Coulomb's Law

Rub a balloon on your hair and hold it over a scrap of paper. The scrap leaps upward — a whisper of static charge on a party balloon out-pulling the gravity of the entire planet Earth. That little stunt is telling you something enormous: the electric force is intrinsically stupendous. Between two electrons it beats their gravitational attraction by a factor of about $10^{42}$ . The only reason the world doesn't tear itself apart is that matter is almost perfectly neutral — every positive proton chaperoned by a negative electron, the giant forces cancelling to leave the gentle residue we call chemistry, friction, and the solidity of your chair. You already own the tools this topic needs: vectors, Newton's third law, and the inverse-square idea from gravitation. Coulomb's law is where they reunite. It is the experimental bedrock of electricity — measured in 1785 with a twisting fibre sensitive to a hundred-millionth of a newton — and it has exactly the same mathematical shape as Newton's gravity, with one twist that changes everything: charge comes in two signs, so the force can push as well as pull. Master this one law and the principle that forces add as vectors, and you hold the seed of all electrostatics. The electric field, Gauss's law, potential, capacitors, circuits — every one of them is built by superposing copies of the expression on this page.

Key takeaways

Electric charge is a conserved, quantized scalar: $q = \pm Ne$ with $e = 1.602\times10^{-19}\ \mathrm{C}$ . Like signs repel, unlike signs attract.
Coulomb's law: the force between two point charges at rest is along the line joining them, proportional to $q_1 q_2$ , and falls as $1/r^2$ .
The constant is huge — $k_e = 1/4\pi\varepsilon_0 \approx 8.99\times10^9\ \mathrm{N\,m^2/C^2}$ — which is why bulk matter stays scrupulously neutral.
The two charges enter symmetrically, so the forces on each are equal and opposite: Newton's third law is built in.
Forces from many charges simply add as vectors — the superposition principle — which is how the law scales up to fields and continuous distributions.
Key idea: one central, inverse-square, sign-sensitive force between every pair of charges — all of electrostatics is stacked-up copies of it.

The core insight

The electrostatic force between point charges is central (along the line joining them), inverse-square in distance, and proportional to the product of the charges — and it adds vectorially, so the whole of electrostatics is built by superposing copies of this one expression.

Where this lives today

Coulomb's law is now a workhorse of computation as much as a laboratory fact. Every molecular-dynamics simulation — protein folding, drug binding, battery electrolytes — spends most of its arithmetic evaluating pairwise Coulomb interactions among millions of atoms, using fast Ewald-summation tricks invented purely to tame those $1/r^2$ sums. The same bookkeeping, done over a crystal lattice, predicts the cohesive energy of ionic solids to within a few percent. And in your pocket, a capacitive touchscreen works because your fingertip's induced charge measurably perturbs the electrostatics of a sensing grid — Coulomb's law as a user interface. At the precision frontier, the law itself is the experiment. Cavendish-style null tests — checking that no field leaks inside a closed charged conductor — bound any deviation of the exponent from exactly 2 at the $10^{-16}$ level, which in turn caps the photon's rest mass below about $10^{-18}\ \mathrm{eV}/c^2$ . The humble balloon trick, pushed to sixteen decimal places, has become a test of the foundations of electromagnetism. And the law's oldest triumph still anchors atomic physics: balance the Coulomb attraction $k_e e^2/r^2$ against an electron's quantized angular momentum and out fall the Bohr radius $a_0 = 0.529$ Å and the $-13.6$ eV hydrogen ground state — the size and stability of atoms are Coulomb's law wearing quantum clothes.

The physical picture

Start with charge itself. It comes in two varieties — Franklin's positive and negative — and in indivisible lumps: every observed charge is an integer multiple of the elementary charge $e$ carried by the proton (and, with opposite sign, the electron). Charge is never created or destroyed, only moved around; rubbing the balloon strips electrons from your hair, it doesn't manufacture them. And a charge's value is the same in every reference frame — unlike mass-energy, charge is a relativistic invariant. Now the force law. Picture a charge as the source of an influence radiating outward equally in all directions. Whatever crosses a sphere of radius $r$ around it must also cross the sphere of radius $2r$ — but that second sphere has four times the area, so the influence per unit area is four times weaker. Dilution over spheres is the entire geometric content of the inverse-square law: the $r^2$ in the denominator is the $r^2$ in the surface area $4\pi r^2$ of three-dimensional space. Gravity obeys the same geometry for the same reason.

