C) / (0.02 m)^2. Make sure you follow along with your own calculator! This gives us E1 = 8.987 × 10^9 * 40 × 10^-6 / 0.0004. Calculate it yourself, and you'll find E1 ≈ 8.987 × 10^9 * 0.1 N/C. This means E1 ≈ 8.987 × 10^8 N/C. The negative sign of the charge means that the direction of the field points toward the negative charge. Are you following me?

For q2 (30 µC), the distance to the midpoint is also 0.02 m. Therefore, the electric field (E2) at the midpoint due to q2 is E2 = (8.987 × 10^9 N⋅m²/C²) * (30 × 10^-6 C) / (0.02 m)^2. Similarly, E2 ≈ 8.987 × 10^9 * 30 × 10^-6 / 0.0004 N/C. This gives us E2 ≈ 8.987 × 10^9 * 0.075 N/C, resulting in E2 ≈ 6.74 × 10^8 N/C. The direction of the field due to q2 points away from the positive charge. Remember that we must add these values to get the resulting total field. Since the directions of the electric fields due to q1 and q2 are opposite (because of the signs of the charges), we need to take the difference between the magnitudes of E1 and E2. Total E = E1 + E2, which is 8.987 x 10^8 + 6.74 x 10^8, so it gives us about 1.57 x 10^9 N/C. Therefore, at the midpoint, the electric field intensity will be E = 1.57 x 10^9 N/C.

Conclusion: Final Answer and Understanding

Alright, we did it! After all those calculations, we can confidently say that the electric field intensity at the midpoint of the line is approximately 1.57 × 10^9 N/C. This result shows the combined effect of both charges on the electric field at that specific point. It’s important to understand the significance of this result. The magnitude of the electric field tells us the strength of the field at that point, and the direction indicates the direction a positive test charge would move if placed at that point. We can see how the interplay of electric fields generated by multiple charges results in a net electric field at a specific location. The electric field is a fundamental concept in physics, and it is crucial for a complete understanding of how charges interact. Hopefully, you now have a solid grasp of how to calculate the electric field intensity at a specific point due to multiple charges. Keep practicing these types of problems, and you'll become a pro in no time! Remember to always pay attention to the signs of the charges and the directions of the fields. So, let's review what we learned. Always remember the fundamental equation E = k * |q| / r^2. Let me know if you want to try another problem, guys! Keep up the great work!