Coulomb's Law Explained With Worked Examples: Static to Atoms

Coulomb's law says that the electric force between two point charges is proportional to the product of the charges and inversely proportional to the square of the distance between them: F = k q₁ q₂ / r², with k = 1/(4πε₀) ≈ 8.988 × 10⁹ N·m²/C². Like charges repel, unlike charges attract. Charles Augustin de Coulomb measured it with a torsion balance and presented it to the French Académie des Sciences in a series of memoirs from 1785, and the same equation explains a sock clinging to a sweater, a chip killed by a spark and the binding of an electron to a proton.

This post goes one level deeper than our tour of famous equations that changed the world. It pairs with Maxwell's equations explained, where Coulomb's law reappears as Gauss's law, with Ohm's law and Kirchhoff's laws, which pick up where charges start to move, and with the laws of gravity from Newton to Einstein, because Newton's law of gravitation has exactly the same shape and makes a useful comparison.

A note on numbers. Constants come from NIST's 2022 CODATA tables, the 2019 SI facts from the BIPM SI Brochure, the history from MacTutor, the Académie's own printed volume and Maxwell's 1879 edition of Cavendish, and ESD figures from the EOS/ESD Association. Inputs we chose ourselves are labelled illustrative, and every calculation was rerun with the Python script in the walkthrough.

Key takeaways

• The law: F = k q₁ q₂ / r². Double the distance and the force falls to a quarter. Double either charge and the force doubles. The force acts along the line joining the charges.
• The constant: k = 1/(4πε₀) = 8.987 551 7862 × 10⁹ N·m²/C², from the CODATA 2022 value ε₀ = 8.854 187 8188(14) × 10⁻¹² F/m. Since 20 May 2019 the elementary charge is exact and ε₀ is a measured quantity with a relative uncertainty of 1.6 × 10⁻¹⁰.
• Electricity dwarfs gravity: in a hydrogen atom the electric attraction between the proton and electron is about 2.27 × 10³⁹ times the gravitational attraction between them.
• A coulomb is enormous: two charges of 1 C held 1 m apart would push on each other with about 9 × 10⁹ N. Everyday static charges are measured in nanocoulombs.
• Superposition makes it general: forces from many charges add as vectors. Turn Coulomb's law into a field, integrate over a closed surface, and you get Gauss's law, the first of Maxwell's four equations.
• It runs real technology: capacitors, laser printers, electrostatic precipitators and the ESD controls in every electronics factory are applied Coulomb's law. Many components can be damaged by less than 100 V, while walking across a carpet in dry air can charge a person to 35,000 V.
• It has limits: the law is exact for charges at rest. Moving charges bring magnetism and radiation, and at atomic scales quantum mechanics takes over the description even though the Coulomb potential is still the input.
• It has been tested hard: the exponent 2 has been confirmed experimentally to about 1 part in 10¹⁶, starting from Cavendish's concentric globes in the 1770s.

What is Coulomb's law, who published it and when?

Rub a balloon on your hair and it will pull your hair toward it and push a second rubbed balloon away. Coulomb's law is the quantitative statement of what is going on. It has three parts.

1. Magnitude depends on the product of the charges. If one charge is doubled, the force doubles. If both are doubled, it goes up four times.
2. Magnitude depends on the inverse square of the distance. Move the charges twice as far apart and the force drops to one quarter. Move them ten times as far and it drops to one hundredth.
3. Direction is along the line joining the charges. Like charges (both positive or both negative) repel. Unlike charges attract.

In symbols, for two point charges q₁ and q₂ a distance r apart in vacuum:

F = k q₁ q₂ / r², with k = 1/(4πε₀)

If you keep the signs of the charges, a positive F means repulsion and a negative F means attraction. In vector form, the force on charge 2 due to charge 1 is F₁₂ = k q₁ q₂ r̂₁₂ / r², where r̂₁₂ is the unit vector pointing from charge 1 to charge 2. Newton's third law holds: the force on charge 1 due to charge 2 has the same size and the opposite direction.

The phrase "point charge" matters. The law is written for charges small compared with the distance between them. For spheres with charge spread uniformly over their surface or through their volume, the outside force is exactly the same as if all the charge sat at the centre, which is a result you can prove with Gauss's law later in this post. For irregular objects you split them into small pieces and add up the forces.

Charles Augustin de Coulomb and the torsion balance

Coulomb was born on 14 June 1736 in Angoulême and died on 23 August 1806 in Paris, according to the MacTutor biography. He trained as a military engineer and was first known for applied mechanics: his memoir on friction and simple machines won the Académie des Sciences Grand Prix in 1781 and led to his election to the Académie. An earlier 1777 memoir on magnetic compasses, which shared a prize, contained his first work on the torsion balance, an instrument that measures a tiny force by the angle through which it twists a fine thread or wire.