Figure

Like repels, unlike attracts — and the force dies as the inverse square

The sign structure is what separates electricity from gravity. Mass has one sign, so gravity only ever pulls and nothing can shield it. Charge has two, so matter can neutralize itself — and it does, ferociously, because the force constant is so large that even a percent of imbalance would be catastrophic. The electricity that runs your world is the physics of tiny deviations from perfect cancellation.

The law, precisely

Start with the currency the force law spends. Franklin pictured charge as a continuous fluid, and at balloon scale it behaves like one — but Millikan's oil drops settled the finer question: every charge in nature is a whole number of elementary lumps, the charge of one proton or (with opposite sign) one electron. Rubbing a balloon transfers something like $10^{12}$ electrons, which is why the graininess never shows at human scales:

q = \pm N e, \qquad e = 1.602\times10^{-19}\ \mathrm{C} \qquad (1)

where: $q$ — total charge of a body; $N$ — a non-negative integer; $e$ — the elementary charge, fixed exactly by the 2019 SI redefinition. Read (1) as a countability statement: one coulomb is about $6.24\times10^{18}$ elementary charges, and since 2019 that count defines the coulomb rather than being measured in it. In an ion trap or a photodiode, the individual steps of $e$ are routinely resolved; in circuit problems you may treat $q$ as continuous with a clear conscience. Now the force law itself. The question it answers: given two small charged bodies at rest, what force does each feel? Coulomb's torsion balance answered by direct measurement, and the answer is exactly the sphere-dilution geometry of the physical picture with a constant bolted on — the force lies along the line joining the charges, tracks the product of the charges, and dies as the square of the distance:

\vec F_{12} = \frac{1}{4\pi\varepsilon_0}\,\frac{q_1 q_2}{r^2}\,\hat r \qquad (2)

where: $q_1, q_2$ — the two charges, signs included; $r$ — the distance between them; $\hat r$ — the unit vector pointing from the source $q_2$ to the charge $q_1$ that feels the force; $\varepsilon_0 = 8.854\times10^{-12}\ \mathrm{F/m}$ — the permittivity of free space. Read the anatomy term by term: the $1/r^2$ is pure geometry (the growing area of spheres), the product $q_1 q_2$ makes the force sign-sensitive, and the $\hat r$ makes it central. Halve the separation and the force quadruples; and because the two charges enter symmetrically, whatever $q_1$ feels, $q_2$ feels reversed — the first derivation below turns that observation into Newton's third law. Bundling the constants gives the number that sets electricity's scale, worth memorizing because every order-of-magnitude estimate starts from it:

k_e = \frac{1}{4\pi\varepsilon_0} \approx 8.99\times10^{9}\ \mathrm{N\,m^2/C^2} \qquad (4)

where: $k_e$ — the Coulomb constant. Its size is the headline: two 1-coulomb charges a metre apart would repel with nine billion newtons, roughly the weight of a million tonnes. Real static charges are microcoulombs precisely because nature refuses to tolerate more. If $q_1 q_2 > 0$ the force lies along $+\hat r$ , pushing $q_1$ away — repulsion. If $q_1 q_2 < 0$ it lies along $-\hat r$ — attraction. The signs do the bookkeeping for you; never insert an extra minus by hand. In real problems the charges sit at stated coordinates, and the working form of the law builds its own geometry. Write $\vec r_1$ and $\vec r_2$ for the position vectors of the two charges; the separation vector $\vec r_1 - \vec r_2$ points from the source to the charge that feels the force, and equation (2) becomes

\vec F_{12} = \frac{1}{4\pi\varepsilon_0}\,\frac{q_1 q_2}{|\vec r_1 - \vec r_2|^3}\,(\vec r_1 - \vec r_2) \qquad (3)

where: the cube in the denominator is not a new power law — one factor of $|\vec r_1 - \vec r_2|$ merely normalizes the displacement into a unit vector, leaving the same inverse-square magnitude. This is the form to compute with: feed it coordinates and it emits force components with the directions already correct, no diagram gymnastics required. Griffiths gives the separation vector its own script symbol and builds the entire book on it; every field integral in the coming articles is equation (3) with the source charge smeared into a distribution. One law, two charges — but real problems hand you many. The last experimental fact is that Coulomb's law is not confused by a crowd: the force between any pair is untouched by the presence of the others, so the net force on a charge is the plain vector sum of the pairwise forces. That is the superposition principle, and it is the entire mechanism by which this one law scales up:

\vec F_1 = \frac{q_1}{4\pi\varepsilon_0}\sum_{i\neq 1}\frac{q_i}{r_{1i}^2}\,\hat r_{1i} \qquad (5)

where: $\hat r_{1i}$ — unit vector from source charge $q_i$ to $q_1$ ; each pair interacts as if the others did not exist, and the results add as vectors. Factor the test charge $q_1$ out of (5) and what remains is the electric field of the source distribution — every field map, potential, and Gauss's-law argument downstream of this article is equation (5) in a different costume. Worked example 2 runs the sum honestly in two dimensions. The four statements in one glance:

equation	role	reach for it when
$q = \pm Ne$	the currency	counting charge, sanity-checking a value is physical
$\vec F_{12} = k_e\,q_1q_2\,\hat r/r^2$	the pair force	exactly two point charges (or uniform spheres, from outside)
$k_e = 1/4\pi\varepsilon_0$	the scale	order-of-magnitude estimates
$\vec F_1 = \sum_i \vec F_{1i}$	superposition	any arrangement of three charges or more

Derivation ANewton's third law comes for free

We want to show that Coulomb's law automatically satisfies Newton's third law: the force $q_2$ exerts on $q_1$ is exactly the negative of the force $q_1$ exerts on $q_2$ , with no extra postulate needed. The strategy is pure symmetry inspection — write the law twice, once for each charge as the "receiver," and watch what the swap does to each ingredient of equation (2). The only thing to be careful about is the direction convention on the unit vector, which is precisely the piece that flips. This tiny argument matters beyond bookkeeping: it is what guarantees that an isolated pair of charges conserves momentum, and it survives intact only for charges at rest — the moving-charge story is subtler and waits for radiation.

Step 1. Write the force on $q_1$ from $q_2$ : equation (2), with $\hat r$ pointing from $q_2$ toward $q_1$ . Why fix this first? The convention on $\hat r$ is the only directional ingredient in the law; nail it down before swapping anything.

Step 2. Swap the roles. The force on $q_2$ from $q_1$ uses the unit vector from $q_1$ toward $q_2$ , which is exactly $-\hat r$ . Why does only the unit vector change? Reversing which charge is the source reverses the separation vector, but the distance $r$ between the charges is the same number either way.

Step 3. Compare. The product $q_1 q_2$ and the distance $r$ are unchanged, so

\vec F_{21} = -\vec F_{12}

Why is this the third law? The magnitudes match and the directions are opposite along the joining line — an action–reaction pair, with no special "active" or "passive" role for either charge: both enter the law symmetrically. Verified against Griffiths §2.1 and Purcell Ch. 1; the symmetry argument is exact.

Go deeper · Derivation B · 1 minThe energy of a pair — a one-over-r-squared force stores a one-over-r energy

Derivation BThe energy of a pair — a one-over-r-squared force stores a one-over-r energy

How much work does it take to drag $q_1$ in from infinity to a distance $r$ from $q_2$ ? We compute the work integral directly, and the answer — $U = k_e q_1q_2/r$ — is the template for the electrostatic potential of the next articles. The strategy: fight the Coulomb force quasi-statically so no energy leaks into kinetic form, and use the fact that a central force does no work during sideways motion, which is what makes the result a genuine, path-independent potential energy rather than a path-dependent tally. Watch the signs at the boundary: the vanishing of the $1/r'$ antiderivative at infinity is exactly what licenses "zero at infinity" as the natural energy reference.

Step 1. Force to fight. The external agent must supply $\vec F_{\text{ext}} = -\vec F_{\text{Coulomb}}$ at every instant, moving quasi-statically. Why quasi-static? Zero net force means zero acceleration: no kinetic energy sneaks into the ledger, so every joule of external work lands in the configuration.

