Coulomb's Law
Two point charges a fixed distance apart exert a definite, calculable force on each other — Coulomb's law tells you exactly what that force is. It also has a strikingly similar shape to Newton's law of gravitation, and comparing the two reveals something important about the relative strength of electric and gravitational forces.
Coulomb's law
The force between two point charges $Q_1$ and $Q_2$, separated by a distance $r$, is given by:
where $\varepsilon_0$ is the permittivity of free space, $\varepsilon_0 = 8.85\times10^{-12}\,\text{F m}^{-1}$ (farads per metre — the farad is introduced properly with capacitance). This force acts along the line joining the two charges, and by Newton's third law acts equally and oppositely on each — $Q_1$ pushes or pulls on $Q_2$ exactly as hard as $Q_2$ does on $Q_1$.
The sign of $F$ tells you attractive or repulsive: same-sign charges give a positive product and a repulsive force; opposite-sign charges give a negative product and an attractive force — matching the familiar rule that like charges repel and unlike charges attract.
The inverse-square law
Coulomb's law is an inverse-square law: force is inversely proportional to $r^2$. Double the distance and the force drops to a quarter; treble it and it drops to a ninth. This follows from field lines spreading out over an ever-larger sphere as you move away from a point charge — flux through the sphere is fixed, but surface area grows as $r^2$, so field strength (and force) falls as $1/r^2$.
Compare Coulomb's law with Newton's law of gravitation, $F=\dfrac{Gm_1m_2}{r^2}$: both are inverse-square laws acting along the line joining two point objects, with a fixed constant of proportionality ($\dfrac{1}{4\pi\varepsilon_0}$ versus $G$). But the forces differ enormously in strength — for two protons, electric repulsion is roughly $10^{36}$ times stronger than their mutual gravitational attraction. Gravity only dominates at the scale of planets and stars because large objects are electrically neutral overall, while even a modest net charge imbalance produces forces gravity can't compete with.
Deriving field strength from Coulomb's law
The previous lesson defined electric field strength generally as $E=F/Q$. Combined with Coulomb's law, this gives an explicit formula for the field strength around a point charge.
This confirms the field-lines picture: field strength around a point charge is inverse-square, matching field lines spreading progressively wider apart with distance.
Find the electric field strength at a distance of 2.0 cm from a point charge of $+5.0\,\text{nC}$.
$E=\dfrac{Q}{4\pi\varepsilon_0 r^2} = \dfrac{5.0\times10^{-9}}{4\pi\times(8.85\times10^{-12})\times(0.020)^2} \approx 1.12\times10^{5}\,\text{N C}^{-1}$, directed radially outward from the charge (since it's positive).
Key takeaways
- Coulomb's law: $F=\dfrac{Q_1Q_2}{4\pi\varepsilon_0 r^2}$, acting along the line joining two point charges, repulsive for like charges and attractive for unlike charges.
- It is an inverse-square law — doubling the separation quarters the force — with $\varepsilon_0$, the permittivity of free space, as the constant of proportionality.
- Coulomb's law has the same mathematical shape as Newton's law of gravitation, but the electric force between two charged particles is vastly stronger than the gravitational force between them (around $10^{36}$ times, for two protons).
- Combining Coulomb's law with $E=F/Q$ gives the field strength around a point charge: $E=\dfrac{Q}{4\pi\varepsilon_0 r^2}$.