Charged Spheres: Attraction To Repulsion Explained

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Hey guys, let's dive into a classic physics problem involving charged spheres! We're gonna break down how these little guys, initially attracting each other, end up repelling. It's a cool illustration of Coulomb's Law and the conservation of charge. Buckle up, because we're about to get our science on!

The Setup: Initial Attraction

So, imagine this: two identical, tiny charged spheres. They're chilling 0.2 meters apart. Initially, they're attracting each other with a force of 4 x 10⁻³ Newtons. This attraction tells us something super important right off the bat: the charges on the spheres must be opposite in sign. One is positive, and the other is negative. That's the basic rule of attraction, remember? Opposites attract! The magnitude of the force is given by Coulomb's Law:

F = k * (|q₁| * |q₂|) / r²

Where:

  • F is the electrostatic force.
  • k is Coulomb's constant (approximately 8.99 x 10⁹ N⋅m²/C²).
  • q₁ and q₂ are the charges on the spheres.
  • r is the distance between the spheres.

At this stage, we're using the absolute values of the charges (|q₁| and |q₂|) because Coulomb's Law deals with the magnitude of the force, not its direction (attraction or repulsion). The negative sign in the force is implicitly handled by the opposite charges causing attraction. The problem doesn't give us the exact values of the charges. But, we can use the information about the final repulsion to figure it out. Also, we know the distance, and the force which means that we can calculate the product of the absolute value of charges. From the initial conditions, we have:

4 x 10⁻³ N = k * (|q₁| * |q₂|) / (0.2 m)²

Let's call the initial charges q₁ and q₂. The fact that they attract means that their product is negative (since one is positive and the other negative). We can solve for the product of the charges |q₁ * q₂| but we can't determine the individual charges. This initial attraction sets the stage for the main event: what happens when the spheres touch?

The Touch: Charge Redistribution

Next, the spheres get cozy and touch each other. When this happens, some charges flow. Since the spheres are identical, the charges will distribute themselves equally. So, the total charge is shared. The charges on each sphere will become the average of the initial charges. Imagine the charges are q₁ and q₂ before they touch, and the resulting charge is q' on both spheres. After they touch, the charge on each sphere will be:

q' = (q₁ + q₂) / 2

This is a direct consequence of the principle of charge conservation. The total charge in the system remains constant, it's just distributed differently. It's similar to mixing two liquids together: the total volume remains the same, but the concentration changes. So, the individual charges rearrange until they reach an equilibrium, where the forces are balanced.

The Aftermath: Repulsion

After they touch, the spheres are separated back to the original distance of 0.2 m. Now, they repel each other with a force of 2.25 x 10⁻³ N. This is our clue that the spheres now have charges of the same sign. This means that after the touch, they ended up with the same charge. The fact that they repel indicates that the new charges are either both positive or both negative. Therefore, the final charge of each sphere is q' = (q₁ + q₂) / 2. We can use Coulomb's law again, but now the charges are the same.

  1. 25 x 10⁻³ N = k * (q'²) / (0.2 m)²

Where q' is the new charge on each sphere.

This repulsion provides us with a key piece of information: the charge on each sphere after they touched. Solving for q', we get: q' = ± √(2.25 x 10⁻³ N * (0.2 m)² / k)

Solving the Problem

Let's go through the detailed steps. First, we use the repulsion force to find the final charge q'. We rearrange the Coulomb's law equation for the repulsion:

|q'| = √(F * r² / k) = √(2.25 x 10⁻³ N * (0.2 m)² / 8.99 x 10⁹ N⋅m²/C²) ≈ 1 x 10⁻⁷ C

This means that after they touched, each sphere has approximately 1 x 10⁻⁷ C of charge. We now know that after they touched, the spheres have the same charge value. The sign depends on the initial charges, however, we know that the magnitude is the same. We know the product of the absolute value of the original charge. We know the charge after they touched each other. We can use this information to calculate the initial charge on each sphere. Let's denote the charge on sphere 1 as q₁, and the charge on sphere 2 as q₂. The average charge will be q' = (q₁ + q₂) / 2.

So, we have two equations:

  1. q₁ * q₂ = - (4 x 10⁻³ N * (0.2 m)² / k)
  2. (q₁ + q₂) / 2 = q' = 1 x 10⁻⁷ C (approximately)

From the second equation, we can find that q₁ + q₂ = 2 x 10⁻⁷ C. Knowing that q₂ = 2 x 10⁻⁷ C - q₁, we can substitute it into the first equation.

q₁ * (2 x 10⁻⁷ C - q₁) = - (4 x 10⁻³ N * (0.2 m)² / k)

Solving this quadratic equation for q₁ will give us the initial charges. This will result in two possible answers for the initial charges, one positive and one negative. Each charge's absolute value would then be:

|q₁| ≈ 2.63 x 10⁻⁷ C and |q₂| ≈ -6.3 x 10⁻⁸ C.

Or approximately:

|q₁| ≈ -6.3 x 10⁻⁸ C and |q₂| ≈ 2.63 x 10⁻⁷ C

Conclusion

So, what did we learn, guys? We saw how two oppositely charged spheres attract, then redistribute their charge when they touch, and finally repel each other because they end up with charges of the same sign. This problem perfectly illustrates Coulomb's law and the principles of charge conservation. It's a neat little puzzle that shows how seemingly simple interactions can reveal fundamental laws of physics. Keep experimenting, keep questioning, and always remember, the universe is full of fascinating stuff waiting to be understood! And that, my friends, is how you tackle a classic physics problem! Pretty cool, right?