Coulomb’s Law: F = k|q₁q₂| / r²
Complete explanation with interactive charge simulation, real-world solved examples, and step-by-step formulas.
I still remember the first time electric charge stopped feeling like a textbook word and started feeling real. It was the moment a balloon stuck to a wall after being rubbed on hair, and suddenly electricity and magnetism felt less like theory and more like something you can actually see happening.
What Is Coulomb’s Law?
Coulomb’s Law tells us how strong the force is between two charged objects. If the charges are at rest and separated by some distance, the force depends on how much charge each object has and how far apart they are.
The simple idea behind it is easy to picture. More charge means more force. More distance means less force. That is why two tiny charged pieces close together can act strongly on each other, while the same charges far apart barely affect each other.
Charge 1
The amount of electric charge on the first object. Measured in coulombs (C).
Charge 2
The amount of electric charge on the second object. Can be positive or negative.
Distance
The separation between the centers of the two charges. Measured in meters (m).
Who Discovered Coulomb’s Law?
Charles-Augustin de Coulomb, a French physicist, introduced this law in 1785 using a torsion balance experiment. He carefully measured the force between charged spheres and discovered the inverse-square relationship that now carries his name.
Coulomb’s work was groundbreaking because it was the first precise measurement of electrostatic forces. His experiments showed that the force between charges follows the same inverse-square pattern as gravity, which later helped unify the study of forces in physics.
The Formula of Coulomb’s Law
Coulomb’s Law is written as:
Understanding the Variables
Here, F is the force between the charges, q₁ and q₂ are the charges, r is the distance between them, and k is Coulomb’s constant. In air or vacuum, k is about 9 × 10⁵ N·m²/C².
Why the Square in the Denominator Matters
The square in the denominator is what makes Coulomb’s Law an inverse-square law. If the distance doubles, the force becomes one fourth. If the distance becomes three times bigger, the force drops to one ninth. That is why even a small change in spacing can make a big difference.
Understanding Charge, Distance, and Force
Every electrostatic interaction depends on three things — the amount of charge, the distance between charges, and whether the charges are alike or opposite.
Like Charges Repel
Two positive charges or two negative charges push away from each other. The force arrow on each charge points outward along the line joining them.
Opposite Charges Attract
A positive and a negative charge pull toward each other. The force arrows point inward along the line joining the two charges.
Inverse-Square Relationship
Force changes with the square of distance. Triple the distance and the force drops to one ninth. This is why distance dominates electrostatic interactions.
Interactive Charge Simulator
Adjust the charges and distance to see how the electrostatic force changes in real time. Like charges repel, opposite charges attract.
Charge q₁
Force & Distance
Charge q₂
Interaction Info
Vector Nature of Coulomb’s Law
Like all forces in physics, the electrostatic force is a vector. This means direction is just as important as magnitude when working with multiple charges.
When two charges interact, the force on each charge acts along the straight line connecting their centers. If you have more than two charges, you must add the force vectors from each pair to find the net force on any single charge.
Direction Rules
Same Signs → Repulsion
Both charges feel a force pushing them apart along the line joining them.
Opposite Signs → Attraction
Both charges feel a force pulling them together along the line joining them.
Solved Example
Two charges are +2 µC and -3 µC, separated by 0.50 m. Find the force between them.
Step 1: Convert to coulombs:
2 µC = 2 × 10⁻⁶ C, 3 µC = 3 × 10⁻⁶ C
Step 2: Apply Coulomb’s Law:
F = (9 × 10⁹) × (2×10⁻⁶)(3×10⁻⁶) / (0.50)²
F = 0.216 N
Since the charges are opposite, the force is attractive. The two objects pull toward each other. This unit conversion is critical — always convert microcoulombs to coulombs before calculating.
Relationships Between Variables
The formula F = k|q₁q₂|/r² establishes two key relationships:
Charge & Force
Directly Proportional: If either charge doubles, the force also doubles. If both charges double, the force becomes four times larger.
Example: Doubling q₁ from +2 µC to +4 µC doubles the force from 0.216 N to 0.432 N.
Distance & Force
Inverse-Square: When distance increases, force decreases by the square of that factor. Half the distance means four times the force.
Example: Moving from 0.50 m to 1.00 m drops the force from 0.216 N to 0.054 N.
Practice Questions
Try these on your own before checking a calculator. They train the eye to see what Coulomb’s Law is doing.
Interactive Multiple Choice Questions
Test your conceptual understanding in real time. Click on your answer choice:
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Coulomb’s Law Calculator
Enter the two charges and distance to calculate the electrostatic force instantly.
Coulomb’s Law vs. Newton’s Law of Gravitation
Both laws follow an inverse-square pattern, but they describe very different forces.
| Aspect | Coulomb’s Law | Newton’s Gravitation |
|---|---|---|
| Force Type | Electrostatic (attractive or repulsive) | Gravitational (always attractive) |
| Depends On | Charge (q₁, q₂) | Mass (m₁, m₂) |
| Constant | k = 9 × 10⁹ N·m²/C² | G = 6.67 × 10⁻¹¹ N·m²/kg² |
| Relative Strength | Extremely strong (10³⁶ × gravity) | Extremely weak |
| Range | Unlimited (but falls as 1/r²) | Unlimited (falls as 1/r²) |
Real Life Applications
Laser Printers
Electrostatic charge attracts toner to paper.
Dust Control
Electrostatic precipitators remove particles.
Atomic Structure
Electrons bound to nucleus by Coulomb force.
Static Electricity
Clothes clinging, hair rising from a comb.
Coulomb’s Law explains how charges behave in printers, on plastic objects, in dust control systems, and even in the tiny interactions inside atoms. A lot of electricity and magnetism becomes clearer once you see that charge is really just a force problem at its core.
Explore Related Topics
Frequently Asked Questions About Coulomb’s Law
Coulomb’s Law tells us how strongly two charges attract or repel each other. The strength depends on how much charge they have and how far apart they are. More charge gives more force, and more distance gives less force.
Because the force decreases with the square of the distance. That means the force drops very quickly as the objects move apart. Even a small increase in distance can make the force much weaker.
It works best for point charges or objects that behave like point charges. In many basic problems, a charged sphere can be treated this way. That is why the formula is so useful in simple physics.
Coulomb’s Law gives the force between charges. The Electric Field tells you what force a charge would feel at a certain point in space. They are closely connected, but they are not the same thing.
Because the force direction depends on the charge signs. Opposite signs produce attraction, while the same signs produce repulsion. It is one of the most important rules in electrostatics.
Conclusion
Coulomb’s Law is one of those physics ideas that looks small at first, but opens the door to a lot of bigger concepts. Once you understand how electric charge interacts through force and distance, the rest starts to feel more connected.