Resultant Coulomb Force Calculation At Point C Of An Equilateral Triangle

Hey guys! Ever wondered how electric charges interact with each other? Let's dive into a fascinating physics problem involving Coulomb's law and the concept of resultant force. We're going to explore how to calculate the net force acting on a charge placed at one of the vertices of an equilateral triangle, with other charges sitting at the remaining vertices. Buckle up, because we're about to unravel the mystery of electrostatic forces!

Understanding the Problem: Charges at the Corners of an Equilateral Triangle

So, here’s the scenario: Imagine an equilateral triangle, ABC, where each side measures 30 cm. Now, picture electric charges placed at each corner of this triangle. At point A, we have a charge of -2µC (microcoulombs), at point B, a charge of +2µC, and at point C, a charge of +3µC. The big question is: what is the resultant Coulomb force acting at point C? In simpler terms, how strongly and in what direction is the charge at C being pushed or pulled by the other charges? To crack this, we'll need to understand Coulomb's law, which is the fundamental principle governing the electrostatic force between charged objects. Coulomb's law states that the force between two point 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, it's expressed as F = k * |q1 * q2| / r^2, where F is the force, k is Coulomb's constant (approximately 8.9875 × 10^9 N⋅m^2/C2), q1 and q2 are the magnitudes of the charges, and r is the distance between them. But, before we jump into the math, let’s visualize what's happening. Since we have both positive and negative charges, we'll have both attractive and repulsive forces at play. The negative charge at A will attract the positive charge at C, while the positive charge at B will repel the positive charge at C. These forces will act along the lines connecting the charges, and since they are vectors, we'll need to consider their directions when we calculate the resultant force. This is where the concept of vector addition comes into play. Remember, forces are vectors, meaning they have both magnitude and direction. To find the resultant force, we can't just add the magnitudes of the forces; we need to consider their directions as well. One common method is to break down the forces into their horizontal and vertical components, add the components separately, and then use the Pythagorean theorem to find the magnitude of the resultant force and trigonometry to find its direction. In our case, the equilateral triangle provides a nice symmetry that we can exploit to simplify the calculations. The angles within the triangle are all 60 degrees, which will be helpful when resolving the forces into components. We'll use these angles to find the horizontal and vertical components of each force. So, to recap, we've got an equilateral triangle with charges at each corner, and we want to find the net force on the charge at point C. We know we'll need to use Coulomb's law to find the individual forces, and we'll need to use vector addition to find the resultant force. Now, let's get into the nitty-gritty of the calculations!

Calculating the Individual Coulomb Forces

Alright, let's get our hands dirty with some calculations! To figure out the resultant Coulomb force at point C, we first need to determine the individual forces exerted on the charge at C by the charges at A and B. We'll be using Coulomb's Law here, which, as we mentioned before, is F = k * |q1 * q2| / r^2. Remember, k is Coulomb's constant (approximately 8.9875 × 10^9 N⋅m^2/C2), q1 and q2 are the magnitudes of the charges, and r is the distance between them. Let's start with the force exerted by the charge at A on the charge at C. We'll call this force F_AC. The charge at A is -2µC, which is -2 × 10^-6 C, and the charge at C is +3µC, which is 3 × 10^-6 C. The distance between A and C is 30 cm, which is 0.3 meters. Plugging these values into Coulomb's law, we get: F_AC = (8.9875 × 10^9 N⋅m^2/C2) * |(-2 × 10^-6 C) * (3 × 10^-6 C)| / (0.3 m)^2. Calculating this, we find that F_AC ≈ 0.6 N. Now, let's think about the direction of this force. Since the charge at A is negative and the charge at C is positive, they will attract each other. So, the force F_AC is an attractive force, pulling the charge at C towards A. Next, let's calculate the force exerted by the charge at B on the charge at C. We'll call this force F_BC. The charge at B is +2µC, which is 2 × 10^-6 C, and the charge at C is still +3µC. The distance between B and C is also 30 cm, or 0.3 meters. Using Coulomb's law again: F_BC = (8.9875 × 10^9 N⋅m^2/C2) * |(2 × 10^-6 C) * (3 × 10^-6 C)| / (0.3 m)^2. This gives us F_BC ≈ 0.6 N. The direction of this force is different, though. Since both the charges at B and C are positive, they will repel each other. So, F_BC is a repulsive force, pushing the charge at C away from B. So, we've now calculated the magnitudes of the two forces acting on the charge at C: F_AC is approximately 0.6 N, and F_BC is also approximately 0.6 N. We also know their directions: F_AC is attractive, pulling C towards A, and F_BC is repulsive, pushing C away from B. Now, the next step is crucial: we need to add these forces as vectors to find the resultant force. Remember, we can't just add the magnitudes because the forces are acting in different directions. We need to consider their components along a coordinate system.

