Calculate Equal Charges Force: Solved Example

Hey guys! Ever wondered how to calculate the value of electric charges? It might sound intimidating, but trust me, it's super fascinating and totally doable. Today, we're diving into a classic physics problem that involves figuring out the value of two equal charges when they're separated by a certain distance and experiencing a specific force. Think of it like this: you've got two invisible magnets pushing or pulling each other, and we need to figure out how strong those magnets actually are. Ready to jump in?

Understanding Coulomb's Law: The Key to Unlocking Charge Values

At the heart of this problem lies Coulomb's Law, a fundamental principle in electrostatics. Coulomb's Law, named after the brilliant French physicist Charles-Augustin de Coulomb, describes the force between electrically charged objects. It's like the backbone of understanding how charges interact. In simple terms, the law states that the force between two charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. Whew, that's a mouthful! Let's break it down a bit.

Imagine you have two charged particles. The bigger the charges are, the stronger the force between them will be – that's the “directly proportional to the product of the magnitudes” part. Now, picture moving those charges further apart. As the distance increases, the force between them weakens drastically – that's the “inversely proportional to the square of the distance” bit. This inverse square relationship is super important and shows up in a lot of physics, like gravity too! Coulomb's Law isn't just some abstract equation; it's a cornerstone of how we understand the electrical world around us. From the sparks you see in static electricity to the forces that hold atoms together, Coulomb's Law is at play. It's essential to grasp this concept if you want to tackle problems involving electric charges and forces. So, let's put this law into action and see how it helps us crack our charge-calculating conundrum!

Mathematically, Coulomb's Law is expressed as:

F = k * (|q1 * q2|) / r²

Where:

• F is the electrostatic force between the charges (in Newtons, N)
• k is Coulomb's constant (approximately 8.9875 × 10⁹ N⋅m²/C²)
• q1 and q2 are the magnitudes of the charges (in Coulombs, C)
• r is the distance between the charges (in meters, m)

This equation might look a little intimidating at first, but don't worry! Once you understand what each part represents, it becomes a powerful tool for solving problems like ours. The absolute value signs around q1 * q2 simply mean that we're only concerned with the magnitude (size) of the charges, not their sign (positive or negative). The r² term highlights that inverse square relationship we talked about – the force decreases rapidly as the distance increases. And k, Coulomb's constant, is just a proportionality factor that makes the units work out correctly. Now that we've got Coulomb's Law in our toolkit, we're ready to apply it to our specific problem.

Applying Coulomb's Law to Our Problem: Setting Up the Equation

Okay, let's get back to our original problem. We have two equal charges separated by 20 inches, experiencing a force of 300 N. Our goal is to find the value of these charges. The first step is always to convert the given information into the correct units for our equation. Since Coulomb's Law uses meters for distance, we need to convert 20 inches into meters. There are approximately 0.0254 meters in an inch, so 20 inches is equal to 20 * 0.0254 = 0.508 meters. Now we're talking!

Next, let's identify what we know and what we're trying to find. We know the force (F = 300 N), the distance (r = 0.508 m), and Coulomb's constant (k ≈ 8.9875 × 10⁹ N⋅m²/C²). We also know that the charges are equal, which means q1 = q2. Let's call the value of each charge 'q'. This simplifies our equation a bit. Now, we can plug these values into Coulomb's Law: 300 = (8.9875 × 10⁹ * |q * q|) / (0.508)².

See how we've taken the words of the problem and translated them into a mathematical equation? This is a crucial skill in physics! We've taken the conceptual understanding of Coulomb's Law and applied it to our specific scenario. By substituting the known values into the equation, we've set up a mathematical relationship that we can now solve for our unknown, 'q'. The next step is to use some algebra magic to isolate 'q' and find its value. Don't worry, we'll walk through it step-by-step!

Solving for the Charge: A Step-by-Step Guide

Alright, guys, we've got our equation set up: 300 = (8.9875 × 10⁹ * |q * q|) / (0.508)². Now comes the fun part – solving for 'q'! This is where our algebra skills come into play. Remember, the goal is to isolate 'q' on one side of the equation. Let's break it down step by step.

First, we want to get rid of the fraction. We can do this by multiplying both sides of the equation by (0.508)²: 300 * (0.508)² = 8.9875 × 10⁹ * |q * q|. Calculate the left side: 300 * (0.508)² ≈ 77.44. Now our equation looks like this: 77.44 = 8.9875 × 10⁹ * |q²|. Remember that q * q is simply q². Next, we want to isolate the q² term. To do this, we'll divide both sides of the equation by 8.9875 × 10⁹: 77.44 / (8.9875 × 10⁹) = |q²|. Doing the division, we get approximately 8.616 × 10⁻⁹ = |q²|.

Now we're getting closer! We have q² isolated, but we want 'q' itself. To undo the square, we need to take the square root of both sides of the equation: √ (8. 616 × 10⁻⁹) = √(|q²|). The square root of 8.616 × 10⁻⁹ is approximately 9.28 × 10⁻⁵. And the square root of |q²| is simply |q|. So, we have |q| ≈ 9.28 × 10⁻⁵ Coulombs. But here's a crucial point: when we take the square root, we need to consider both the positive and negative solutions. This is because both a positive charge and a negative charge, when squared, will give a positive result. Therefore, q ≈ ± 9.28 × 10⁻⁵ C.

Interpreting the Result: What Does the Charge Value Tell Us?

Awesome! We've crunched the numbers and found that the value of the charges is approximately ± 9.28 × 10⁻⁵ Coulombs. But what does this actually mean? It's important not just to get the answer but also to understand what it tells us about the physical situation. First, let's focus on the magnitude of the charge: 9.28 × 10⁻⁵ Coulombs. A Coulomb is the standard unit of electric charge, and this value represents the amount of charge on each of our particles. The exponent of 10⁻⁵ tells us that this is a relatively small amount of charge, but it's enough to produce a significant force of 300 N when the charges are separated by about half a meter.

Now, let's think about the ± sign. This tells us that the charges could be either positive or negative. However, since the problem states that the charges are equal and they are experiencing a repulsive force (300 N), we know that they must have the same sign. Like charges repel, and opposite charges attract. If one charge were positive and the other negative, they would be pulling towards each other, not pushing away. Therefore, both charges are either positive or both charges are negative. We can't determine the exact sign without more information, but we know they are the same. The magnitude of the charge tells us how “strongly” charged the particles are, and the sign tells us about the nature of the interaction between them.

This result highlights the power of Coulomb's Law. It allows us to quantitatively relate the force, charge, and distance in electrostatic interactions. By understanding and applying this law, we can predict and explain a wide range of phenomena, from the behavior of simple charged objects to the complex interactions within electronic devices. So, next time you see static cling or a spark of electricity, remember that Coulomb's Law is at work behind the scenes!

Real-World Applications of Coulomb's Law: Beyond the Textbook

So, we've successfully calculated the value of equal charges using Coulomb's Law. But you might be wondering,