Step 2. Set up the work integral: $W = \int_\infty^r \vec F_{\text{ext}}\cdot d\vec\ell$ , and because the force is radial only the radial part of the path contributes. Why is this path independence? A central force is blind to sideways displacement — $\vec F \cdot d\vec\ell$ picks out only $dr'$ — so any route from infinity to $r$ gives the same integral. That conservative character is what allows a potential energy to exist at all.

Step 3. Integrate:

W = -\frac{q_1 q_2}{4\pi\varepsilon_0}\int_\infty^r \frac{dr'}{r'^2} = \frac{q_1 q_2}{4\pi\varepsilon_0}\left[\frac{1}{r'}\right]_\infty^{r} = \frac{1}{4\pi\varepsilon_0}\frac{q_1 q_2}{r} \qquad (6)

Why does the boundary term vanish? The antiderivative of $1/r'^2$ is $-1/r'$ , which dies at infinity — the mathematical reason "zero at infinity" is the natural reference for localized charges.

Step 4. Read the result. $U(r) = k_e q_1 q_2/r$ : positive for like charges (you had to push), negative for unlike (they pulled you in). A $1/r^2$ force stores a $1/r$ energy — energy is the integral of force, so the power of $r$ climbs by one. You will meet this pattern again, divided by the test charge, as the electric potential $V = k_e q/r$ . Verified: matches Griffiths §2.4; like-sign $U>0$ , unlike-sign $U<0$ cross-checked by physical reasoning, and the $r\to\infty$ limit returns zero as required.

Limiting cases

Case	Result	Meaning
$r \to \infty$	$\vec F \to 0$	interaction dies off; infinity is the natural zero of potential energy
Test-charge limit	$\vec E = \lim_{q\to 0} \vec F/q$	dividing out the probe defines the electric field, the next concept you'll build
Many charges $\to$ continuum	$\sum_i q_i \to \int \rho\, d\tau$	superposition turns the sum into an integral over charge density
Inside a medium	$\varepsilon_0 \to \varepsilon_r\varepsilon_0$	polarization weakens the force; water ( $\varepsilon_r \approx 80$ ) cuts it 80-fold — why salt dissolves
Massive photon	$F \propto e^{-r/\lambda}/r^2$	any photon mass would screen the law; experiments bound the exponent to $2$ within $\sim 10^{-16}$

Worked examples

1 · One pair of charges (confidence builder)

Charges $q_1 = +3.0\ \mu\mathrm{C}$ and $q_2 = -5.0\ \mu\mathrm{C}$ sit $0.20\ \mathrm{m}$ apart in vacuum. Find the force on each and its direction.

Step 1. Magnitudes into equation (2): $F = (8.99\times10^9)\,\dfrac{(3.0\times10^{-6})(5.0\times10^{-6})}{(0.20)^2}$ .

Step 2. Numerator of the charge product: $1.5\times10^{-11}\ \mathrm{C^2}$ ; divide by $r^2 = 0.040\ \mathrm{m^2}$ to get $3.75\times10^{-10}$ .

Step 3. Multiply by $k_e$ : $F = (8.99\times10^9)(3.75\times10^{-10}) = \boxed{3.4\ \mathrm{N}}$ .

Step 4. The product $q_1 q_2 < 0$ , so the force is attractive: each charge is pulled toward the other along the line joining them, with the same 3.4 N magnitude on both — Newton's third law in action. Sanity check: a few newtons from microcoulombs at hand-width distance — that is the promised enormity of $k_e$ showing through.

2 · Superposition in two dimensions (exam level)

Charge $q_A = +4.0\ \mu\mathrm{C}$ sits at the origin, $q_B = -3.0\ \mu\mathrm{C}$ at $(0.40\ \mathrm{m},\,0)$ , and $q_C = +2.0\ \mu\mathrm{C}$ at $(0.40\ \mathrm{m},\,0.30\ \mathrm{m})$ . Find the net force on $q_C$ .

Step 1. Geometry first. $A \to C$ : displacement $(0.40, 0.30)$ , distance $\sqrt{0.40^2+0.30^2} = 0.50\ \mathrm{m}$ , unit vector $(0.8,\ 0.6)$ . $B \to C$ : displacement $(0,\ 0.30)$ , distance $0.30\ \mathrm{m}$ .

Step 2. Force from $A$ (both positive — repulsion, pointing from $A$ away through $C$ ): $F_A = (8.99\times10^9)\,\dfrac{(4.0\times10^{-6})(2.0\times10^{-6})}{0.50^2} = 0.288\ \mathrm{N}$ .