A light horizontal rod hangs from a thin fibre with a small charged ball on one end. Bring a second charged ball near it and the repulsion turns the rod, twisting the fibre, whose restoring torque grows in proportion to the angle. By twisting the top of the fibre to push the balls back to a set separation, Coulomb could compare forces at different distances. A fine fibre resists twisting so weakly that even the minute forces between charged pith balls produced readable angles.

MacTutor records that Coulomb submitted seven treatises on electricity and magnetism to the Académie between 1785 and 1791. In them he established the inverse square law of attraction and repulsion between electric charges, extended the same law to magnetic poles, and studied how charge distributes itself on conductors. The Académie printed its proceedings in annual volumes that ran a few years behind: the volume for the year 1786, printed in Paris by the Imprimerie royale in 1788, carries his "Quatrième Mémoire sur l'Électricité". It opens by noting that the three preceding memoirs had determined the law of repulsion between like electric fluid and the law of attraction between unlike fluids, and goes on to show that on a conductor the charge spreads over the surface and does not penetrate the interior.

Cavendish got there first, but nobody knew

Henry Cavendish carried out a series of electrical experiments between 1771 and 1781 and published only a small part of them. One of the unpublished results was a test of the inverse square law. The experiment used a globe enclosed within two hemispheres: charge the outer conductor while it is connected to the inner globe, then check whether any charge ended up on the inner globe. If the force law were exactly inverse square, none would. James Clerk Maxwell edited Cavendish's manuscripts and published them in 1879. In his introduction Maxwell wrote that the experiment proved the electric force varies inversely as the square of the distance, or at least cannot differ from that ratio by more than a fiftieth part. Maxwell's own repetition with a quadrant electrometer could, he wrote, detect a deviation not exceeding one in 72,000.

So the honest answer to "who discovered Coulomb's law" is that Cavendish had a strong null test in the 1770s and kept it in a drawer, and Coulomb published direct force measurements from 1785 onward. The law carries Coulomb's name because his work was the one the scientific world could read and build on.

Timeline: from torsion balance to the redefined SI

What is the Coulomb constant, and how does it relate to ε₀?

The constant k converts "coulombs squared per metre squared" into newtons. Writing it as 1/(4πε₀) looks like extra clutter, but it pays off. The 4π is the surface area of a unit sphere, and putting it here removes it from Gauss's law and from the formula for a parallel plate capacitor, which are used far more often in engineering. The SI system made that choice, and it is why ε₀ appears everywhere in electromagnetism.

From the CODATA 2022 value of the vacuum electric permittivity:

• ε₀ = 8.854 187 8188(14) × 10⁻¹² F m⁻¹, relative standard uncertainty 1.6 × 10⁻¹⁰
• k = 1/(4πε₀) = 8.987 551 7862 × 10⁹ N m² C⁻²

For hand calculations, k ≈ 8.99 × 10⁹ N·m²/C² is fine, and that is the rounded value OpenStax University Physics uses. Many textbooks also round ε₀ to 8.85 × 10⁻¹² C²/(N·m²).

There is a neat cross check. The permittivity and the magnetic permeability of vacuum are linked through the speed of light: ε₀ μ₀ = 1/c². That means k can also be written as μ₀c²/(4π). With the CODATA 2022 value μ₀ = 1.256 637 061 27(20) × 10⁻⁶ N A⁻² and the exact c = 299 792 458 m/s, both routes give the same k to ten significant figures.

What did the 2019 SI redefinition change?

Before 2019, the ampere was defined by the force between two infinitely long parallel wires carrying current. As the BIPM SI Brochure explains, that definition had the effect of fixing μ₀ at exactly 4π × 10⁻⁷ H/m. Because ε₀ = 1/(μ₀c²) and c was already exact, ε₀ and therefore k were exact numbers too. The elementary charge e, on the other hand, had to be measured.

The 26th meeting of the General Conference on Weights and Measures in November 2018 changed that, with effect from 20 May 2019. The ampere is now defined by fixing the elementary charge at exactly e = 1.602 176 634 × 10⁻¹⁹ C, so one ampere is the current carried by 1/(1.602 176 634 × 10⁻¹⁹) elementary charges per second. NIST's ampere page describes the old wire definition as "a completely hypothetical situation", which is one reason the change was welcome.

The BIPM spells out the trade. Fixing e instead of μ₀ means μ₀ must be determined experimentally, and so must ε₀, the characteristic impedance of vacuum Z₀ and the admittance Y₀. They all carry the same relative uncertainty as μ₀. The product ε₀μ₀ = 1/c² stays exact. At the time of the change μ₀ was equal to 4π × 10⁻⁷ H/m within a relative standard uncertainty of 2.3 × 10⁻¹⁰.

