Coulomb’s law

Welcome to Electric Fields: Coulomb’s Law

Have you ever rubbed a balloon on your hair and watched it stick to a wall, or felt a tiny zap when touching a metal doorknob? These everyday moments are caused by electric forces. In this chapter, we are going to look at the "rulebook" that governed those forces: Coulomb’s Law.

Don’t worry if Physics feels like a lot of math right now. Think of this chapter as learning how particles "talk" to each other across space without even touching!

1. Prerequisite: What is Charge?

Before we dive into the law, let’s quickly recap what we’re dealing with:

• Electric charge is a property of matter.
• There are two types: positive and negative.
• The "golden rule" of charges: Opposites attract, and likes repel.
• Charge is measured in Coulombs (C).

2. Point Charges and Spheres

In Physics problems, we often talk about point charges. A point charge is a charge that exists at a single mathematical point with no size.

Did you know? Even though a van de Graaff generator or a charged metal ball is quite large, we can model a uniformly charged sphere as a point charge located at its very center. This makes our calculations much simpler because we can just measure the distance from the center of the sphere!

Quick Review: The Point Charge Model

Treat any uniformly charged sphere as if all its charge is concentrated at the center.

3. Coulomb’s Law: The Big Equation

Coulomb’s Law calculates the electric force (F) between two point charges.

The formula is: \( F = \frac{Qq}{4\pi\epsilon_0 r^2} \)

Let’s break down what these symbols mean:

• \( F \): The force between the charges (measured in Newtons, N).
• \( Q \) and \( q \): The magnitude of the two charges (measured in Coulombs, C).
• \( r \): The distance between the centers of the two charges (measured in meters, m).
• \( \epsilon_0 \): This is the permittivity of free space. It is a constant that represents how easily an electric field passes through a vacuum (roughly \( 8.85 \times 10^{-12} \text{ F m}^{-1} \)).
• \( 4\pi \): This appears because the electric field spreads out in all directions like a sphere.

Understanding the Relationships

1. Directly Proportional to Charge: If you double the charge of one particle, the force doubles. Think of it like magnets: stronger magnets mean a stronger pull.

2. The Inverse Square Law (\( 1/r^2 \)): This is the tricky part! The force is inversely proportional to the square of the distance.

• If you double the distance (\( 2r \)), the force becomes 4 times weaker (\( 1/2^2 = 1/4 \)).
• If you triple the distance (\( 3r \)), the force becomes 9 times weaker (\( 1/3^2 = 1/9 \)).

Memory Aid: The "Social Distancing" Rule

The further away charges get, the force doesn't just get weaker—it gets much weaker very quickly because of that squared symbol on the bottom!

4. Electric Field Strength (\( E \))

If Coulomb's Law tells us the force between two things, Electric Field Strength tells us how "strong" the field is at a certain point, even if there isn't another charge there to feel it.

The general definition is: \( E = \frac{F}{q} \)

This means the field strength is the force per unit positive charge. If you imagine a tiny "+1 Coulomb" test charge, the force it feels is the field strength.

Field Strength for a Point Charge

By combining the definition of \( E \) with Coulomb's Law, we get: \( E = \frac{Q}{4\pi\epsilon_0 r^2} \)

Notice that this formula only needs one charge (Q). It tells you the strength of the field created by that single charge at a distance \( r \).

Quick Review: Key Differences

Force (F): Needs TWO charges to exist. Measured in Newtons (N).
Field Strength (E): Created by ONE charge. Measured in Newtons per Coulomb (N/C).

5. Comparing Electric and Gravitational Fields

You might notice that Coulomb’s Law looks a lot like Newton’s Law of Gravitation (\( F = \frac{GMm}{r^2} \)). They are both inverse square laws, but they have some big differences!

Similarities:

• Both involve a "field" that acts across a distance without touching.
• Both follow the inverse square law (\( F \propto 1/r^2 \)).
• Both involve a constant (\( G \) for gravity, \( \epsilon_0 \) for electricity).

Differences:

• Mass vs. Charge: Gravity depends on mass (always positive); Electric force depends on charge (can be positive or negative).
• Attraction vs. Repulsion: Gravity is always attractive. Electric forces can be attractive OR repulsive.
• Strength: Electric forces are significantly stronger than gravitational forces on a subatomic level.

6. Common Mistakes to Avoid

• Forgetting to square \( r \): This is the most common error in exams. Always check your calculator work!
• Using the wrong units: Distances must be in meters (m), not cm or mm. Charges are often given in micro-Coulombs (\( \mu\text{C} \)), which means \( \times 10^{-6} \text{ C} \).
• Mixing up \( F \) and \( E \): If the question asks for "Force," you need two charges. If it asks for "Field Strength," you are looking for the effect of one charge at a specific point.

Key Takeaways for Revision

1. Electric fields are created by charges.
2. Uniformly charged spheres act like point charges at their centers.
3. Coulomb’s Law calculates force: \( F = \frac{Qq}{4\pi\epsilon_0 r^2} \).
4. Electric Field Strength is force per unit charge: \( E = \frac{Q}{4\pi\epsilon_0 r^2} \).
5. Doubling distance reduces force/field strength by four times.