Electric Charge and Coulomb's Law

1. The Nature of Electric Charge

Long before the discovery of the electron, scientists observed that rubbing certain materials together caused them to attract or repel other objects. This invisible property was named "electric charge."

To get an intuitive feel for this phenomenon and its history, let's start with a classic lecture by MIT Professor Walter Lewin. His demonstrations bring the concepts of attraction, repulsion, and charging to life.

As you saw, there are two types of charge, which Benjamin Franklin arbitrarily named positive (+) and negative (-). The core rule governing their interaction is simple:

• Like charges repel each other (+ repels +, - repels -).
• Opposite charges attract each other (+ attracts -).

Today, we know that these charges are carried by subatomic particles. Let's formalize this with a reading from a physics textbook.

Here are the key definitions from that reading:

• Atomic Basis of Charge: The proton has a positive charge (+e), and the electron has a negative charge (-e). A neutral object has an equal number of protons and electrons.
• Quantization of Charge: Electric charge is quantized, meaning it comes in integer multiples of a fundamental unit, the elementary charge, $e$. $$e=1.602\times 10^{-19}\text{C}$$ You can have a charge of $-e$ or $+10e$, but never $0.5e$.
• The Coulomb (C): This is the SI unit for charge. It is a very large amount of charge. One coulomb is the magnitude of charge of approximately $6.24\times 10^{18}$ electrons.
• Conservation of Charge: The net electric charge in an isolated system is constant. When you rub a rod and it becomes negative, the material you rubbed it with becomes equally positive.

2. Coulomb's Law: The Force Between Charges

We know that charges exert forces on each other, but how strong are these forces? In the 18th century, Charles Coulomb precisely measured these forces and formulated a law that describes them perfectly. This law is fundamental to all of electrostatics.

The law states that the magnitude of the electric force, $F$, between two point charges, $q_1$ and $q_2$, is:

Where:

• $q_1$ and $q_2$ are the quantities of the charges in Coulombs (C).
• $r$ is the distance between the charges in meters (m).
• $k$ is Coulomb's constant, approximately $8.99\times 10^9\text{N}\cdot {\text{m}}^2/{\text{C}}^2$.

Notice the striking similarity to Newton's Law of Universal Gravitation: $F_g=G\frac{m_1{m}_2}{r^2}$. Both are inverse-square laws, but the electric force depends on charge instead of mass.

The next video segment formally introduces Coulomb's Law and makes this important comparison to gravity.

As Prof. Lewin demonstrated, the electric force is astonishingly stronger than gravity at the atomic scale (by about 36 orders of magnitude!). This is why it, not gravity, governs the structure of atoms, molecules, and everything you interact with daily.

3. Applying Coulomb's Law

Let's work through a simple example to see how the law is used.

Problem: A proton ($q_p=+1.602\times 10^{-19}$ C) and an electron ($q_e=-1.602\times 10^{-19}$ C) are separated by a distance of $5.3\times 10^{-11}$ m in a hydrogen atom. What is the magnitude of the electric force between them?

Solution:
We use Coulomb's Law: $F=k\frac{|q_1{q}_2|}{r^2}$.

1. Identify the values:

   • $k\approx 8.99\times 10^9\text{N}\cdot {\text{m}}^2/{\text{C}}^2$
   • $q_1=+1.602\times 10^{-19}$ C
   • $q_2=-1.602\times 10^{-19}$ C
   • $r=5.3\times 10^{-11}$ m

2. Plug them into the formula:

   $$F=(8.99\times 10^9)\frac{|(+1.602\times 10^{-19})(-1.602\times 10^{-19})|}{(5.3\times 10^{-11}{)}^2}$$ $$F=(8.99\times 10^9)\frac{2.566\times 10^{-38}}{2.809\times 10^{-21}}$$ $$F\approx 8.2\times 10^{-8}\text{N}$$

This may seem like a tiny force, but given the incredibly small mass of an electron, this force produces an immense acceleration! Since the charges are opposite, the force is attractive.

4. The Superposition Principle

What happens when there are more than two charges? The answer is beautifully simple: the net force on any single charge is the vector sum of the individual forces exerted on it by all other charges. This is called the principle of superposition.

For a charge $q_0$, the total force ${\vec{F}}_{\text{total}}$ is:

The next video works through a clear, one-dimensional example showing how to apply this principle.

This principle is extremely important. It allows us to analyze complex arrangements of charges by breaking the problem down into a series of simple two-charge calculations.

Conclusion

In this lesson, we moved from the abstract idea of electrons in a material to the concrete, calculable forces they exert on one another.

Here are the main takeaways:

• Electric charge is a fundamental property of matter, measured in Coulombs (C). It is quantized in units of the elementary charge $e$.
• The governing rule of interaction is simple: likes repel, opposites attract.
• Coulomb's Law, $F=k\frac{|q_1{q}_2|}{r^2}$, provides the mathematical tool to calculate the magnitude of the electrostatic force between any two point charges.
• The electrostatic force is a vector. Its direction is along the line connecting the charges.
• The principle of superposition allows us to find the net force on a charge from a system of multiple charges by calculating the vector sum of the individual forces.

We have now established the fundamental force law of electrostatics. In the next lesson, we will introduce a new concept: the electric field. Instead of thinking about forces between two distant charges, the electric field describes how a single charge modifies the space around it. This concept will lead us directly to voltage, the driving "pressure" in an electrical circuit.