Electric Force

Force between point electric charges according to Coulomb's law. Theory, formulas, worked examples and interactive graphs.

Conceptual context

The electric force is the fundamental interaction between objects that possess electric charge. Before Coulomb measured it precisely in 1785, natural philosophers knew that charges attracted or repelled each other, but could not quantify that action. Coulomb's law put numbers to the intuition and placed electricity at the same level of mathematical rigor as Newtonian gravitation.

Within the Phyxio tree, the electric force occupies the hinge position between the concept of charge (which describes a property of the object) and the concept of the electric field (which describes a property of space). Understanding the electric force means understanding why two charges "feel" each other at a distance and how strongly they do so.

The practical relevance is enormous: the electric force keeps electrons bound to atomic nuclei, determines the mechanical strength of crystalline solids, and controls the chemistry of covalent and ionic bonds. Without Coulomb's force there would be no condensed matter as we know it.

🟢 Essential level

Two particles with non-zero $q_1$ and $q_2$ attract or repel each other. If both carry the same type of charge, the force $F_C$ pushes them apart; if they carry opposite types, the force pulls them together. This behavior is universal and requires no contact between the particles: it acts at a distance.

The strength of that action depends on how large $q_1$ and $q_2$ are, and on how far apart they sit. The most important intuitive point is that doubling the distance does not halve the force — it reduces it to one quarter: the dependence on $r$ is quadratic.

[!NOTE] The electric force is always mutual: if $q_1$ pushes $q_2$ with a certain force, $q_2$ pushes $q_1$ with exactly the same force in the opposite direction. This is Newton's third law applied to electricity.

A quick self-check before calculating: decide first whether the charges share the same sign or opposite signs, and anticipate whether the force should be attractive or repulsive before substituting any number.

🔵 Formal level

Basic formulas

The mathematical relation between $F_C$, $q_1$, $q_2$ and $r$ is Coulomb's law:

Coulomb's Law

where $k_e$ is the Coulomb constant, whose value in vacuum is approximately 8.988×10⁹ N·m²·C⁻². The formula shows that $F_C$ is proportional to the product of the charges and proportional to the inverse square of the distance.

The sign of $F_C$ follows directly from the product $[$q_1$](var:q1) \cdot [$q_2$](var:q2)$: if both charges are positive or both are negative, the product is positive and the force is repulsive; if they have opposite signs, the product is negative and the force is attractive.

Real student questions

Why does the force decrease with the square of the distance and not simply with the distance?

The quadratic dependence is not arbitrary: it reflects how the interaction "dilutes" in three-dimensional space. The "flux" of the force spreads over a spherical surface that grows with $r$², so the force density falls with $r$². This same logic explains gravitation and the intensity of light.

Why multiply the charges rather than add them?

Coulomb's force is an interaction between pairs of charges. Doubling either charge doubles the number of elementary interactions contributing to the total force. This is precisely what the product $q_1$·$q_2$ expresses.

Does the electric force exist in absolute vacuum, without any intermediary particle?

From the classical physics perspective, yes. Quantum field theory explains the interaction via the exchange of virtual photons, but the macroscopic result is identical to Coulomb's law. At the formal level, the force acts through vacuum without needing any medium.

How do I know whether to use $F_C$ or $E_C$ to solve a problem?

If the problem asks for the force on a specific charge, use $F_C$. If it asks what acceleration or force any charge would experience (without specifying it), the correct answer involves $E_C$, which does not depend on the test charge. If the problem gives $E_C$ and asks for $F_C$, simply multiply by $q_2$.

Final synthesis

Coulomb's law is the quantitative relation governing the force between point charges: proportional to the product of $q_1$ and $q_2$, and inversely proportional to the square of $r$. Its inverse-square structure is the same as gravity, but the presence of signs in $q_1$ and $q_2$ adds the possibility of repulsion, which gravitation lacks. This simple formula is the starting point of all classical electrostatics.