Vector Addition: Finding the Resultant Force

Okay, guys, this is where things get a bit more interesting! We've calculated the individual forces, F_AC and F_BC, acting on the charge at point C. Now, we need to find the resultant force, which is the vector sum of these two forces. Since forces are vectors, we can't just add their magnitudes; we need to consider their directions as well. One common and effective way to do this is by breaking down each force into its horizontal (x) and vertical (y) components. Let's imagine a coordinate system where the x-axis runs horizontally and the y-axis runs vertically. We can align this coordinate system so that, for example, the line BC lies along the x-axis. This will make it easier to find the components of the forces. Now, let's think about the directions of our forces. F_AC acts along the line connecting A and C, and F_BC acts along the line connecting B and C. Since ABC is an equilateral triangle, the angle between these two forces is 60 degrees. This is a crucial piece of information! To find the x and y components of F_AC, we'll use trigonometry. Let's call the angle between F_AC and the x-axis θ (theta). In our case, θ is 120 degrees (since the angle between BC and AC is 60 degrees, and we're measuring from the positive x-axis). The x-component of F_AC, which we'll call F_ACx, is given by F_AC * cos(θ), and the y-component of F_AC, which we'll call F_ACy, is given by F_AC * sin(θ). Plugging in our values, we get: F_ACx = 0.6 N * cos(120°) = 0.6 N * (-0.5) = -0.3 N. F_ACy = 0.6 N * sin(120°) = 0.6 N * (√3/2) ≈ 0.52 N. Now, let's consider F_BC. This force acts along the x-axis, so it only has an x-component. The y-component of F_BC is zero. The x-component of F_BC, which we'll call F_BCx, is simply the magnitude of F_BC, which is 0.6 N. Now that we have the x and y components of both forces, we can add them separately to find the x and y components of the resultant force. The x-component of the resultant force, F_Rx, is the sum of F_ACx and F_BCx: F_Rx = F_ACx + F_BCx = -0.3 N + 0.6 N = 0.3 N. The y-component of the resultant force, F_Ry, is the sum of F_ACy and the y-component of F_BC (which is zero): F_Ry = F_ACy + 0 = 0.52 N. Great! We've found the x and y components of the resultant force. Now, to find the magnitude of the resultant force, we use the Pythagorean theorem: F_R = √(F_Rx^2 + F_Ry^2) = √((0.3 N)^2 + (0.52 N)^2) ≈ 0.6 N. So, the magnitude of the resultant force acting on the charge at C is approximately 0.6 N. But, we're not quite done yet! We also need to find the direction of this force. We can do this using trigonometry. Let's call the angle between the resultant force and the x-axis α (alpha). We can find α using the arctangent function: α = arctan(F_Ry / F_Rx) = arctan(0.52 N / 0.3 N) ≈ 60 degrees. This means the resultant force is acting at an angle of approximately 60 degrees with respect to the x-axis. So, to summarize, we've found that the resultant Coulomb force acting on the charge at C has a magnitude of approximately 0.6 N and acts at an angle of approximately 60 degrees with respect to the line BC. Phew! That was a lot of calculations, but we've successfully solved the problem. Remember, the key is to break down the problem into smaller steps: calculate the individual forces using Coulomb's law, break the forces into components, add the components, and then use the Pythagorean theorem and trigonometry to find the magnitude and direction of the resultant force.

Final Result and Direction of the Force

Alright, let's wrap things up and talk about the grand finale: the resultant Coulomb force acting at point C. We've crunched the numbers, wrestled with vectors, and now we're ready to state our findings. We found that the magnitude of the resultant force is approximately 0.6 N. That's the strength of the push or pull felt by the charge at C due to the other charges at A and B. But force isn't just about magnitude; it's also about direction. We determined that the direction of this resultant force is approximately 60 degrees with respect to the line BC. Now, what does this mean in terms of the physical situation? Imagine standing at point C and looking at the triangle. The force is pushing you (or pulling you, depending on the signs of the charges) with a strength of 0.6 N, and it's doing so at an angle of 60 degrees relative to the side of the triangle connecting B and C. To visualize this even better, you can draw a vector diagram. Draw the triangle ABC, and then draw the individual force vectors F_AC and F_BC at point C. Then, draw the resultant force vector, which is the diagonal of the parallelogram formed by F_AC and F_BC. You'll see that the resultant force vector points in a direction that is consistent with our calculations. So, there you have it! We've successfully calculated the resultant Coulomb force at point C. We've used Coulomb's law to find the individual forces, broken them down into components, added the components to find the resultant force, and then determined the magnitude and direction of the resultant force. This problem is a great example of how to apply fundamental physics principles to solve real-world problems. It also highlights the importance of understanding vectors and vector addition. Remember, forces are vectors, and you can't just add their magnitudes; you need to consider their directions as well. This problem also demonstrates the power of breaking down complex problems into smaller, more manageable steps. By calculating the individual forces first, and then adding them as vectors, we were able to find the resultant force without getting overwhelmed by the complexity of the situation. I hope this explanation has been helpful and has given you a better understanding of Coulomb's law and resultant forces. If you have any questions, feel free to ask! And remember, physics is all about understanding the world around us, so keep exploring and keep asking questions.

repair-input-keyword: What is the resultant Coulomb force at point C of an equilateral triangle with charges -2µC, +2µC, and +3µC at its vertices and sides of 30cm?

title: Calculating Resultant Coulomb Force at a Triangle's Vertex