In practice this changes nothing you will notice with a calculator. The CODATA 2022 value of μ₀ divided by 4π × 10⁻⁷ is 0.999 999 999 87, a difference in the tenth decimal place. What changed is which numbers are defined and which are measured. Before 2019, k was exact and e was measured. Since 2019, e is exact and k is measured, through the fine structure constant α = e²/(2ε₀hc), which experiments pin down with great precision.

How big is a coulomb?

One coulomb is the charge of about 6.24 × 10¹⁸ electrons (1 divided by e). Plug two 1 C charges at 1 m into Coulomb's law and you get 8.99 × 10⁹ N, the weight of roughly 900,000 tonnes. Even 1 km apart they would push with almost 9,000 N. That is why everyday static charges are measured in nanocoulombs (10⁻⁹ C) and microcoulombs (10⁻⁶ C).

Worked example 1: two charged spheres, and what happens when the distance doubles

Take two small metal spheres, each carrying +50 nC (illustrative values, typical of a well rubbed balloon or a charged sphere in a teaching lab). Their centres are 10 cm apart in air. OpenStax lists the dielectric constant of dry air at 1 atm as 1.00059, so for this purpose air behaves like vacuum.

Step 1: convert to SI units. q₁ = q₂ = 50 × 10⁻⁹ C = 5.0 × 10⁻⁸ C. r = 0.10 m.

Step 2: apply the law.

F = k q₁ q₂ / r² = (8.988 × 10⁹) × (5.0 × 10⁻⁸)² / (0.10)²

(5.0 × 10⁻⁸)² = 2.5 × 10⁻¹⁵, and dividing by 0.01 gives 2.5 × 10⁻¹³. Multiply by k:

F ≈ 2.247 × 10⁻³ N, about 2.2 millinewtons.

Both charges are positive, so the force is repulsive. To make it tangible: 2.2 mN is the weight of about 0.23 g, a little less than a paper clip. That is enough to visibly swing two pith balls on threads apart, which is exactly what the classic electroscope demonstration shows.

Step 3: double the distance. Move the centres to 20 cm. The charges are unchanged, r² goes from 0.01 m² to 0.04 m², four times larger, so the force falls to one quarter:

F ≈ 5.617 × 10⁻⁴ N, about 0.56 mN.

Step 4: flip a sign. Make the second sphere −50 nC. The magnitude is unchanged at each distance; only the direction flips from repulsion to attraction.

Step 5: what if the spheres touch? If you touch an uncharged identical sphere to one of the charged ones, the charge shares equally (by symmetry), leaving 25 nC on each. Now one sphere has 50 nC and the other 25 nC, so the force at 10 cm halves to about 1.12 mN. This charge sharing trick is how Coulomb produced known fractions of charge without any way to measure charge absolutely.

A practical caveat: real spheres close together are not point charges. When conducting spheres are only a few radii apart, the charges on each sphere push each other toward the far sides (repulsion) or near sides (attraction), which changes the effective separation. The point charge formula is accurate when the separation is large compared with the sphere radii, and engineers use numerical field solvers when it is not.

Worked example 2: electric versus gravitational force in hydrogen

This is the classic comparison, and OpenStax University Physics uses it as its first Coulomb's law example. A hydrogen atom is one proton and one electron. In the Bohr model, the electron in the ground state sits at the Bohr radius a₀. Modern quantum mechanics replaces the orbit with a probability cloud, but a₀ remains the most probable distance and the natural length scale.

Constants (CODATA 2022):

• e = 1.602 176 634 × 10⁻¹⁹ C (exact)
• a₀ = 5.291 772 105 44(82) × 10⁻¹¹ m
• mₑ = 9.109 383 7139(28) × 10⁻³¹ kg
• mₚ = 1.672 621 925 95(52) × 10⁻²⁷ kg
• G = 6.674 30(15) × 10⁻¹¹ m³ kg⁻¹ s⁻²

Step 1: the electric force.

Fₑ = k e² / a₀² = (8.988 × 10⁹) × (1.602 × 10⁻¹⁹)² / (5.292 × 10⁻¹¹)²

e² = 2.567 × 10⁻³⁸ C². a₀² = 2.800 × 10⁻²¹ m². So:

Fₑ ≈ 8.239 × 10⁻⁸ N, attractive.

That looks small, but it acts on an electron with a mass of about 10⁻³⁰ kg. Divide the force by the mass and the electron's acceleration toward the proton is around 9 × 10²² m/s².

Step 2: the gravitational force.

F_g = G mₑ mₚ / a₀² = (6.674 × 10⁻¹¹) × (9.109 × 10⁻³¹) × (1.673 × 10⁻²⁷) / (2.800 × 10⁻²¹)

F_g ≈ 3.632 × 10⁻⁴⁷ N, also attractive.

Step 3: the ratio.

