Coulomb's Law Formula

The Formula

F = k\frac{|q_1||q_2|}{r^2} where k \approx 8.99 \times 10^9 N m^2/C^2.

When to use: Like gravity between masses, but for charges. Double the distance and the force drops to one quarter. Double either charge and the force doubles.

Quick Example

Two charges of 2\,\mu\text{C} and 3\,\mu\text{C} separated by 0.50 m exert a force of about 0.22 N on each other: F = kq_1q_2/r^2.

Notation

F is the force in newtons, q_1 and q_2 are the charges in coulombs, r is the separation in metres, k is Coulomb's constant, and \epsilon_0 is the permittivity of free space.

What This Formula Means

Coulomb's law gives the electric force between two point charges. The force gets larger when the charges are larger and gets smaller with the square.

Like gravity between masses, but for charges. Double the distance and the force drops to one quarter. Double either charge and the force doubles.

Formal View

Coulomb's law in vector form is \vec{F}_{12} = \frac{1}{4\pi\epsilon_0}\frac{q_1 q_2}{r^2}\hat{r}_{12}, where \hat{r}_{12} points from charge 1 to charge 2. The constant k = 1/(4\pi\epsilon_0) \approx 8.99 \times 10^9 N·m²/C².

Worked Examples

Example 1

medium
Two charges q_1 = 3 \times 10^{-6} \text{ C} and q_2 = 5 \times 10^{-6} \text{ C} are separated by 0.2 \text{ m}. What is the electrostatic force between them? Use k = 9 \times 10^9 \text{ N m}^2/\text{C}^2.

Solution

  1. 1
    Apply Coulomb's law: F = k\frac{|q_1 q_2|}{r^2}
  2. 2
    F = 9 \times 10^9 \times \frac{3 \times 10^{-6} \times 5 \times 10^{-6}}{(0.2)^2}
  3. 3
    F = 9 \times 10^9 \times \frac{15 \times 10^{-12}}{0.04} = 9 \times 10^9 \times 3.75 \times 10^{-10} = 3.375 \text{ N}

Answer

F = 3.375 \text{ N}
Coulomb's law describes the electrostatic force between two point charges. Like gravity, it follows an inverse-square law, but it can be attractive or repulsive depending on the signs of the charges.

Example 2

hard
Two identical charges experience a force of 0.1 \text{ N} when separated by 0.3 \text{ m}. What is the magnitude of each charge?

Common Mistakes

  • Forgetting the inverse-square dependence — halving the distance quadruples the force, not just doubles it.
  • Using the distance between the surfaces of two charged spheres instead of the distance between their centres — Coulomb's law uses centre-to-centre distance for point charges and uniform spheres.
  • Dropping the sign of the charges and getting the force direction wrong — like charges repel (positive force away) and unlike charges attract (force toward each other).

Common Mistakes Guide

If this formula feels simple in isolation but keeps breaking during real problems, review the most common errors before you practice again.

Coulomb's Law Mistakes

Why This Formula Matters

Coulomb's law is the foundation of electrostatics. It explains how charged particles interact in atoms, electric fields, capacitors, and many devices from photocopiers to particle accelerators. It also gives the starting point for electric field and electric potential calculations.

Frequently Asked Questions

What is the Coulomb's Law formula?

Coulomb's law gives the electric force between two point charges. The force gets larger when the charges are larger and gets smaller with the square.

How do you use the Coulomb's Law formula?

Like gravity between masses, but for charges. Double the distance and the force drops to one quarter. Double either charge and the force doubles.

What do the symbols mean in the Coulomb's Law formula?

F is the force in newtons, q_1 and q_2 are the charges in coulombs, r is the separation in metres, k is Coulomb's constant, and \epsilon_0 is the permittivity of free space.

Why is the Coulomb's Law formula important in Physics?

Coulomb's law is the foundation of electrostatics. It explains how charged particles interact in atoms, electric fields, capacitors, and many devices from photocopiers to particle accelerators. It also gives the starting point for electric field and electric potential calculations.

What do students get wrong about Coulomb's Law?

Like charges repel, unlike charges attract — the sign of the force tells you the direction.

What should I learn before the Coulomb's Law formula?

Before studying the Coulomb's Law formula, you should understand: electric charge, electric field.

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