Coulomb's Law

Fundamental law governing force between static electric charges. Theory, formulas, worked examples and interactive graphs.

Conceptual context

Coulomb's law is the starting point of all classical electrostatics. Before this law existed, electricity was a collection of experimental observations without a quantitative model: it was known that charged bodies attract or repel each other, but the magnitude of the force could not be predicted. Charles-Augustin de Coulomb formalized in 1785 what we now call the first fundamental law of classical electromagnetism.

This law occupies in electromagnetism the same place that the law of universal gravitation occupies in mechanics. Both describe long-range forces that fall off as the square of the distance, but the electric force is intrinsically far more intense: between a proton and an electron, the Coulomb force exceeds the gravitational force by roughly 39 orders of magnitude.

Coulomb's law is not merely a historical result. It is the foundation from which the electric field, the electric potential, and, through Gauss's law, all of the electrostatics of continuous distributions are built. Understanding its dependencies — linear in each charge, inverse square in distance — is a prerequisite for any further analysis in electromagnetism.

🟢 Essential level

Imagine two small charged spheres. If they carry the same type of charge — both positive or both negative — they repel; if they carry opposite charges, they attract. This everyday experience has a precise quantitative structure.

The Coulomb force $F_C$ between two charges depends on three factors: how much charge each object carries (magnitudes $q_1$ and $q_2$), how far apart they are ($r_{12}$), and the medium they are in. In vacuum, the force is larger when the charges are larger and smaller when they are farther apart. The most important aspect — and the one that causes the most confusion — is that the force does not vary linearly with distance: if you double the separation, the force drops to one quarter, not one half.

Coulomb's law says that two typical laboratory charges (of the order of microcoulombs) separated by a centimeter attract or repel each other with a force of tens of Newtons. It is an enormously intense interaction at the atomic scale and still appreciable at the macroscopic scale.

🔵 Formal level

Basic formulas

The mathematical formulation of Coulomb's law for the magnitude of the force is:

Coulomb force between two point charges

where $k_e$ is Coulomb's constant (approximately nine times ten to the ninth power in SI units), $q_1$ and $q_2$ are the charges of the two objects, and $r_{12}$ is the distance between them. The absolute value signs in the numerator ensure that the force magnitude is always positive: the attractive or repulsive character is not given by the magnitude but by the sign of the product of the charges.

The constant $k_e$ can also be written as one divided by four pi times the permittivity of free space. Both notations are equivalent and the student will encounter both in textbooks.

Real student questions

Why does the force decrease with the square of the distance and not with the distance itself? The quadratic dependence has a geometric interpretation: the "flux" of electric action is distributed uniformly over a sphere. The area of a sphere grows with the square of the radius, so the density of action falls with the square of the distance. This is the same reason why the intensity of a light bulb follows the same law.

Why is the force on $q_1$ equal to the force on $q_2$ even if one charge is much larger? Newton's third law is universal: the magnitude of $F_C$ is unique for the pair. What changes is the resulting acceleration, which depends on each object's mass.

If I separate the charges to double the distance, does the force drop to half? No. $r_{12}$ is squared in the denominator: doubling the distance reduces $F_C$ to one quarter.

What is the difference between $F_C$ and $E_C$? $F_C$ depends on both charges $q_1$ and $q_2$. $E_C$ is $F_C$ divided by the test charge: it depends only on $q_1$ and position. The field is a property of space; the force is what a specific charge feels in it.

Final synthesis

Coulomb's law establishes that the force $F_C$ between two point charges $q_1$ and $q_2$ is proportional to the product of their absolute values and inversely proportional to the square of $r_{12}$, with a proportionality constant $k_e$ that sets the scale in the SI. Its domain of validity is classical electrostatics with objects whose extent is small compared to their separation, and its conceptual heritage is the superposition principle and the definition of the electric field as force per unit charge.