Fₑ / F_g ≈ 2.27 × 10³⁹

The distance cancels, because both laws are inverse square. The ratio depends only on the constants: k e² / (G mₑ mₚ). For two protons the same calculation gives about 1.24 × 10³⁶, and for two electrons about 4.17 × 10⁴².

What the ratio means. Gravity plays no measurable part inside atoms. It dominates at large scales only because matter is almost perfectly neutral: positive and negative charges cancel to extraordinary precision, so large objects carry almost no net charge, while mass always adds. The laws of gravity post covers what happens at those scales.

Step 4: two cross checks you get for free. The potential energy of the electron at a₀ is U = −k e² / a₀ = −4.3597 × 10⁻¹⁸ J. Divide by e and that is −27.211 eV. The CODATA 2022 Hartree energy, the atomic unit of energy, is 4.359 744 722 2060(48) × 10⁻¹⁸ J, and it is defined as exactly this quantity, so the numbers match. Set the Coulomb force equal to the centripetal force mₑv²/a₀ and solve for v, and you get about 2.19 × 10⁶ m/s. Divide by c and the answer is 0.007 297, which is the CODATA fine structure constant α = 7.297 352 5643(11) × 10⁻³. In other words, α is the speed of the Bohr model electron as a fraction of the speed of light, a useful way to remember what the "coupling strength of the electromagnetic force" means.

How do superposition, the electric field and Gauss's law extend Coulomb's law?

Coulomb's law describes two charges. Real problems have many. Three ideas make the law general.

Superposition: forces add as vectors

The force on a charge from a collection of other charges is the vector sum of the forces each one would exert alone. The charges do not modify each other's influence. This principle of superposition is an experimental fact about electromagnetism in vacuum, not a mathematical given, and it is what turns a two body law into a tool for any arrangement.

Worked example 3: three charges and a vector sum

Place two charges of +4 nC on the x axis at x = 0 and x = 0.30 m. A third charge of −1 nC sits at (0.15 m, 0.20 m), directly above the midpoint. What is the net force on the third charge? (All values illustrative.)

Step 1: distances. From each positive charge to the third charge, the horizontal offset is 0.15 m and the vertical offset 0.20 m. r = √(0.15² + 0.20²) = 0.25 m. This is a 3, 4, 5 triangle scaled by 0.05.

Step 2: magnitude from each charge. F = k × (4 × 10⁻⁹) × (1 × 10⁻⁹) / 0.25² = 8.988 × 10⁹ × 4 × 10⁻¹⁸ / 0.0625 ≈ 5.752 × 10⁻⁷ N. The third charge is negative and the others positive, so each force is attractive and points from the third charge toward the positive charge that exerts it.

Step 3: components. By symmetry, the horizontal components cancel: one force pulls left and down, the other right and down. Each vertical component is F × (0.20/0.25) = F × 0.8 ≈ 4.602 × 10⁻⁷ N, both pointing down toward the axis.

Step 4: add. Net force = 2 × 4.602 × 10⁻⁷ ≈ 9.20 × 10⁻⁷ N, pointing straight down toward the x axis.

Without symmetry you would compute x and y components for each source separately and add them. That is what a computer does, at scale, in an electrostatic simulation.

The electric field: force per unit charge

The electric field at a point is the force a small positive test charge would feel there, divided by its charge: E = F/q. For a single point charge Q, Coulomb's law gives E = k Q / r², pointing away from a positive Q and toward a negative one. Fields from many charges add by superposition, just like forces.

In worked example 3, the field at the third charge's location, produced by the two +4 nC charges, is the net force divided by 1 nC: about 920 N/C, pointing up away from the axis (the field points the way a positive test charge would be pushed; the negative charge feels a force in the opposite direction). In hydrogen, the proton's field at a₀ is about 5.14 × 10¹¹ N/C, which is why atoms are so hard to pull apart with laboratory fields.

The field picture splits the problem in two: sources set up a field, and any charge placed in it responds.

Gauss's law: the inverse square law seen from outside

Draw any closed surface around a point charge and add up the field passing through it (the electric flux). For a sphere of radius r centred on the charge, the field is E = Q/(4πε₀r²) everywhere on the surface and the area is 4πr², so the flux is exactly Q/ε₀. The r² in the area cancels the r² in the field. For any other closed shape the result is the same, because the inverse square law is exactly what makes flux independent of distance.

That is Gauss's law: the total electric flux through a closed surface equals the enclosed charge divided by ε₀. It is equivalent to Coulomb's law for static charges, and it is much faster for symmetric problems. OpenStax works through the standard cases:

• Outside a uniformly charged sphere the field is the same as for a point charge at the centre. This is why worked examples 1 and 2 can treat spheres and nuclei as points.
• Inside a conductor at equilibrium the field is zero, and any excess charge sits on the surface. This is the result of Coulomb's fourth memoir, now a one line proof.
• Near a large flat sheet of charge density σ, the field is σ/(2ε₀), independent of distance. Between two oppositely charged plates the fields add to σ/ε₀, which is the starting point for every capacitor calculation.

