Section 02
The Idea, Step by Step
Rub a balloon on your hair and it sticks to the wall; socks fresh out of the dryer cling together. That clinging is electric charge at work. The whole of Coulomb's law grows from two everyday rules: opposite charges pull together, like charges push apart — and the closer the charges get, the dramatically harder they tug.
Put numbers on it
Give each object a charge $q$, measured in coulombs (C). Set two charges $q_1$ and $q_2$ a distance $r$ apart, and the force between them is
Simplest form
$$F=\frac{k\,q_1 q_2}{r^2},\qquad k=8.99\times10^9\;\text{N·m}^2/\text{C}^2$$
Work one out. Two tiny spheres, each carrying $q=10\;\text{nC}=10\times10^{-9}\;\text{C}$, held $r=0.10\;\text{m}$ apart. Then $F = (8.99\times10^9)(10^{-8})(10^{-8})/(0.10)^2 \approx 9\times10^{-5}\;\text{N}$ — about a tenth of a milli-newton. Small, but easily measured, and it grows fast as you push the spheres together.
The precise picture
The "inverse-square" part is the heart of it: double the separation and the force drops to a quarter; halve it and the force quadruples. The force points along the line joining the two charges, so it is really a vector. The sign of the product $q_1 q_2$ tells you the direction: positive (two likes) means push apart, negative (two opposites) means pull together. When the charges sit in a material instead of vacuum, divide by the medium's dielectric constant $\kappa$ — water's $\kappa\approx80$ weakens the force eighty-fold, which is exactly why salt dissolves. With several charges present, the net force is just the vector sum of the one-on-one Coulomb forces (superposition), and the field each charge makes is $E=kq/(\kappa r^2)$.
The sliders map straight onto these symbols: $q_1$ and $q_2$ set the two charges (slide negative to flip a sign), $r$ sets the separation, and $\kappa$ sets the medium.
Try this in the sim above
Set $q_1$ and $q_2$ to the same sign and watch the arrows flip outward to repulsion. Then halve the separation $r$ and watch the force reading jump by roughly four times — the inverse-square law made visible. Finally drag $\kappa$ up toward 80 (water) and watch the once-strong force nearly collapse.
Section 03
Equations & Derivation
Coulomb's Law
$$F=\frac{kq_1q_2}{\kappa r^2},\quad k=\frac{1}{4\pi\varepsilon_0}=8.988\times10^9\;\text{N\,m}^2\text{C}^{-2}$$
Electric Field & Potential Energy
$$E=\frac{kq}{\kappa r^2},\quad U=\frac{kq_1q_2}{\kappa r},\quad V=\frac{kq}{\kappa r}$$
Superposition
$$\vec{E}_{\text{net}}=\vec{E}_1+\vec{E}_2+\cdots=k\sum_i\frac{q_i}{r_i^2}\hat{r}_i$$
Shell Theorem
$$\text{Spherical charge distribution: behaves as point charge at centre}$$
Symbol Definitions
| Symbol | Quantity | SI Unit |
|---|
| $k=8.988\times10^9$ | Coulomb constant | N m² C⁻² |
| $\varepsilon_0=8.854\times10^{-12}$ | Permittivity of vacuum | C² N⁻¹ m⁻² |
| $\kappa$ | Relative permittivity (dielectric constant) | dimensionless |
| $r$ | Separation | m |
| $U$ | Potential energy | J |
| $V$ | Electric potential | V |
1
Coulomb's Law in media. In a medium with dielectric constant $\kappa$: $F=kq_1q_2/(\kappa r^2)$. Water ($\kappa=80$): force reduced 80×. This explains why ions dissolve in water — electrostatic attraction is drastically weakened, entropy wins.
2
Field lines. Start on $+$, end on $-$ charges. Density $\propto$ field strength. Never cross. Perpendicular to conductors. For a point charge: uniform density in all directions, radially outward (for $+$).
3
Superposition principle. Forces and fields add as vectors. To find $\vec{E}$ from multiple charges: compute each $\vec{E}_i$ and add vectorially. This is a fundamental experimental result of electrostatics, not derivable from Coulomb's Law alone.
4
Inverse-square law. Electrostatic force is $10^{36}$ times stronger than gravity at the same separation for two protons (for an electron–proton pair the ratio is even larger, $\sim10^{39}$). Yet gravity dominates cosmically because matter is electrically neutral overall. The small imbalance can create Van der Waals forces (residual electrostatics).
Ref: Halliday, Resnick & Walker 10th Ed., Ch. 21–22; Griffiths — Introduction to Electrodynamics (4th Ed.), §2.1–2.2.
Section 05
Common Misconceptions
❌ Coulomb's Law applies only to stationary charges.
✅ Coulomb's Law strictly applies to static charges. For moving charges, magnetic forces also arise, and there are retardation effects (the field propagates at finite speed $c$). For slow-moving charges ($v\ll c$), Coulomb's Law is an excellent approximation. The full treatment requires Maxwell's equations.
📖 HRW 10th Ed., §21-1.
❌ Like charges always repel.
✅ Like charges repel via Coulomb's Law. However, there are quantum mechanical situations where two protons in a nucleus are held together despite electrostatic repulsion — the strong nuclear force is $\sim100\times$ stronger at fm scales and overwhelms electrostatics. At large distances, strong force vanishes, leaving electrostatic repulsion.
📖 HRW 10th Ed., §21-1.
❌ Coulomb's constant $k$ is the same in all materials.
✅ In a medium, Coulomb's Law uses $k_\text{medium}=k/\kappa$ where $\kappa$ is the relative permittivity. In water ($\kappa=80$), the electrostatic force between ions is 80× weaker than in vacuum. This drastically affects dissolution, chemical equilibria, and biological function (most biology occurs in aqueous solution).
📖 HRW 10th Ed., §25-4; Griffiths — Introduction to Electrodynamics §4.4.
Misconception research: McDermott — Tutorials in Introductory Physics; Maloney et al. (2001), Am. J. Phys. Suppl.