Resultant Force Calculation: Electric Charges & Distance

Hey guys! Let's dive into a cool physics problem involving electric charges and the forces they exert on each other. We're going to figure out the resultant force on a charge positioned between two other charges. It's like a tug-of-war, but with electricity! This scenario is a fundamental concept in electrostatics, and understanding it is crucial for grasping more complex electromagnetic phenomena. This type of problem is super common in introductory physics courses, so nailing the concept will definitely help you ace your exams. Plus, it's just plain interesting to see how these invisible forces interact! So, grab your calculators and let's get started. We'll break down the problem step-by-step to make sure everyone understands the process. We will also talk about how to apply Coulomb's Law effectively. Remember, the key to solving these problems is to carefully consider the direction of the forces involved. Let's make sure we're on the same page with units. We'll be using microcoulombs (µC) for charge and meters (m) for distance. Make sure to convert everything to the correct units if needed. Let's get to the problem. We have two charges, and they are -8µC each. They are separated by a distance of 0.12 meters. The question asks us to find the resultant force on a third charge of -4µC, placed in the middle of the other two charges. Let's start with a visual representation. Imagine the two -8µC charges on either side, with the -4µC charge right in the middle. Now, the fun part – figuring out the forces.

The Setup and Coulomb's Law

Okay, so the stage is set: we have two equal negative charges and a third negative charge nestled right in the center. The core of this problem revolves around Coulomb's Law, which quantifies the electrostatic force between charged particles. Coulomb's Law 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. Mathematically, it's represented as: F = k * |q1 * q2| / r². Where F is the force, k is Coulomb's constant (approximately 8.9875 x 10⁹ Nm²/C²), q1 and q2 are the magnitudes of the charges, and r is the distance between them. The absolute value signs ensure that we're only considering the magnitude of the force. The direction of the force is determined by the sign of the charges. Like charges repel, and unlike charges attract. This is super important to remember! Now, let's look at our setup. The -4µC charge is in the middle, and the other two are on either side. Since all three charges are negative, they will all repel each other. This is the first key aspect in finding the force. Because of this, the two -8µC charges will push the -4µC charge in opposite directions. So, the forces exerted on the -4µC charge by the -8µC charges will be equal in magnitude but opposite in direction. This means that they will cancel each other out. That's right, the net force on the -4µC charge is zero. Pretty neat, huh? Let's break this down further with a bit more detail. Imagine the left -8µC charge, it will push the -4µC charge to the right. The right -8µC charge will push the -4µC charge to the left. Since all distances are equal, the forces will be equal in magnitude. Since they are in opposite directions, they will cancel out. So the final answer is zero.

Step-by-Step Calculation of Forces

Alright, let's crunch some numbers and see how this all comes together. We'll apply Coulomb's Law to find the force exerted by each of the -8µC charges on the -4µC charge. First, let's think about the distances. The -4µC charge is placed in the exact middle, so it's 0.06 meters from each of the -8µC charges (because 0.12 m / 2 = 0.06 m). Now, we can plug the values into Coulomb's Law. Remember, we need to convert the charges to Coulombs: -8µC = -8 x 10⁻⁶ C and -4µC = -4 x 10⁻⁶ C. Let's calculate the force from one of the -8µC charges. Let’s say we're calculating the force exerted by the left -8µC charge on the -4µC charge. F = k * |q1 * q2| / r² --> F = (8.9875 x 10⁹ Nm²/C²) * |(-8 x 10⁻⁶ C) * (-4 x 10⁻⁶ C)| / (0.06 m)². Calculate the values. We get F = 0.089875 N. Since both the charges are negative, we know that the forces will be repulsive. So the force on the -4µC charge from the left -8µC charge is to the right, and the force on the -4µC charge from the right -8µC charge is to the left. Now, let’s calculate the force from the right -8µC charge on the -4µC charge. It will be the same calculation, since the values are all the same, and the distance is the same. The answer is also F = 0.089875 N, but in the opposite direction. Since they have the same magnitude but different directions, we can just say that they cancel each other out. So, the net force on the -4µC charge is zero.

Determining the Resultant Force

Okay, so we've calculated the forces exerted by each of the -8µC charges on the -4µC charge. Now, we need to find the resultant force. This means we need to combine all the forces acting on the -4µC charge to find the total force. Since the two -8µC charges are exerting forces in opposite directions and with equal magnitude, the resultant force is the vector sum of these forces. Because the forces are equal in magnitude but opposite in direction, they will cancel each other out. So, the resultant force is zero. If the -4µC charge wasn't exactly in the middle, then we'd have to do some more complex calculations. We'd have to consider the different distances and apply Coulomb's Law for each pair of charges. Then, we'd need to add the forces as vectors, taking into account their directions. This might involve using trigonometry to break down the forces into their components (x and y). This is especially true if the charges are arranged in a more complex configuration, like a triangle. Always remember that the direction of the force is crucial. If the charges are of the same sign, the force is repulsive (pushing them apart). If the charges are of opposite signs, the force is attractive (pulling them together). A good way to keep track of this is to draw a diagram with arrows showing the direction of the forces. Label each force clearly and be sure to include the magnitudes you've calculated. Understanding how to calculate resultant forces is a fundamental skill in physics. It not only applies to electrostatics but also to other areas like mechanics (forces on objects) and even fluid dynamics. So, the next time you encounter a similar problem, remember to break it down step-by-step, use Coulomb's Law, consider the directions of the forces, and find the vector sum. You've got this, guys!