Electrical Force Calculation Using Coulomb's Law Step-by-Step Solution

Hey everyone! Let's dive into a classic physics problem involving Coulomb's Law. We're going to figure out the electrical force between two charges, each with a magnitude of 1 Coulomb (1C), separated by a distance of 1 meter. It sounds intense, right? But don't worry, we'll break it down step by step so it's super clear. Understanding these fundamental concepts is crucial, whether you're a student tackling homework or just curious about the forces that govern the universe around us. We'll explore how Coulomb's Law helps us quantify this force, the significance of the constant involved, and what the magnitude of the force actually tells us about the interaction between these charges. Get ready to put on your thinking caps and explore the fascinating world of electromagnetism!

Understanding Coulomb's Law

To calculate the electrical force, we'll use Coulomb's Law. This law is a cornerstone of electromagnetism, describing the force between electrically charged objects. It's similar in some ways to Newton's Law of Universal Gravitation, but instead of dealing with masses, we're dealing with charges. Let's break down the formula:

• Formula: F = k * (|q1 * q2|) / r²

  • Where:

    • F is the electrical force.
    • k is Coulomb's constant, approximately 8.99 x 10^9 N⋅m²/C².
    • q1 and q2 are the magnitudes of the charges.
    • r is the distance between the charges.

• Key Concepts Explained

  • Electrical Force (F): This is the force we're trying to find! It tells us how strongly the charges are either attracting or repelling each other. A larger value of F means a stronger force.
  • Coulomb's Constant (k): This constant is super important. It's like a conversion factor that makes sure our units work out correctly. The value 8.99 x 10^9 N⋅m²/C² might seem intimidating, but it's just a number we plug into the formula.
  • Magnitudes of the Charges (q1 and q2): These are the amounts of charge each object has. In our case, both charges are 1C. The absolute value signs (| |) mean we only care about the size of the charge, not whether it's positive or negative for this part of the calculation.
  • Distance Between Charges (r): This is the separation between the two charges, which is 1 meter in our problem. Notice that the distance is squared in the formula (r²), meaning the force changes dramatically with distance. If you double the distance, the force becomes four times weaker!

The Role of Coulomb's Constant in Determining Electrical Force: The magnitude of Coulomb's constant (k) plays a crucial role in determining the strength of the electrical force. A larger value of k indicates a stronger force for the same amount of charge and separation distance. In the case of the given problem, where two 1C charges are separated by 1 meter, the electrical force is directly proportional to Coulomb's constant. This means that the force will be approximately 8.99 x 10^9 N, highlighting the immense strength of electrical forces at this scale. The constant essentially scales the force to the appropriate level, given the charges and the distance between them, underscoring its importance in accurately calculating electrical interactions.

The significance of distance in influencing the electrical force: The distance between charged objects dramatically influences the electrical force, as described by Coulomb's Law. The force is inversely proportional to the square of the distance, meaning that if you double the distance, the force decreases by a factor of four. This inverse square relationship has profound implications. At close range, electrical forces can be incredibly strong, but they rapidly weaken as the separation increases. This is why, in our example, even though the charges are only 1 meter apart, the resulting force is enormous due to the large value of Coulomb's constant and the inverse square relationship with distance. Understanding this relationship is crucial for predicting and manipulating electrical interactions in various scenarios, from electronics to particle physics.

Understanding the interplay between charge magnitude and distance: The interplay between charge magnitude and distance is crucial in determining the electrical force between two objects, as described by Coulomb's Law. While the force is directly proportional to the product of the charges (q1 and q2), it is inversely proportional to the square of the distance (r) between them. This means that both the amount of charge and the separation distance significantly impact the strength of the electrical force. For instance, doubling either charge will double the force, whereas doubling the distance will reduce the force by a factor of four. In the case of two 1C charges separated by 1 meter, the force is immensely strong due to the large magnitude of the charges and the relatively small distance. However, even if the charges were smaller, bringing them closer together could still result in a significant force. Understanding this relationship is essential for manipulating and controlling electrical forces in various applications, from designing electronic devices to studying the behavior of charged particles.