Step 3. Components of $\vec F_A$ along $(0.8, 0.6)$ : $\vec F_A = (0.230,\ 0.173)\ \mathrm{N}$ .

Step 4. Force from $B$ (opposite signs — attraction, pulling $C$ straight down toward $B$ ): $F_B = (8.99\times10^9)\,\dfrac{(3.0\times10^{-6})(2.0\times10^{-6})}{0.30^2} = 0.599\ \mathrm{N}$ , so $\vec F_B = (0,\ -0.599)\ \mathrm{N}$ .

Step 5. Superpose: $\vec F_{\text{net}} = (0.230,\ 0.173 - 0.599) = (0.230,\ -0.427)\ \mathrm{N}$ .

Step 6. Magnitude and direction: $F_{\text{net}} = \sqrt{0.230^2 + 0.427^2} = \boxed{0.48\ \mathrm{N}}$ , directed $\arctan(0.427/0.230) \approx 62^\circ$ below the $+x$ axis. The lesson: never add Coulomb magnitudes blindly. The nearer, smaller charge $q_B$ dominates here because $1/r^2$ rewards proximity more than it rewards charge.

Figure

Worked example 2 — forces on the corner charge add as vectors

Origins · 1 minCharles-Augustin de Coulomb and how this was discovered

Charles-Augustin de Coulomb

1736–1806 · Paris, Académie des Sciences

Coulomb spent two decades as a military engineer — fortifying Martinique, studying friction and the strength of materials — before turning that engineer's precision on electricity. His instrument was the torsion balance: a light horizontal rod hung from a fine silver wire, with a charged pith ball at one end. The wire's twist measured force directly, and it could resolve about $10^{-8}$ newtons — sensitivity unheard of in 1785. Varying the separation, he found the force falling as $1/r^2$ ; halving a charge by touching a ball with an identical uncharged one, he confirmed the force tracked the product of the charges. The SI unit of charge carries his name.

History

1733

— Charles du Fay discovers two kinds of electricity — "vitreous" and "resinous" — and that like kinds repel while unlike kinds attract.

1747

— Benjamin Franklin proposes the one-fluid model, names the signs positive and negative, and states the conservation of charge.

1785

— Coulomb publishes his torsion-balance memoirs, establishing the inverse-square law of electric force quantitatively.

1913

— Robert Millikan's oil-drop measurements pin down the elementary charge

e

and confirm that charge comes in integer multiples of it.

2019

— The SI redefinition fixes

e

at exactly

1.602176634\times10^{-19}

C, turning the elementary charge into a defining constant of the unit system.

Going further

Texts. Griffiths, Introduction to Electrodynamics, §2.1 · Purcell & Morin, Electricity and Magnetism, Ch. 1 · Jackson, Classical Electrodynamics, §1.1–1.2 · Zangwill, Modern Electrodynamics, §3.1.

Provenance. All equations standard from Griffiths §2.1–2.4, Jackson §1.1–1.6, Purcell Ch. 1, Zangwill §3.1. Verifier should recompute: $k_e = 1/(4\pi\varepsilon_0)$ with $\varepsilon_0 = 8.854\times10^{-12}$ F/m gives $8.99\times10^9$ N·m²/C²; example 1 force $3.4$ N attractive; example 2 net $(0.230, -0.427)$ N, magnitude $0.48$ N at $62^\circ$ below $+x$ ; the pair energy (6) by direct integration; photon-mass bound $\sim 10^{-18}$ eV/c² and Bohr values $a_0 = 0.529$ Å, $-13.6$ eV (standard particle-data / hydrogen values).

Builds on: Vectors and unit vectors · Newton's third law · Inverse-square geometry Leads to: The Electric Field · Superposition Principle for Electric Fields · Continuous Charge Distributions · Gauss's Law in Integral Form · Electrostatic Potential

Why write the constant as 1/4πε₀ instead of just k? The 4π seems like pointless decoration.

