Lesson 6.2: Electric Fields and Coulomb's Law

Introduction

Welcome to Lesson 6.2! In this lesson, we will explore electric fields and Coulomb's law. Our goal is for you to understand the forces between point charges, how electric fields work, and how to analyze the motion of charged particles in these fields. If you've ever wondered how your phone charges or how lightning strikes occur, you're in the right place! ⚡

Learning Objectives

By the end of this lesson, you should be able to:

• Explain Coulomb's law and the force between point charges.
• Define electric field strength $E$ and draw field lines for point charges and parallel plates.
• Describe the motion of a charged particle in a uniform field, including the concept of the electronvolt.
• Compare gravitational and electric fields, noting their similarities and key differences.
• Calculate the force and field strength due to point charges.

Coulomb's Law and Electric Forces

Coulomb's law describes the force between two point charges. It states that the force $F$ between two charges $q_1$ and $q_2$ is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance $r$ between them. This can be expressed mathematically as:

$$

F = k $\cdot$ $\frac{|q_1 \cdot q_2|}{r^2}$

$$

where $k$ is Coulomb's constant, approximately equal to $8.99 \times 10^9 \, \text{N m}^2/\text{C}^2$.

Let's break this down:

• If the charges are of opposite signs, the force is attractive (the charges pull towards each other).
• If the charges are of the same sign, the force is repulsive (the charges push away from each other).

Example 1: Calculating Force

Imagine two charges: $q_1 = +2 \, \mu C$ and $q_2 = -3 \, \mu C$ located $0.5 \, m$ apart. We can calculate the force between them using Coulomb's law:

1. Convert $q_1$ and $q_2$ into coulombs:

• $q_1 = 2 \times 10^{-6} \, C$
• $q_2 = -3 \times 10^{-6} \, C$

1. Calculate the force:

$$

F = $8.99 \times 10^9$ $\cdot$ $\frac{|2 \times 10^{-6} \cdot -3 \times 10^{-6}|}{(0.5)^2}$

$$

This gives:

$$

$F \approx 2.16 \, N $

$$

The negative charge pulls the positive charge with a force of approximately $2.16 \, N$ toward it! 🎯

Electric Field Strength

The electric field $E$ at a point in space due to a point charge $q$ is defined as the force $F$ experienced by a positive test charge $q_0$ placed at that point, divided by the magnitude of the test charge:

$$

$E = \frac{F}{q_0}$

$$

For a point charge, we can relate this to Coulomb's law. The electric field can also be expressed by rearranging Coulomb's law as:

$$

$E = k \cdot \frac{|q|}{r^2}$

$$

This formula tells us that the strength of the electric field decreases with the square of the distance $ r $.

Electric Field Lines

Electric field lines provide a visual representation of the electric field around a charge. The direction of the lines is away from positive charges and towards negative charges. Here’s how to visualize it:

• Point Charge: If you have a positive point charge, the lines will radiate outward. For a negative charge, the lines point inward.
• Parallel Plates: For two parallel plates with opposite charges, the field lines between them are straight and uniform, showing a constant electric field.

Example 2: Electric Field around a Point Charge

Let's say we have a charge $q = +1 \, \mu C$ located at the origin. At a point $ r = 0.1 \, m $, the electric field strength can be calculated as follows:

$$

E = $8.99 \times 10^9$ $\cdot$ $\frac{|1 \times 10^{-6}|}{(0.1)^2}$

$$

This results in:

$$

E $\approx 8$.$99 \times 10^5$ \, N/C

$$

This means that at a distance of $0.1 \, m$, the electric field is very strong, about $899,000 \, N/C$! 🌟

Motion of Charged Particles

When a charged particle moves in an electric field, it experiences a force that influences its motion. The force acting on a charge in an electric field is given by:

$$

$F = q \cdot E$

$$

From Newton's second law, we know:

$$

F = m $\cdot$ a$$

Equating the two expressions gives us:

$$

ma = qE \implies a = $\frac{qE}{m}$

$$

This equation means that the acceleration of a charged particle is directly proportional to the electric field strength and its charge, while inversely proportional to its mass.

Electronvolt (eV)

The electronvolt is a unit of energy defined as the amount of energy gained by an electron when it is accelerated through a potential difference of one volt. One electronvolt is equal to:

$$

1 \, eV = $1.6 \times 10^{-19}$ \, J

$$

Example 3: Acceleration of a Charged Particle

Suppose an electron (charge $ q = -1.6 \times 10^{-19} \, C $, mass $ m = 9.1 \times 10^{-31} \, kg $) is placed in a uniform electric field of strength $ E = 2 \, N/C $. We can calculate the acceleration:

$$

a = $\frac{(-1.6 \times 10^{-19})(2)}{9.1 \times 10^{-31}}$ $\approx$ -$3.52 \times 10^{11}$ \, m/s^2

$$

The negative sign indicates the direction of the acceleration is opposite to the field direction since the electron has a negative charge. 💨

Comparison of Electric and Gravitational Fields

Both electric and gravitational fields act at a distance, but there are key differences:

• Nature of Force: Electric fields can be both attractive and repulsive, while gravitational fields are always attractive.
• Strength: Electric forces are typically much stronger than gravitational forces. For example, the electric force between two electrons is approximately $10^{42}$ times stronger than the gravitational force between them.

Key Differences

| Property | Electric Field | Gravitational Field |

|---------------------|-----------------------------------|----------------------------------|

| Nature of Force | Attractive/Repulsive | Attractive only |

| Relative Strength | Strong (varies) | Weak (constant) |

| Charge | Depends on charge | Depends on mass |

Conclusion

In this lesson, we explored electric fields, Coulomb's law, and the forces acting on charged particles in those fields. You learned to calculate electrical forces, electric field strengths, and how to visualize electric fields using lines. Understanding these concepts will greatly aid you in comprehending more advanced topics in electromagnetism later on!

Study Notes

• Coulomb's Law: $F = k \cdot \frac{|q_1 \cdot q_2|}{r^2}$
• Electric Field Strength: $E = k \cdot \frac{|q|}{r^2}$
• Force on a charge in an electric field: $F = q \cdot E$
• Acceleration of a charged particle: $a = \frac{qE}{m}$
• 1 electronvolt: $1 \, eV = 1.6 \times 10^{-19} \, J$
• Electric fields can be attractive or repulsive, while gravitational fields are always attractive.