Gauss's law also explains why Cavendish's concentric globe test is so sensitive. If the force law were 1/r^(2+q) with q not zero, a charged outer shell would produce a small field inside, and charge would flow to the inner globe. Finding none limits q. Every modern test of the inverse square law uses some version of that null method.

How Coulomb's law sits inside Maxwell's equations

Maxwell's equations are four statements that together describe all classical electric and magnetic phenomena. OpenStax notes that the four basic laws had been found experimentally through the work of Oersted, Coulomb, Gauss and Faraday before Maxwell unified them and corrected a logical inconsistency in Ampère's law. In differential form:

1. Gauss's law for electricity: ∇·E = ρ/ε₀. This is Coulomb's law in local form. For a static point charge it gives back E = kQ/r² exactly.
2. Gauss's law for magnetism: ∇·B = 0. There are no magnetic charges.
3. Faraday's law: ∇×E = −∂B/∂t. A changing magnetic field makes a circulating electric field.
4. Ampère's law with Maxwell's correction: ∇×B = μ₀J + μ₀ε₀ ∂E/∂t.

For static charges, the time derivatives vanish, ∇×E = 0, and the electric field is fully determined by the first equation. That is electrostatics, and Coulomb's law is its complete solution for point sources. Once charges move, the other three equations switch on. Notice that ε₀ and μ₀ both appear in the fourth equation, and their product is 1/c². Coulomb's constant and the speed of light are tied together, which is exactly the relation the 2019 SI preserves as exact. Our Maxwell's equations post follows that thread to radio, Wi-Fi and fibre, and our electromagnetic spectrum post covers what the resulting waves do.

Which tool should you reach for?

Where does Coulomb's law show up in technology?

Several everyday technologies are Coulomb's law with an engineering wrapper, and one of them, electrostatic discharge, is a real cost in every electronics factory and on every hardware workbench.

Capacitors

A capacitor is two conductors separated by an insulator. Charge one plate with +Q and the other with −Q, and Coulomb attraction holds the charges on the facing surfaces. For parallel plates of area A and gap d in vacuum, Gauss's law gives the field between them as E = σ/ε₀ = Q/(ε₀A), and the voltage as V = Ed, so the capacitance is C = Q/V = ε₀A/d. Filling the gap with a dielectric of dielectric constant κ multiplies it by κ, because the material's own charges partly shield the field.

Illustrative numbers. Plates of 1 cm² (10⁻⁴ m²) with a 10 µm vacuum gap: C = 8.854 × 10⁻¹² × 10⁻⁴ / 10⁻⁵ ≈ 88.5 pF. At 5 V the stored charge is Q = CV ≈ 4.43 × 10⁻¹⁰ C. The plates pull on each other with F = Q²/(2ε₀A) ≈ 1.1 × 10⁻⁴ N, a tenth of a millinewton. Real multilayer ceramic capacitors reach microfarads in millimetre packages by stacking many thin layers with high κ dielectrics, but the physics is still Coulomb's law between the plates. The Ohm's law post picks up how capacitors charge and discharge through resistors.

Laser printers and photocopiers

OpenStax describes xerography in detail. A photoconducting drum (selenium on aluminium in the classic design) is sprayed with charge by a corona device. In the dark, the drum holds its charge. Light makes the photoconductor conduct, so the illuminated areas lose their charge. What remains is an invisible charge image. Toner particles, charged with the opposite sign, are attracted by Coulomb forces only to the charged areas, transferred to paper by a stronger charge on the paper, then fused with heat. A laser printer uses the same process with a laser writing the image onto the drum under computer control. OpenStax also notes that ink jet printers give droplets an electrostatic charge and steer them with charged plates.

Electrostatic precipitators

The same textbook section describes precipitators: charge the smoke, dust and pollen in a stream of air, then pass it across an oppositely charged grid that attracts and holds the particles. OpenStax reports that large industrial precipitators remove over 99% of particles from stack gas emissions from burning coal and oil.

Electrostatic discharge damage to chips

Static electricity is Coulomb's law at its most practical. Rub two materials together and separate them, and electrons transfer from one to the other. The EOS/ESD Association's ESD fundamentals guide lists a typical triboelectric series, from materials that tend to charge positive (rabbit fur, glass, human hair, nylon, wool) to those that tend to charge negative (polyester, silicon, the fluoropolymer used in non stick coatings). Which way and how much depends on the materials, their surfaces, the speed of separation and the humidity.

The voltages are larger than most people expect. The ESDA's Table 1 gives these typical static generation levels:

Source: EOS/ESD Association, ESD Fundamentals Part 1. The ESDA stresses that charge generation still occurs at high humidity, just less of it.