Applying Coulomb's Law to Our Problem

Now, let's plug in the values from our problem into Coulomb's Law:

• q1 = 1 C
• q2 = 1 C
• r = 1 m
• k = 8.99 x 10^9 N⋅m²/C²

So, the formula becomes:

F = (8.99 x 10^9 N⋅m²/C²) * (|1 C * 1 C|) / (1 m)²

Let's simplify this:

F = (8.99 x 10^9 N⋅m²/C²) * (1 C²) / (1 m²)

F = 8.99 x 10^9 N

Therefore, the electrical force between the two charges is 8.99 x 10^9 Newtons.

Step-by-step breakdown of substituting values into the formula: Let's walk through how we plugged the values into Coulomb's Law. First, we identified the given information: two charges (q1 and q2) of 1 Coulomb each, separated by a distance (r) of 1 meter. We also know Coulomb's constant (k) is approximately 8.99 x 10^9 N⋅m²/C². Now, we simply substitute these values into the formula F = k * (|q1 * q2|) / r². This gives us F = (8.99 x 10^9 N⋅m²/C²) * (|1 C * 1 C|) / (1 m)². This step is crucial because it sets up the calculation correctly, ensuring we use the appropriate values in their designated places within the equation. From here, we can simplify the expression to find the magnitude of the electrical force between the charges. It's like following a recipe – get the ingredients (values) right, and the dish (answer) will turn out perfectly!

Detailed simplification of the equation to arrive at the final answer: Okay, guys, let's break down the simplification process step-by-step. We started with F = (8.99 x 10^9 N⋅m²/C²) * (|1 C * 1 C|) / (1 m)². First, we tackle the numerator. The absolute value of 1 C times 1 C is simply 1 C². So, we have F = (8.99 x 10^9 N⋅m²/C²) * (1 C²) / (1 m)². Next, we simplify the denominator. 1 meter squared is just 1 m², so our equation becomes F = (8.99 x 10^9 N⋅m²/C²) * (1 C²) / (1 m²). Now, we can cancel out the units. The C² in the numerator cancels with the C² in Coulomb's constant, and the m² in Coulomb's constant cancels with the m² in the denominator. This leaves us with F = 8.99 x 10^9 N * (1) / (1), which simplifies to F = 8.99 x 10^9 N. Voila! We've simplified the equation to find the electrical force. It's like solving a puzzle, where each step brings us closer to the final picture.

Highlighting the role of units in ensuring the correctness of the result: Let's talk about units – they're the unsung heroes of physics calculations! In our problem, paying close attention to the units is crucial for ensuring the correctness of our result. When we plugged the values into Coulomb's Law, we had units like Coulombs (C) for charge, meters (m) for distance, and the complex unit N⋅m²/C² for Coulomb's constant. As we simplified the equation, we saw the magic happen: the C² units in the numerator canceled out the C² in Coulomb's constant, and the m² units in Coulomb's constant canceled out the m² in the denominator. This cancellation isn't just a math trick; it's a sign that we're on the right track! If the units didn't cancel out properly, we'd know we'd made a mistake somewhere. The fact that we ended up with Newtons (N), the unit of force, tells us that our final answer is dimensionally correct. It's like having the right ingredients and following the recipe correctly – the units are the taste test that tells us if we've cooked up the right result!

The Answer and Its Significance

So, the correct answer is:

• d) 8.99 x 10^9 N

This is a huge force! It's equivalent to the weight of roughly 900,000 metric tons – that's like lifting the weight of a fleet of aircraft carriers! This result highlights the immense strength of the electrical force. Even though we're only dealing with charges of 1 Coulomb, which might not sound like much, the force is incredibly strong because of the magnitude of Coulomb's constant. This is why we don't typically encounter such large forces in everyday situations; charges tend to balance each other out.

Comparing the magnitude of the calculated force with everyday experiences: Let's put this force into perspective, guys. We calculated an electrical force of 8.99 x 10^9 Newtons between those two 1C charges separated by a meter. That's like trying to lift something that weighs around 900,000 metric tons! Think about it – a fully loaded aircraft carrier weighs about 100,000 metric tons, so we're talking about the equivalent of lifting nine aircraft carriers! This enormous force is way beyond anything we experience in our daily lives. When you pick up a book or push a door open, you're dealing with forces of a few Newtons, maybe a few hundred if you're pushing something heavy. But 8.99 x 10^9 Newtons? That's on a whole different scale! This comparison really drives home just how incredibly strong the electrical force can be, especially when dealing with significant amounts of charge. It's like comparing a gentle breeze to a category 5 hurricane – they're both wind, but their magnitudes are vastly different.

