Problem 5 - Fundamentals of Physics

Question

A particle of charge \(+3.00 \times 10^{-6} \mathrm{C}\) is \(12.0 \mathrm{~cm}\) distant from a second particle of charge \(-1.50 \times 10^{-6} \mathrm{C}\). Calculate the magnitude of the electrostatic force between the particles.

Step-by-Step Solution

Answer

The magnitude of the electrostatic force is \(2.81 \times 10^{-9} \, \mathrm{N}\).

Step 1: Understanding Coulomb's Law

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. The formula is given by:\[ F = k \frac{|q_1 q_2|}{r^2} \]where \( k \) is Coulomb's constant (\(8.99 \times 10^9 \, \mathrm{N}\cdot\mathrm{m}^2/\mathrm{C}^2\)), \( q_1 \) and \( q_2 \) are the charges, and \( r \) is the distance between the charges.

Step 2: Identifying the values in the problem

Identify the given values that will be used in the formula:- \( q_1 = +3.00 \times 10^{-6} \, \mathrm{C} \)- \( q_2 = -1.50 \times 10^{-6} \, \mathrm{C} \)- \( r = 12.0 \, \mathrm{cm} = 0.12 \, \mathrm{m} \) (converted from cm to meters)

Step 3: Applying Coulomb's Law

Plug the values into Coulomb's Law formula to find the force:\[ F = (8.99 \times 10^9) \times \frac{|3.00 \times 10^{-6} \times (-1.50 \times 10^{-6})|}{(0.12)^2} \]

Step 4: Calculating the product of charges

Calculate the product of the charges:\[ |q_1 \, q_2| = |(3.00 \times 10^{-6}) \times (-1.50 \times 10^{-6})| = 4.50 \times 10^{-12} \, \mathrm{C^2} \]

Step 5: Calculating the square of the distance

Calculate the square of the distance between the particles:\[ r^2 = (0.12)^2 = 0.0144 \, \mathrm{m^2} \]

Step 6: Calculating the electrostatic force

Insert the results from Steps 4 and 5 back into the formula:\[ F = (8.99 \times 10^9) \times \frac{4.50 \times 10^{-12}}{0.0144} \]\[ F = (8.99 \times 10^9) \times 3.13 \times 10^{-10} \]\[ F = 2.81 \times 10^{-9} \, \mathrm{N} \]

Step 7: Interpreting the result

The magnitude of the electrostatic force between the two charges is \( 2.81 \times 10^{-9} \, \mathrm{N} \). The force is attractive because the charges have opposite signs.

Key Concepts

Electrostatic Force

The electrostatic force is a fundamental interaction between two charged particles or objects. It is a type of force that can attract or repel objects, depending on the nature of their charges. Key Characteristics:

• The force can be either attractive or repulsive. If the charges are opposite, the force is attractive, while like charges produce a repulsive force.
• Electrostatic forces are governed by Coulomb's Law, which provides a mathematical expression to quantify this interaction.

Understanding electrostatic force is crucial in fields like physics and electrical engineering, as it helps explain how charged particles behave in different environments. It is also the driving principle behind many technologies, such as capacitors and electrostatic precipitation systems.

Point Charges

Point charges are hypothetical charges located at a singular point in space. These are used in theoretical physics to simplify complex charge interactions and to make calculations feasible. Nature of Point Charges:

• A point charge has a negligible size but a significant amount of charge.
• They are treated as having no spatial extent, which allows us to apply mathematical formulas like Coulomb's Law directly.
• In our exercise, the charges of the particles are considered as point charges for ease of calculation.

Using point charges, we ignore real-world details like the shape and size of the charged objects to easily predict electrostatic interactions mathematically.

Inverse Square Law

The inverse square law is a key principle in physics that describes how the strength of certain forces diminishes with distance. Specifically, for electrostatic force, this law states that the force between two point charges is inversely proportional to the square of the distance between them. How It Works:

• The further apart two charges are, the weaker the force between them becomes, rapidly diminishing with distance.
• This decrease follows a square relation: doubling the distance results in a force that is one-fourth its original strength.

For instance, in the given exercise, as the distance of 12 cm between two charges grows, the resulting force decreases accordingly, following this law. This principle is vital for understanding forces in electromagnetic fields and assessments of gravitational forces as well.