Now compare with what parts can survive. The same ESDA guide says many electronic components are susceptible to ESD damage below 100 V, and many disk drive components withstand less than 10 V. Shrinking transistors make the problem worse because thinner gate insulators break down at lower voltages. The guide notes that a damaged part can cost anything from a few cents for a simple diode to thousands of dollars for a complex integrated circuit, before repair, rework and shipping costs are added.

Worked example 4: how much charge is in a carpet shock?

Treat a person as a conductor with a capacitance to the surroundings. We assume 100 pF as a round illustrative figure; real values depend on body size, footwear and the floor. Use the ESDA's dry air carpet voltage of 35,000 V.

Charge: Q = CV = 100 × 10⁻¹² × 35,000 = 3.5 × 10⁻⁶ C, or 3.5 µC.

Number of electrons: 3.5 × 10⁻⁶ / 1.602 × 10⁻¹⁹ ≈ 2.2 × 10¹³ electrons. That sounds like a lot, but it is a tiny imbalance in a body made of an enormous number of neutral atoms, which is why you feel nothing until the spark.

Energy stored: ½CV² = ½ × 100 × 10⁻¹² × 35,000² ≈ 0.061 J. Released in a fraction of a microsecond through a fingertip spark, that is what you feel.

Why there is a spark at all: OpenStax gives the dielectric strength of dry air at 1 atm as 3.0 × 10⁶ V/m. Above that field the air ionises and conducts. As a rough uniform field estimate, 35,000 V can jump about 35,000 / 3.0 × 10⁶ ≈ 1.2 cm, which is why a dry winter shock arcs across a visible gap before your finger touches the metal. Real gaps between a fingertip and a door handle are not uniform, so treat the figure as an order of magnitude.

At high humidity: the same person at 1,500 V carries 0.15 µC and stores about 0.11 mJ. At the 100 V a worker might reach at a bench in humid air, the charge is 10 nC and the energy 0.5 µJ, and you would feel nothing.

Why that matters for chips: a discharge you cannot feel is still enough to exceed the sub 100 V tolerance the ESDA describes. The current flows through a path a few micrometres wide inside the chip, and the energy density is high enough to melt a junction or punch through an oxide. Sometimes the device fails at once. Sometimes it is weakened and fails weeks later in the field, which is harder to diagnose.

ESD protection in electronics manufacturing

The industry manages ESD with programmes and standards. The EOS/ESD Association's standards page lists the main documents, including:

• ANSI/ESD S20.20, the standard for developing an ESD control programme (the ESDA's adoption notice names the 2021 edition, S20.20-2021). The ESDA also notes that the US Department of Defense cancelled its own military standard MIL-STD-1686C and moved to S20.20, and moved from MIL-HDBK-263 to the ESD TR20.20 handbook.
• ANSI/ESDA/JEDEC JS-001, the human body model (HBM) device test, listed in its 2024 edition, and JS-002 for the charged device model (CDM), listed in its 2022 edition. HBM simulates a charged person touching a part; CDM simulates a charged part discharging when it touches a conductor.
• The IEC 61340 series, the international counterpart, which the ESDA references in its standards news.

The ESDA guide describes three event models: a person discharging into a device (HBM), a charged metal tool or fixture discharging (the machine model), and a charged device discharging into ground (CDM). Each produces a different current waveform, and parts are rated against them.

The controls follow straight from Coulomb's law and charge conservation:

• Ground the conductors. Wrist straps, grounded worksurfaces, conductive flooring and footwear give charge on people and tools a slow, controlled path to ground through a resistor, so it bleeds away instead of sparking.
• Keep insulators away or neutralise them. Insulators cannot be grounded, because charge does not move through them. Ionisers blow balanced positive and negative ions over the work area so the field from a charged insulator attracts ions of the opposite sign and neutralises itself.
• Shield in transit. Static shielding bags use a conductive layer that, by Gauss's law, keeps the field inside near zero when the outside is charged, the same principle as Coulomb's charge on the surface of a conductor.
• Design protection into the chip. Integrated circuits include on chip ESD protection structures on their pins that clamp a fast spike and steer it to the supply rails. Device ratings such as HBM and CDM classes tell board designers how much handling margin they have.

At a home workbench the same principles apply: a grounded mat and wrist strap, parts kept in shielding bags until needed, and plastic packaging kept away from exposed boards. Our post on how a printed circuit board is made shows where those boards come from.

What does Coulomb's law not cover?

Coulomb's law is exact for its domain. Knowing the edges of that domain is how you avoid using it where it gives wrong answers.

