Calculating Attractive Force Between Point Charges
Hey guys! Today, we're diving into a classic physics problem involving electric charges and the forces they exert on each other. We'll be calculating the attractive force between two point charges in a vacuum. This is a fundamental concept in electromagnetism, so let's get started!
The Problem: Two Point Charges in a Vacuum
Let's break down the problem we're tackling. We have two point charges, which means we're treating them as if all their charge is concentrated at a single point. This is a common simplification in physics that makes calculations much easier. These charges are in a vacuum, so we don't have to worry about any interference from other materials. The charges are:
- q₁ = -1.0 × 10⁻⁶ C (Coulombs)
- q₂ = -2.0 × 10⁻⁶ C (Coulombs)
Notice that both charges are negative. This is crucial because charges with the same sign repel each other, while charges with opposite signs attract. Since both our charges are negative, they will repel each other. However, the problem asks for the attractive force, which seems like a contradiction! Don't worry, we'll address this shortly.
The distance separating these charges is:
- r = 3.0 cm = 0.03 m (We need to convert centimeters to meters for consistency in our units)
And finally, we're given the electrostatic constant:
- k = 9 × 10⁹ N⋅m²/C²
This constant is a fundamental part of Coulomb's Law, which we'll use to calculate the force.
Understanding Coulomb's Law
Coulomb's Law is the cornerstone of electrostatics, describing the force between charged objects. It's expressed mathematically as:
F = k * |q₁ * q₂| / r²
Where:
- F is the magnitude of the electrostatic force
- k is Coulomb's constant (approximately 9 × 10⁹ N⋅m²/C²)
- q₁ and q₂ are the magnitudes of the charges
- r is the distance between the charges
Key takeaways from Coulomb's Law include the force being directly proportional to the product of the charges' magnitudes. Meaning, if you increase either charge, the force increases proportionally. The force is also inversely proportional to the square of the distance. Increase the distance, and the force drops dramatically (squared!).
Solving for the Force of Attraction
Now, let's plug in the values we have into Coulomb's Law. Remember, we're looking for the magnitude of the force, so we'll use the absolute values of the charges:
F = (9 × 10⁹ N⋅m²/C²) * |(-1.0 × 10⁻⁶ C) * (-2.0 × 10⁻⁶ C)| / (0.03 m)²
Let's break this down step-by-step:
- Multiply the charges: |-1.0 × 10⁻⁶ C * -2.0 × 10⁻⁶ C| = 2.0 × 10⁻¹² C²
- Square the distance: (0.03 m)² = 9.0 × 10⁻⁴ m²
- Now we have: F = (9 × 10⁹ N⋅m²/C²) * (2.0 × 10⁻¹² C²) / (9.0 × 10⁻⁴ m²)
- Multiply the numerator: (9 × 10⁹ N⋅m²/C²) * (2.0 × 10⁻¹² C²) = 1.8 × 10⁻² N⋅m²
- Finally, divide by the denominator: F = (1.8 × 10⁻² N⋅m²) / (9.0 × 10⁻⁴ m²) = 20 N
So, the magnitude of the force of attraction between the two charges is 20 Newtons. But wait! We said earlier that like charges repel. What's going on?
