Calculating Electrostatic Charge On Plastic Balls Using Coulomb's Law
In the realm of physics, understanding electrostatic forces is crucial. This article delves into a classic problem involving two negatively charged plastic balls and the repulsive force between them. We will explore how to calculate the charge on each ball under two different scenarios: first, when the charges are equal, and second, when one ball possesses twice the charge of the other. This exploration will not only solidify your understanding of Coulomb's Law but also demonstrate its practical application in determining electric charges.
Introduction to Electrostatic Forces
Before diving into the calculations, it's essential to grasp the fundamental concept of electrostatic forces. These forces, governed by Coulomb's Law, describe the interaction between electrically charged objects. Coulomb's Law states that the 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. Mathematically, this is expressed as:
F = k * |q1 * q2| / r^2
Where:
- F is the electrostatic force
- k is Coulomb's constant (approximately 8.9875 × 10^9 N⋅m2/C2)
- q1 and q2 are the magnitudes of the charges
- r is the distance between the charges
The force is repulsive if the charges have the same sign (both positive or both negative) and attractive if the charges have opposite signs. In our scenario, we are dealing with two negatively charged balls, implying a repulsive force between them. The given information includes the distance between the balls (50 cm or 0.5 m) and the magnitude of the repulsive force (12 N). Our goal is to determine the charge on each ball under different conditions.
Scenario 1: Equal Charges on Both Balls
In this first scenario, we assume that both plastic balls carry the same amount of negative charge. Let's denote this charge as 'q'. This simplifies our problem significantly, as we now have only one unknown variable. We can directly apply Coulomb's Law and solve for 'q'. The key here is to understand how the symmetry of the situation simplifies the calculation. The force each ball exerts on the other is equal in magnitude, and this symmetry is reflected in our mathematical setup.
Applying Coulomb's Law
We start with the formula:
F = k * |q1 * q2| / r^2
Since q1 = q2 = q, the formula becomes:
F = k * q^2 / r^2
We are given F = 12 N and r = 0.5 m. Plugging in the values, we get:
12 N = (8.9875 × 10^9 N⋅m2/C2) * q^2 / (0.5 m)^2
Now, we solve for q^2:
q^2 = (12 N * (0.5 m)^2) / (8.9875 × 10^9 N⋅m2/C2)
q^2 ≈ 3.338 × 10^-10 C^2
Calculating the Charge
To find the charge 'q', we take the square root of both sides:
q = √ (3.338 × 10^-10 C^2)
q ≈ 1.827 × 10^-5 C
Since the balls are negatively charged, the charge on each ball is approximately -1.827 × 10^-5 Coulombs. This result demonstrates how Coulomb's Law allows us to quantitatively determine the electric charge when the force and distance are known. This symmetrical case serves as a building block for understanding more complex scenarios where charges may not be equal.
Scenario 2: One Ball Has Twice the Charge of the Other
Now, let's consider a more intricate scenario where the charges on the two balls are unequal. Specifically, we are told that one ball has twice the charge of the other. This introduces an asymmetry into the problem, which we must carefully account for in our calculations. Denoting the charge on the first ball as 'q', the charge on the second ball will be '2q'. While the fundamental principles of Coulomb's Law remain the same, the algebraic manipulation required to solve for 'q' becomes slightly more involved.
Setting Up the Equation
Again, we start with Coulomb's Law:
F = k * |q1 * q2| / r^2
In this case, q1 = q and q2 = 2q. Substituting these values, we get:
F = k * |q * 2q| / r^2
F = 2 * k * q^2 / r^2
We are given F = 12 N and r = 0.5 m. Plugging in these values, we have:
12 N = 2 * (8.9875 × 10^9 N⋅m2/C2) * q^2 / (0.5 m)^2
Solving for the Unknown Charge
Now, we isolate q^2 and solve for it:
q^2 = (12 N * (0.5 m)^2) / (2 * 8.9875 × 10^9 N⋅m2/C2)
q^2 ≈ 1.669 × 10^-10 C^2
Taking the square root of both sides to find 'q':
q = √ (1.669 × 10^-10 C^2)
q ≈ 1.292 × 10^-5 C
Determining Both Charges
This value represents the charge on the first ball. Since the second ball has twice the charge, its charge is:
2q ≈ 2 * (1.292 × 10^-5 C)
2q ≈ 2.584 × 10^-5 C
Therefore, the charges on the two balls are approximately -1.292 × 10^-5 Coulombs and -2.584 × 10^-5 Coulombs, respectively. This result highlights the importance of carefully considering the relationships between charges when applying Coulomb's Law. The asymmetry in the charge distribution leads to a different numerical result compared to the equal charge scenario.
Conclusion: The Significance of Coulomb's Law
In summary, we have successfully calculated the charges on two plastic balls under different charge distribution scenarios, leveraging the power of Coulomb's Law. This fundamental law of electrostatics allows us to quantitatively analyze the interactions between charged objects. The calculations demonstrate that understanding the relationship between force, charge, and distance is crucial for solving electrostatic problems. Whether the charges are equal or unequal, Coulomb's Law provides the framework for determining the magnitudes of these charges.
This exercise underscores the importance of Coulomb's Law in various applications, ranging from basic physics problems to advanced engineering designs. A solid grasp of this principle is essential for anyone studying electromagnetism or working with electrical systems. The ability to accurately calculate electrostatic forces and charges is a cornerstone of understanding the behavior of electric fields and their interactions with matter. As we continue to explore the world of electromagnetism, the foundational knowledge provided by Coulomb's Law will remain invaluable.