Moving charges and magnetism

The law assumes charges at rest, or at least moving slowly enough that the field has time to settle. When charges move, two things change. First, a moving charge produces a magnetic field, and a second moving charge feels a magnetic force on top of the electric one. Two parallel wires carrying current in the same direction attract each other even though each wire is electrically neutral overall; that is the effect the old definition of the ampere was built on. Second, the electric field of a moving charge is not the simple kQ/r² pattern. Changes in a charge's position propagate outward at the speed of light, so the field at a distant point reflects where the charge was, not where it is. An accelerating charge radiates electromagnetic waves. All of this lives in the full Maxwell's equations, not in Coulomb's law.

For slow charges the corrections are of order (v/c)². That is why Coulomb's law still works in hydrogen, where the electron moves at about α ≈ 1/137 of the speed of light, as worked example 2 showed.

Materials

In a material, the charges of the atoms rearrange in response to a field. A dielectric reduces the force between two embedded charges by its dielectric constant κ. OpenStax gives water a dielectric constant of about 80, so in water the attraction between two ions is roughly eighty times weaker than in vacuum. That is a big part of why salts dissolve. Coulomb's law with ε₀ replaced by κε₀ is a good approximation for a uniform material on scales much larger than its molecules, and it breaks down on scales comparable to them.

Quantum effects at short range

At atomic scales, electrons do not follow classical orbits. The Coulomb potential −ke²/r is still the correct input for the hydrogen atom, and solving the Schrödinger equation with it reproduces the energy levels very well. But the electron's position is a probability distribution, and quantities like "the force at a definite distance" are no longer what you measure. Inside an atomic nucleus, the strong nuclear force overwhelms the Coulomb repulsion between protons at distances around 1 fm.

To see how strong that repulsion is, plug two protons at 1 fm (10⁻¹⁵ m) into Coulomb's law: F ≈ 231 N. That is the weight of a 23 kg mass acting on a particle with a mass of about 1.7 × 10⁻²⁷ kg. Something much stronger has to hold nuclei together, and that is the strong force, which falls off very quickly beyond a few femtometres. The Coulomb number is a useful order of magnitude, not a precise nuclear calculation.

How precisely has the inverse square law been tested?

Williams, Faller and Hill expressed their result as a limit on a photon rest mass, because a massive photon would make the electric field fall off slightly faster than 1/r². Their null result extended the validity of Coulomb's law, in their words, "by two orders of magnitude" over earlier work.

Table: everyday and atomic scale scenarios with forces

All forces computed with k = 8.987 551 7862 × 10⁹ N·m²/C² and CODATA 2022 constants. Charges and distances marked illustrative are our own round choices.

Walkthrough: reproduce every number in Python

You can check every figure in this post with standard Python. The script uses only the standard library and hard codes the CODATA 2022 values, so nothing needs installing for the calculations. Scientific libraries also publish physical constants, but check which CODATA edition your installed version uses before you mix its values with the ones here; constants change slightly between CODATA releases.

Step 1: create a clean environment

mkdir coulomb-check && cd coulomb-check
python3 -m venv .venv
source .venv/bin/activate
python -m pip install --upgrade pip
python -m pip freeze > requirements.txt

A reproducible calculation should not depend on whatever is installed globally. If you later add a library, pin an exact version and freeze again.

Step 2: write the calculations

# coulomb.py
import math

# CODATA 2022, from physics.nist.gov/cuu
e    = 1.602176634e-19     # C, exact since 20 May 2019
eps0 = 8.8541878188e-12    # F/m, measured
mu0  = 1.25663706127e-6    # N/A^2, measured
c    = 299792458.0         # m/s, exact
a0   = 5.29177210544e-11   # m
me   = 9.1093837139e-31    # kg
mp   = 1.67262192595e-27   # kg
G    = 6.67430e-11         # m^3 kg^-1 s^-2

k = 1 / (4 * math.pi * eps0)

def force(q1, q2, r):
    """Coulomb force in newtons; positive = repulsive."""
    return k * q1 * q2 / r**2

print(f"k = {k:.10e}  (via mu0 c^2 / 4pi: {mu0 * c**2 / (4 * math.pi):.10e})")

# Worked example 1: two 50 nC spheres
for r in (0.05, 0.10, 0.20, 0.40, 1.0):
    print(f"50 nC pair at {r:4.2f} m: {force(50e-9, 50e-9, r):.4e} N")

# Worked example 2: hydrogen
Fe = k * e**2 / a0**2
Fg = G * me * mp / a0**2
print(f"hydrogen Fe = {Fe:.4e} N, Fg = {Fg:.4e} N, ratio = {Fe / Fg:.4e}")
U = -k * e**2 / a0
v = math.sqrt(k * e**2 / (me * a0))
print(f"U = {U:.6e} J = {U / e:.4f} eV; v/c = {v / c:.6f}")

# Worked example 3: superposition
q, q3 = 4e-9, -1e-9
r = math.hypot(0.15, 0.20)
each = abs(force(q, q3, r))
net = 2 * each * (0.20 / r)
print(f"three charges: each {each:.4e} N, net {net:.4e} N toward the axis")

