Charged Spheres: Force Calculation After Contact

Hey guys! Let's dive into an interesting physics problem involving charged spheres and the forces between them. We're going to break down how to calculate the force when two spheres with different charges come into contact and then get separated. This is a classic example that uses Coulomb's Law, and understanding it will really solidify your grasp of electrostatics. So, buckle up and let's get started!

Understanding the Problem: Initial Charges and Contact

First, let's clearly define the situation. We have two spheres, one with a positive charge (q1 = +80 μC) and another with a negative charge (q2 = -18 μC). The crucial part here is what happens when these spheres come into contact. When they touch, the charges will redistribute themselves until both spheres have the same charge. Think of it like pouring water between two glasses – the water level will equalize.

So, how do we calculate this equalized charge? It’s pretty straightforward: we simply add the charges together and divide by two. This gives us the average charge that each sphere will have after contact. Mathematically, it looks like this:

q_final = (q1 + q2) / 2

Plugging in our values, we get:

q_final = (+80 μC + (-18 μC)) / 2 = 62 μC / 2 = +31 μC

Therefore, after contact, both spheres will have a charge of +31 μC. This is a key piece of information because it’s the charge we'll use in Coulomb's Law to calculate the force. Remember, the units are important! We’re working with microcoulombs (μC), which is 10^-6 Coulombs. This conversion will be necessary when we use Coulomb's constant.

Understanding this charge redistribution is the foundation for solving the problem. Without it, we'd be using the wrong values and get a completely incorrect answer. So, take a moment to make sure you're comfortable with this concept before moving on. We're essentially finding the equilibrium charge state after the interaction.

Applying Coulomb's Law: Calculating the Force

Now that we know the final charge on each sphere (+31 μC), and we know the distance separating them (3 cm), we can use Coulomb's Law to calculate the force between them. Coulomb's Law is the fundamental equation that describes the electrostatic force between two charged objects. It states that the force is directly proportional to the product of the charges and inversely proportional to the square of the distance between them.

The mathematical form of Coulomb's Law is:

F = k * |q1 * q2| / r^2

Where:

• F is the electrostatic force (in Newtons)
• k is Coulomb's constant (approximately 8.9875 × 10^9 N⋅m^2/C2)
• q1 and q2 are the magnitudes of the charges (in Coulombs)
• r is the distance between the charges (in meters)

Notice the absolute value signs around q1 * q2. This is because we're interested in the magnitude of the force, not its direction. The sign of the charges will tell us whether the force is attractive or repulsive.

Before we plug in our numbers, we need to make sure our units are consistent. Our charges are in microcoulombs (μC), and the distance is in centimeters (cm). We need to convert these to Coulombs (C) and meters (m), respectively:

• +31 μC = 31 × 10^-6 C
• 3 cm = 0.03 m

Now we have all the pieces we need! Let’s plug the values into Coulomb's Law:

F = (8.9875 × 10^9 N⋅m^2/C2) * |(31 × 10^-6 C) * (31 × 10^-6 C)| / (0.03 m)^2

This looks a bit intimidating, but let's break it down step by step. First, multiply the charges:

(31 × 10^-6 C) * (31 × 10^-6 C) = 9.61 × 10^-10 C^2

Then, square the distance:

(0.03 m)^2 = 9 × 10^-4 m^2

Now, plug these back into the equation:

F = (8.9875 × 10^9 N⋅m^2/C2) * (9.61 × 10^-10 C^2) / (9 × 10^-4 m^2)

Multiply the numerator:

(8.9875 × 10^9 N⋅m^2/C2) * (9.61 × 10^-10 C^2) = 8.637 × 10^0 N⋅m^2

Finally, divide by the denominator:

F = (8.637 N⋅m^2) / (9 × 10^-4 m^2) = 9596.67 N

So, the magnitude of the force between the spheres is approximately 9596.67 Newtons. That's a pretty significant force! But remember, we're dealing with relatively large charges and a small distance.

Determining the Nature of the Force: Repulsive or Attractive?

We've calculated the magnitude of the force, but we also need to determine its nature – is it attractive or repulsive? This is where the signs of the charges come into play. Remember, like charges repel, and opposite charges attract.

After contact, both spheres have a positive charge (+31 μC). Since they have the same sign, the force between them will be repulsive. This means the spheres will push away from each other.

It's important to state both the magnitude and the nature of the force to fully describe the interaction between the charged spheres. So, our final answer is:

The force between the two spheres is approximately 9596.67 Newtons, and it is repulsive.

Key Takeaways and Conceptual Understanding

Let’s recap what we've done and highlight some key concepts:

1. Charge Redistribution: When charged objects come into contact, charges redistribute to equalize the potential. We calculated the final charge by averaging the initial charges.
2. Coulomb's Law: This is the fundamental law governing electrostatic forces. It states that the force is proportional to the product of the charges and inversely proportional to the square of the distance.
3. Units: Always pay attention to units! We needed to convert microcoulombs to Coulombs and centimeters to meters before applying Coulomb's Law.
4. Magnitude and Nature: We calculated the magnitude of the force using Coulomb's Law and determined its nature (attractive or repulsive) based on the signs of the charges.

Understanding these concepts is crucial for tackling other electrostatics problems. This example highlights the importance of both the mathematical calculations and the underlying physics principles.

Practice Problems and Further Exploration

To really solidify your understanding, try working through similar problems with different charges and distances. You can also explore how the force changes if you change the medium between the spheres (this involves the dielectric constant).

Here are a few practice problems you can try:

1. Two spheres have charges of +40 μC and -20 μC. They come into contact and are then separated by a distance of 5 cm. Calculate the force between them.
2. Two spheres have equal charges of +10 μC and are separated by a distance of 2 cm in a vacuum. Calculate the force between them. How would the force change if the spheres were submerged in water (dielectric constant ≈ 80)?

Working through these problems will help you build confidence and a deeper understanding of electrostatics. Physics is all about practice, so don't be afraid to get your hands dirty with calculations!

Conclusion: Mastering Electrostatic Forces

So there you have it! We've successfully calculated the force between two charged spheres after they came into contact. By understanding charge redistribution and applying Coulomb's Law, you can tackle a wide range of electrostatic problems. Remember to always pay attention to units, and don't forget to consider the nature of the force (attractive or repulsive).

Keep practicing, keep exploring, and you'll become a master of electrostatic forces in no time! Good luck, guys, and happy calculating!