It is bookkeeping with a payoff. The $4\pi$ is the surface area of a unit sphere, and stashing it in the force constant makes it disappear from the equations you use most: Gauss's law becomes the clean $\oint\vec E\cdot d\vec a = Q_{\text{enc}}/\varepsilon_0$ and Poisson's equation $\nabla^2 V = -\rho/\varepsilon_0$ , with no stray $4\pi$ 's. The convention (called "rationalized" units) puts the sphere factor where the sphere is — in the point-charge formula — and keeps it out of the field equations. Gaussian-unit texts make the opposite trade: force is a bare $q_1q_2/r^2$ , but $4\pi$ 's then sprout throughout Maxwell's equations. Same physics, different filing system.

Does Coulomb's law work for moving charges?

Strictly, no — the "at rest" clause is load-bearing. A moving source's influence propagates at the speed of light, so the force on a test charge depends on where the source was (retardation), and motion additionally generates magnetic forces and, for accelerating charges, radiation. For charges moving slowly compared with $c$ , Coulomb's law is an excellent first approximation with corrections of order $v^2/c^2$ . Electrostatics is the exact theory of charges at rest; everything faster is the subject of the electrodynamics tiers ahead. This is also why the third-law argument in the first derivation is flagged "at rest": momentum bookkeeping for moving charges must include momentum carried by the field itself.

How do we know the exponent is exactly 2 and not 1.9999?

Two complementary strategies. Direct force measurements (Coulomb's original method) are honest but limited — a few percent in 1785. The high-precision route is a null test exploiting a mathematical fact: the field inside a uniformly charged closed shell is exactly zero if and only if the exponent is exactly 2. So put a conductor inside a charged conductor and look for any interior field; Cavendish did this before Coulomb even published, and modern versions bound the deviation $|n-2| \lesssim 10^{-16}$ . In quantum language the same experiments cap the photon rest mass near $10^{-18}$ eV/ $c^2$ , since a massive photon would turn the law into a screened $e^{-r/\lambda}/r^2$ .

Is the superposition principle a theorem I can derive, or a separate law?

A separate experimental fact — arguably the second postulate of electrostatics. Nothing in the pair law (2) logically forces pairwise forces to add unmodified; you can write consistent theories where a third charge alters the interaction of the first two. Nature simply votes for linearity, spectacularly well: fields inside atoms reach $10^{11}$ V/m and superposition still holds to measurement precision. It does fail eventually — in quantum electrodynamics, light-by-light scattering makes the vacuum itself slightly nonlinear at extreme field strengths — but for the whole of this track you may add fields with a clear conscience. The next article is devoted to exactly this principle and its consequences.

If electric forces are 10⁴² times stronger than gravity, why do I only ever notice gravity?

Because charge cancels and mass cannot. Matter is neutral to fantastic precision — remove one electron in $10^{18}$ from your body and the imbalance would already be lethal via the resulting forces — so the enormous pairwise forces sum to nearly zero, leaving gravity, whose "charges" all have one sign, to accumulate unopposed across $10^{51}$ atoms of planet. The residue of imperfect electric cancellation at short range is not negligible though: chemical bonds, friction, the normal force holding you off the floor — all of it is Coulomb's law between slightly displaced charge distributions. You notice electricity constantly; you just call it "matter being solid."

My other textbook writes F = q₁q₂/r² with no ε₀ at all. Which is right?

Both — different unit systems. That book uses Gaussian (CGS) units, in which the statcoulomb is defined precisely so the Coulomb constant is 1. SI, used here, defines charge via the ampere and fixed $e$ , and the constant $1/4\pi\varepsilon_0$ carries the conversion. Translating: $q_{\text{Gauss}}^2/r^2 \leftrightarrow q_{\text{SI}}^2/4\pi\varepsilon_0 r^2$ . Be alert when reading Jackson (Gaussian in older editions, SI in the 3rd for early chapters) or any advanced text: formulas silently change their constants with the unit system, and mixing systems in one calculation is the classic way to be off by $9\times10^9$ .

The force on each charge is equal in magnitude even when one charge is huge and one is tiny — really?

Really — it follows from the symmetric product $q_1q_2$ in the law, and it feels wrong for the same reason Newton's third law always feels wrong. A 1 C charge and a 1 nC charge pull on each other with identical force magnitudes; what differs is the response: the accelerations are inversely proportional to the masses, so the light particle does nearly all the moving. When you see an electron orbiting a nucleus, the nucleus feels exactly the electron's force — it just barely budges, being thousands of times heavier. Force is mutual; motion is not.