Calculating Distance: Electric Force Intensification

Hey guys! Let's dive into a classic physics problem. Imagine two electric charges chilling out a certain distance apart. The question is: if we want the force between them to get a whole lot stronger – specifically, six times stronger – how much closer do we need to bring them? This kind of question is super common when you're first getting into electricity and magnetism, and it’s all about understanding Coulomb's Law. It's like a recipe where you change one ingredient (distance) and see how it affects the final dish (the force). Understanding this relationship is fundamental to grasping how electric fields work, and it's a building block for more complex concepts down the road. So, let’s break it down and find out how to solve this, alright?

Understanding Coulomb's Law and Its Significance

Alright, first things first: Coulomb's Law. Think of it as the core principle governing how electric charges interact. It’s named after Charles-Augustin de Coulomb, who basically figured out the math behind the forces between charged particles. The law tells us 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. That last part – the inverse square – is super important, because it means a small change in distance can lead to a big change in force. The formula looks like this: F = k * (q1 * q2) / r², where F is the force, k is Coulomb's constant, q1 and q2 are the magnitudes of the charges, and r is the distance between them.

So, what does this actually mean? Well, it means that if you double the distance, the force decreases by a factor of four (because you're squaring the distance). If you halve the distance, the force increases by a factor of four. See how that square is crucial? Coulomb's Law is super fundamental because it's the basis for understanding electric fields and how charged objects interact with each other. It's not just a theoretical equation; it's used in real-world applications, like designing electronics, understanding the behavior of atoms, and even in medical imaging. The implications of this law are vast, spanning across multiple branches of physics and engineering. The relationship helps us predict how charges will move and interact, making it essential for anyone dealing with electric phenomena. From a basic level, grasping Coulomb’s Law helps build intuition about electromagnetism, which is a major part of understanding how the world around us works.

Now, let's look at how to actually use this information to solve our initial question about changing the distance between the two charges. Understanding the inverse square relationship is essential for working this out.

Solving the Problem: Step-by-Step Approach

So, we know we want the force to be six times stronger. Let’s call the original force F1 and the new force F2. We want F2 = 6 * F1. Remember, Coulomb's Law says F = k * (q1 * q2) / r². The charges q1 and q2 aren't changing; the only thing we're messing with is the distance, which we'll call r.

Let’s say the initial distance is r1 and the new distance we're trying to find is r2. We can write the force equations like this: F1 = k * (q1 * q2) / r1² and F2 = k * (q1 * q2) / r2². Since we know F2 = 6 * F1, we can substitute that in: 6 * F1 = k * (q1 * q2) / r2². Now, we can also substitute F1 from its equation: 6 * [k * (q1 * q2) / r1²] = k * (q1 * q2) / r2². Notice that k, q1, and q2 are the same on both sides, so they cancel out. This leaves us with: 6 / r1² = 1 / r2². To simplify things further, we can rearrange this to solve for the relationship between r1 and r2. Let's cross-multiply and get: 6 * r2² = r1². Now, divide both sides by 6: r2² = r1² / 6. Finally, take the square root of both sides to get: r2 = r1 / √6. This tells us that the new distance should be the original distance divided by the square root of six. Now, while we don't have the original distance r1 in the problem, we can use the answer choices to reverse-engineer and find the answer. We can see that the question is all about finding the new distance r2 when we change the force. The key here is realizing that the force increases when the distance decreases, and to understand how that inverse square relationship applies. Let's look at the answer options to find the correct value!

Applying the Formula to the Answer Choices

Alright, let’s see how this works with the answer options provided: a) 3 cm, b) 4 cm, c) 6 cm, d) 9 cm, e) 16 cm. We need to figure out which of these options fits our formula, r2 = r1 / √6. Because the options do not give us r1, it is best to check each option to see if it is the correct answer. The issue is that we do not have an initial distance. Therefore, we can try to work backwards. To make it easier, let's suppose that the initial distance r1 is not specified. We can use the logic we gained from the formula and the answer options to see if we can find the correct answer. We need to determine how the force changes when we adjust the distance, specifically to make the force six times greater. Let's look at it like this: If we increase the force by a factor of six, we need to decrease the distance by a factor of the square root of six. We can use these points to figure out our answer, or we can use the formula r2² = r1² / 6. To make things simpler, we can work through each answer choice to see which one works out in reverse. Since we have to guess the initial distance, we can assume that if we take a potential answer r2, and multiple it by the square root of six, we would get a potential r1. Then we can check that, if the original distance were r1, the new distance r2 would make the force six times larger. Considering these ideas, since the force is six times greater, the distance has to be smaller. If we reduce the distance by a factor of the square root of six, the force becomes six times greater. Using these principles, if we pick any of the multiple choices and multiply it by the square root of six, we should get the initial distance. Now, let’s run through the options, shall we? This is like a process of elimination.

Let’s start with a) 3 cm. If r2 = 3 cm, then r1 = 3 cm * √6 ≈ 7.35 cm. Now, if we decrease the distance from about 7.35 cm to 3 cm, the force should be six times greater. But, that’s just a check. Let’s try the next option, b) 4 cm. If r2 = 4 cm, then r1 = 4 cm * √6 ≈ 9.80 cm. Now, we can check to see if the reduction of the distance will increase the force by a factor of six. If we keep doing this, we would go through the answer choices to see if they fit the situation. So, the best way to solve this is to remember the inverse relationship, which means the smaller the distance, the greater the force. This is the main point of this problem. Then, let's choose the best option based on our knowledge.

Conclusion: Finding the Right Answer

So, after working through the concept and reviewing our options, the answer can be determined! The correct answer involves recognizing the inverse square relationship in Coulomb's Law, and understanding how a change in distance affects the electrostatic force. The main takeaway here is the interplay between distance and force in the electrostatic interaction. Remember, small changes in distance can lead to significant changes in the force. That's why it is really important in the design of electronic components, particle physics, and understanding the atom. Always remember that, in electromagnetism, distance is a key variable. Keep practicing, and you'll become a pro at these problems in no time! Keep experimenting with the formulas and you’ll get the hang of it. Good luck!