Coulomb's Law — Electric Force Formula & Examples

Coulomb's law gives the electric force between two charges. For two point charges in vacuum, the magnitude of the force is

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

where $q_1$ and $q_2$ are the charges, $r$ is the distance between them, and $k \approx 8.99 \times 10^9\ \mathrm{N \cdot m^2/C^2}$. Like charges repel. Unlike charges attract.

This is the electric force formula most students need first. It works directly when the charges can be treated as point charges, or when spherically symmetric charge distributions are far enough away that center-to-center distance is the right model. In introductory physics, air is usually treated as close enough to vacuum unless the problem says otherwise.

What Coulomb's Law Means

The force gets stronger when the charges are larger. It gets weaker when the charges are farther apart. The key pattern is the inverse-square dependence: the force scales like $1/r^2$, not $1/r$.

That means doubling the distance makes the force one-fourth as large. Halving the distance makes the force four times as large.

The force acts along the line joining the two charges. Each charge feels a force of the same magnitude, but in opposite directions.

Coulomb's Law Formula And Variables

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

• $F$ is the magnitude of the electric force.
• $q_1$ and $q_2$ are the charges in coulombs.
• $r$ is the separation distance in meters.
• $k$ is Coulomb's constant in vacuum.

The absolute value in $|q_1 q_2|$ is there because this formula gives magnitude. The signs of the charges tell you the direction:

• same sign $\rightarrow$ repulsion
• opposite sign $\rightarrow$ attraction

When The Formula Applies

Use Coulomb's law directly if the problem involves point charges, or if larger charged objects can be approximated by point charges from far away. For extended objects with messy shapes or charge spread through a material, this formula may not be enough by itself.

Be careful with distance. The $r$ in the formula is the separation between the charges, usually measured center to center.

Coulomb's Law Example

Suppose

• $q_1 = 2.0 \times 10^{-6}\ \mathrm{C}$
• $q_2 = -3.0 \times 10^{-6}\ \mathrm{C}$
• $r = 0.50\ \mathrm{m}$

Find the electric force magnitude and decide whether it is attractive or repulsive.

Start with Coulomb's law:

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

Substitute the values:

$$F = (8.99 \times 10^9)\frac{|(2.0 \times 10^{-6})(-3.0 \times 10^{-6})|}{(0.50)^2}$$

Multiply the charges:

$$|q_1 q_2| = 6.0 \times 10^\{-12\}\ \mathrm\{C^2\}$$

Square the distance:

$$r^2 = 0.25\ \mathrm{m^2}$$

Now compute the force:

$$F = (8.99 \times 10^9)\frac{6.0 \times 10^{-12}}{0.25} \approx 0.216\ \mathrm{N}$$

So the force magnitude is about $0.22\ \mathrm{N}$. Because the charges have opposite signs, the force is attractive.

Common Coulomb's Law Mistakes

• Forgetting to convert microcoulombs to coulombs before substituting values.
• Using $r$ instead of $r^2$ in the denominator.
• Measuring distance from surface to surface instead of center to center.
• Treating the formula as exact for any large or irregular charged object.
• Mixing up magnitude and direction. The formula above gives size; the charge signs tell you attraction or repulsion.

Where Coulomb's Law Is Used

Coulomb's law is a basic tool in electrostatics. It is used to calculate forces between charged particles, to build the idea of electric field, and to analyze simple charge configurations before moving to more advanced methods.

It also helps with quick proportional reasoning. If the charges stay the same and the distance triples, the force becomes $1/9$ of the original value. That kind of scaling is often more useful than a full calculation.

Coulomb's Law Vs. Electric Field

Coulomb's law tells you the force between charges. Electric field tells you the force per unit charge at a location. The two ideas are closely connected, but they are not the same thing.

If you already know the electric field $E$ at a point, then the force on a charge $q$ there is $F = qE$. If you are starting from two charges directly, Coulomb's law is usually the first step.

Try A Similar Problem

Keep the same charges and change the distance from $0.50\ \mathrm{m}$ to $1.0\ \mathrm{m}$. Solve for the new force and compare it with $0.22\ \mathrm{N}$. That one change is enough to make the inverse-square pattern feel concrete. A useful next step is to explore electric potential and compare its $1/r$ dependence with Coulomb force's $1/r^2$ dependence.