5.1.3. Coulomb's Law

Table of Contents

Electric interaction between point charges

Coulomb's law gives the electric force between two point charges. It is one of the basic laws of electricity, just as Newton's law of gravitation is a basic law of gravity. It tells us how strong the force is and in what direction it acts.

If two small charged objects can be treated as point charges, with charges $q_1$ and $q_2$, separated by a distance $r$, then the magnitude of the electric force between them is

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

Here, $k$ is Coulomb's constant. In vacuum,

$$
k \approx 8.99 \times 10^9 \, \text{N m}^2/\text{C}^2
$$

The unit of charge is the coulomb, written as $\text{C}$.

For two point charges in vacuum, the magnitude of the electric force is
$$
F = k \frac{|q_1 q_2|}{r^2}
$$
The force decreases with the square of the distance. If the distance doubles, the force becomes one fourth as large.

Attraction and repulsion

The sign of the charges determines whether the force is attractive or repulsive.

If the two charges have the same sign, both positive or both negative, they repel each other.

If the two charges have opposite signs, one positive and one negative, they attract each other.

This means that the formula for magnitude uses $|q_1 q_2|$, while the physical interpretation comes from the signs.

Charge pair	Interaction
$+$ and $+$	Repulsion
$-$ and $-$	Repulsion
$+$ and $-$	Attraction

Direction of the force

Coulomb's law does not only give the size of the force, it also gives its direction. The force acts along the straight line joining the two charges.

If the charges repel, each charge is pushed away from the other along that line.

If the charges attract, each charge is pulled toward the other along that line.

The two forces have equal magnitude and opposite directions. This is consistent with Newton's third law.

Rendering scientific sketch…

Attraction and repulsion of point charges

Inverse square dependence

A very important feature of Coulomb's law is the factor $1/r^2$. This is called an inverse square law.

It means that the force changes rapidly with distance. A small increase in separation can greatly reduce the force.

For example, if all else stays the same:

New distance	New force
$2r$	$F/4$
$3r$	$F/9$
$4r$	$F/16$
$r/2$	$4F$

Coulomb force follows an inverse square law:
$$
F \propto \frac{1}{r^2}
$$
This is one of the most important patterns in electrostatics.

Vector form

Because force is a vector, Coulomb's law can also be written in vector form. If $\hat{r}$ is a unit vector pointing from one charge toward the other, then one common form is

$$
\vec{F}_{12} = k \frac{q_1 q_2}{r^2} \hat{r}
$$

This equation includes both magnitude and direction. The sign of $q_1 q_2$ automatically determines whether the force points along $\hat{r}$ or opposite to it.

For beginners, it is often easiest to use the magnitude formula first, then decide separately whether the force is attraction or repulsion.

Comparison with gravitational force

Coulomb's law looks mathematically similar to Newton's law of gravitation:

$$
F_g = G \frac{m_1 m_2}{r^2}
$$

Both are inverse square laws. But there is an important physical difference. Gravitational force is always attractive, while electric force can be either attractive or repulsive.

Also, electric forces are usually much stronger than gravitational forces between small particles.

Medium and permittivity

The constant $k$ given above applies in vacuum. In materials, the electric force is usually weaker than in vacuum. This effect is described using the permittivity of the medium.

In vacuum,

$$
k = \frac{1}{4\pi \varepsilon_0}
$$

where $\varepsilon_0$ is the permittivity of free space:

$$
\varepsilon_0 \approx 8.85 \times 10^{-12} \, \text{C}^2/(\text{N m}^2)
$$

So Coulomb's law may also be written as

$$
F = \frac{1}{4\pi \varepsilon_0} \frac{|q_1 q_2|}{r^2}
$$

For many introductory problems, you can assume vacuum or air, where this form works very well.

Simple example

Suppose two point charges are separated by $0.50 \, \text{m}$. Let

$$
q_1 = 2.0 \times 10^{-6} \, \text{C}, \qquad q_2 = 3.0 \times 10^{-6} \, \text{C}
$$

Then the force magnitude is

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

Substitute the values:

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

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

$$
F \approx 0.216 \, \text{N}
$$

Since both charges are positive, the force is repulsive.

When Coulomb's law applies

Coulomb's law is exact for point charges at rest. It is also a very good approximation for small charged objects when the distance between them is much larger than their size.

If the charge is spread out over a larger object, the calculation can become more complicated. Those cases are handled later using electric fields and continuous charge distributions.

Use Coulomb's law directly when charges can be treated as point charges, or when spherical charge distributions behave like point charges at large distances.

Key ideas to remember

Coulomb's law describes the electric force between two point charges. The force depends on the product of the charges and varies inversely with the square of the distance. Like charges repel, unlike charges attract, and the force acts along the line joining the charges.

Essential facts about Coulomb's law:
$$
F = k \frac{|q_1 q_2|}{r^2}
$$
Same signs repel, opposite signs attract.
The force acts along the line between the charges.
In vacuum,
$$
k = \frac{1}{4\pi\varepsilon_0} \approx 8.99 \times 10^9 \, \text{N m}^2/\text{C}^2
$$

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