Discussing the implications of such a strong force in the context of electrostatics: Now, let's think about what this massive force means in the world of electrostatics. We've calculated that two 1C charges separated by just 1 meter exert a force of nearly 9 billion Newtons on each other. That's an astonishing amount of force! So why don't we see things flying apart all the time due to these forces? The key is that, in most everyday situations, objects are electrically neutral. They have roughly equal amounts of positive and negative charge, which cancel each other out. It's like a tug-of-war where both sides are pulling with equal strength – there's no net movement. But when we have a significant imbalance of charge, like in our hypothetical scenario, the forces become enormous. This highlights why static electricity, like the shock you get from touching a doorknob on a dry day, can be surprisingly powerful. It also underscores the importance of grounding and other safety measures when dealing with high-voltage systems. The immense forces at play in electrostatics, as demonstrated by our calculation, remind us of the fundamental power of electrical interactions.

Exploring why such large forces are not commonly observed in everyday scenarios: Okay, guys, so we've established that the force between two 1C charges a meter apart is mind-bogglingly huge. But it begs the question: why don't we see things exploding all the time due to these massive electrical forces? The answer lies in the fact that, in the vast majority of everyday situations, matter is electrically neutral. This means that objects have an equal balance of positive and negative charges. Think of it like a perfectly balanced scale – the positive and negative charges are in equilibrium, so there's no net force. However, when there's an imbalance of charge, that's when things get interesting (and potentially dangerous!). For example, when you rub a balloon on your hair, you're transferring electrons, creating a charge imbalance. This is why the balloon can then stick to the wall – the electrical force of attraction is at play. But even in these situations, the amount of charge involved is tiny compared to 1 Coulomb. That's why the forces, while noticeable, aren't on the scale of billions of Newtons. So, while electrical forces are fundamental and powerful, the neutrality of most matter keeps these huge forces in check in our daily lives. It's like having a sleeping giant – the potential is there, but it only awakens under specific circumstances.

Conclusion

In conclusion, using Coulomb's Law, we've determined that the electrical force between two 1C charges separated by 1 meter is a staggering 8.99 x 10^9 N. This exercise demonstrates the sheer power of electrical forces and the importance of understanding the principles that govern them. Remember, physics isn't just about formulas; it's about understanding the world around us. So keep exploring, keep questioning, and keep learning!

Recap of the problem-solving process using Coulomb's Law: Alright, let's do a quick recap of our journey through this problem. We started with the question: what's the electrical force between two 1C charges 1 meter apart? To tackle this, we reached into our physics toolbox and pulled out Coulomb's Law, the fundamental law governing electrostatic forces. We carefully defined each term in the formula – the force (F), Coulomb's constant (k), the charges (q1 and q2), and the distance (r). Then, we substituted the given values into the equation, making sure to keep track of our units. We simplified the equation step-by-step, and boom! We arrived at the answer: 8.99 x 10^9 N. But we didn't stop there. We put that number into perspective, realizing just how immense that force is. We also explored why we don't see such forces causing chaos in our everyday lives. By breaking down the problem and understanding each step, we've not only solved it but also gained a deeper appreciation for the power and intricacies of Coulomb's Law. It's like learning to ride a bike – once you get the hang of it, you can go anywhere!

Emphasizing the significance of understanding fundamental physics principles: Let's wrap this up by emphasizing something super important: the significance of understanding fundamental physics principles. You see, this problem wasn't just about plugging numbers into a formula. It was about grasping the concept of electrical force, understanding how Coulomb's Law works, and appreciating the relationships between charge, distance, and force. When you truly understand the principles, you can apply them to a wide range of situations. You're not just memorizing equations; you're developing a way of thinking about the world. It's like learning the rules of grammar – once you know them, you can write anything! So, whether you're studying for an exam, building a robot, or just trying to figure out why your socks stick together in the dryer, a solid understanding of fundamental physics principles will take you far. Keep digging deeper, keep asking questions, and keep building that foundation of knowledge – it's the key to unlocking the mysteries of the universe!