Calculating Electrical Force Magnitude And Direction A Physics Problem

Hey everyone! Let's dive into the fascinating world of electrical forces! We're going to explore a scenario involving two charges and calculate the force between them. This is a fundamental concept in physics, and understanding it will open doors to grasping more complex electromagnetic phenomena. So, buckle up, and let's get started!

The problem we're tackling involves a negative charge, lovingly named $q_1$, boasting a charge of $6 extit{μC}$ (that's micro Coulombs, guys!). This little fella is hanging out 0.002 meters north of a positive charge, $q_2$, which has a charge of $3 extit{μC}$. The big question is: what's the magnitude and direction of the electrical force, $F_{e}$, that $q_1$ exerts on $q_2$?

To crack this, we need to bring in the big guns: Coulomb's Law. This law is the cornerstone of electrostatics, describing the force between stationary charged particles. It's a beautiful equation that perfectly captures how charges interact. Coulomb's Law states that the electrical 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. Woah, that's a mouthful, right?

In simpler terms, this means:

• Bigger charges mean bigger force: The more charge each particle has, the stronger the force between them.
• Distance is key: The farther apart the charges are, the weaker the force. But it's not just a simple decrease – it's an inverse square relationship. This means if you double the distance, the force becomes four times weaker! (2 squared is 4, get it?).

The formula for Coulomb's Law looks like this:

$F = k * (|q_1 * q_2|) / r^2$

Where:

• $F$ is the magnitude of the electrical force.
• $k$ is Coulomb's constant, approximately $8.9875 × 10^9 N⋅m^2/C^2$ (a crucial number to remember!).
• $q_1$ and $q_2$ are the magnitudes of the charges.
• $r$ is the distance between the charges.
• The absolute value signs | | around $q_1 * q_2$ mean we only care about the magnitude (the numerical value) of the charges, not their sign, when calculating the force's magnitude. The sign will come into play when determining the direction of the force later.

Now that we've armed ourselves with Coulomb's Law, let's plug in the values from our problem. We've got:

• $q_1 = 6 extit{μC} = 6 × 10^{-6} C$ (Remember to convert micro Coulombs to Coulombs!)
• $q_2 = 3 extit{μC} = 3 × 10^{-6} C$
• $r = 0.002 m$
• $k = 8.9875 × 10^9 N⋅m^2/C^2$

Let's substitute these values into Coulomb's Law equation. Grab your calculators, guys, because we're about to do some math!

$F = (8.9875 × 10^9 N⋅m^2/C^2) * (|6 × 10^{-6} C * 3 × 10^{-6} C|) / (0.002 m)^2$

First, let's calculate the product of the charges:

$|6 × 10^{-6} C * 3 × 10^{-6} C| = 18 × 10^{-12} C^2$

Next, let's square the distance:

$(0.002 m)^2 = 4 × 10^{-6} m^2$

Now, we can plug these values back into the equation:

$F = (8.9875 × 10^9 N⋅m^2/C^2) * (18 × 10^{-12} C^2) / (4 × 10^{-6} m^2)$

Let's multiply the numerator:

$(8.9875 × 10^9 N⋅m^2/C^2) * (18 × 10^{-12} C^2) = 0.161775 N⋅m^2$

Finally, divide by the denominator:

$F = 0.161775 N⋅m^2 / (4 × 10^{-6} m^2) = 40443.75 N$

So, the magnitude of the electrical force is approximately 40.44 N. That's a pretty hefty force for such small charges!

But we're not done yet! We need to determine the direction of this force. Remember, force is a vector quantity, meaning it has both magnitude and direction. To figure out the direction, we need to consider the signs of the charges.

This is where things get interesting! The direction of the electrical force is determined by the signs of the charges involved. Remember the fundamental rule: opposites attract, and likes repel.

In our case, $q_1$ is negative, and $q_2$ is positive. This means they're going to attract each other. The question asks for the force applied by $q_1$ on $q_2$. So, $q_1$ is pulling $q_2$ towards itself.

Since $q_1$ is north of $q_2$, the force exerted by $q_1$ on $q_2$ will be in the northward direction. Think of it like a magnet pulling another magnet – they're drawn together!

Therefore, the direction of the electrical force $F_{e}$ is northward.

Let's recap our findings. We've calculated that the magnitude of the electrical force exerted by $q_1$ on $q_2$ is approximately 40.44 N, and the direction of this force is northward. Awesome! We've successfully solved the problem!

Let's take a moment to solidify our understanding by reviewing the key concepts and takeaways from this problem.

• Coulomb's Law is King: This law is the foundation for calculating the electrical force between charged particles. Mastering it is crucial for understanding electrostatics. The formula $F = k * (|q_1 * q_2|) / r^2$ should be your new best friend!
• Inverse Square Law: The force decreases rapidly as the distance between charges increases. Remember, doubling the distance reduces the force by a factor of four! This inverse square relationship is a common theme in physics, popping up in gravity and light intensity as well.
• Opposites Attract, Likes Repel: This simple rule governs the direction of the electrical force. It's a fundamental principle that makes the world of electromagnetism tick.
• Vector Nature of Force: Always remember that force is a vector quantity. This means you need to consider both magnitude and direction when describing a force. Don't just give a number – tell us where it's pointing!
• Units are Crucial: Pay close attention to units! Make sure you're using consistent units throughout your calculations (e.g., Coulombs for charge, meters for distance). A little unit conversion can save you from big errors.

Now that you've conquered this problem, let's think about how we can expand our knowledge further. Here are a few ideas:

• Try Different Scenarios: What happens if we change the magnitudes of the charges? What if we change the distance between them? Experiment with different values to see how the force changes.
• Explore Repulsive Forces: What if both charges were positive or both were negative? How would the direction of the force change? Think about how the charges would interact in this scenario.
• Multiple Charges: What if there were three or more charges? How would you calculate the net force on a particular charge? This introduces the concept of vector addition, which is a powerful tool in physics.
• Electric Fields: Electrical force is closely related to the concept of electric fields. Dive into electric fields to gain a deeper understanding of how charges interact.
• Real-World Applications: Electrical forces are at play all around us, from the static electricity that makes your hair stand on end to the technology that powers our devices. Think about how these forces are used in everyday life.

By exploring these questions and concepts, you'll build a stronger foundation in electromagnetism and physics as a whole. Keep questioning, keep exploring, and keep learning!

We've journeyed through the world of electrical forces, tackled a problem involving charged particles, and uncovered the secrets of Coulomb's Law. We've learned how to calculate the magnitude and direction of the electrical force between charges, and we've explored the key concepts that govern these interactions.

Remember, guys, the key to mastering physics is to break down complex problems into smaller, manageable steps. By understanding the fundamental principles and practicing applying them, you can conquer any challenge that comes your way. So, keep up the great work, and keep exploring the wonders of physics! This is just the beginning of an exciting adventure into the realm of electromagnetism.

To really solidify your understanding, here are a couple of practice problems for you to tackle:

1. A charge of +$4 extit{μC}$ is located 0.005 m east of a charge of -$8 extit{μC}$. Calculate the magnitude and direction of the electrical force exerted by the -$8 extit{μC}$ charge on the +$4 extit{μC}$ charge.
2. Two identical positive charges of $2 extit{μC}$ are placed 0.01 m apart. What is the magnitude and direction of the electrical force acting on each charge?

Give these a try, and don't hesitate to review the concepts we've discussed if you get stuck! Remember, practice makes perfect!