# Worked example 4: carpet shock with an illustrative 100 pF body
C = 100e-12
for V in (35000, 1500, 100):
    Q = C * V
    print(f"{V:>6} V: Q = {Q:.2e} C, electrons = {Q / e:.2e}, energy = {0.5 * C * V**2:.2e} J")

Step 3: run it and compare

python coulomb.py

Expected output:

k = 8.9875517862e+09  (via mu0 c^2 / 4pi: 8.9875517862e+09)
50 nC pair at 0.05 m: 8.9876e-03 N
50 nC pair at 0.10 m: 2.2469e-03 N
50 nC pair at 0.20 m: 5.6172e-04 N
50 nC pair at 0.40 m: 1.4043e-04 N
50 nC pair at 1.00 m: 2.2469e-05 N
hydrogen Fe = 8.2387e-08 N, Fg = 3.6315e-47 N, ratio = 2.2687e+39
U = -4.359745e-18 J = -27.2114 eV; v/c = 0.007297
three charges: each 5.7520e-07 N, net 9.2033e-07 N toward the axis
 35000 V: Q = 3.50e-06 C, electrons = 2.18e+13, energy = 6.12e-02 J
  1500 V: Q = 1.50e-07 C, electrons = 9.36e+11, energy = 1.12e-04 J
   100 V: Q = 1.00e-08 C, electrons = 6.24e+10, energy = 5.00e-07 J

If your hydrogen ratio comes out near 2.3 × 10³⁹ but not exactly 2.2687, check that you used the CODATA 2022 masses and not older rounded values. If k is off in the fourth digit, you probably typed ε₀ as 8.85 × 10⁻¹² instead of the full value.

Step 4: keep the script's dependencies healthy

A calculation script stays small only until someone adds a plotting library, a notebook server or a field solver binding. Each of those pulls in more open source packages, and those packages collect vulnerabilities over time. Scan the folder with the Safeguard CLI and read the JSON results it writes under ./output:

safeguard scan --dir .
ls output/

Run it again whenever you add a library, and commit requirements.txt so a colleague who reruns your numbers installs the same versions.

Common mistakes

• Forgetting to square the distance. The most common error. Halving the distance multiplies the force by four, not two.
• Mixing units. Charges must be in coulombs and distances in metres. Writing 50 for 50 nC, or 10 for 10 cm, gives answers wrong by many orders of magnitude.
• Using a sign convention inconsistently. Either compute magnitudes and decide the direction from "like repel, unlike attract", or keep signs and remember that a negative product means attraction. Mixing the two gives forces pointing the wrong way.
• Adding forces as numbers instead of vectors. Superposition is a vector sum. Two equal forces at an angle do not add to twice the force; in worked example 3 the horizontal components cancel.
• Treating close conductors as point charges. Charges on nearby conductors rearrange. The point charge formula is accurate only when the separation is large compared with the objects.
• Ignoring the medium. In water, oil or a capacitor dielectric the force is reduced by the dielectric constant. Vacuum formulas give the wrong answer inside materials.
• Assuming humidity makes ESD safe. The ESDA's own figures show reduced but not zero charge generation at 65 to 90% relative humidity, and many parts fail below 100 V.
• Applying Coulomb's law to fast moving or accelerating charges. Once velocities are a meaningful fraction of c, or the question is about radiation, you need Maxwell's equations.

FAQ

What is Coulomb's law in simple terms?

The electric force between two charged objects gets stronger as the charges get bigger and weaker as they move apart, falling with the square of the distance. Like charges push apart and unlike charges pull together.

What is the formula for Coulomb's law?

F = k q₁ q₂ / r², where q₁ and q₂ are the charges in coulombs, r is the distance between them in metres, and k = 1/(4πε₀) ≈ 8.988 × 10⁹ N·m²/C². The force acts along the line joining the charges.

What is the value of Coulomb's constant?

Using the CODATA 2022 value of the vacuum electric permittivity, k = 8.987 551 7862 × 10⁹ N·m²/C². For homework, 8.99 × 10⁹ is accurate enough.

Who discovered Coulomb's law?

Charles Augustin de Coulomb published it, based on torsion balance measurements, in memoirs submitted to the French Académie des Sciences from 1785. Henry Cavendish had tested the inverse square law in the 1770s but did not publish; Maxwell published his notes in 1879.

Why is Coulomb's law an inverse square law?

The field of a point charge spreads equally in all directions through spheres whose area grows with r², so its strength falls as 1/r². Gauss's law formalises this, and experiments confirm the exponent 2 to about 1 part in 10¹⁶.

How is Coulomb's law similar to Newton's law of gravitation?

Both are inverse square laws. Gravity only attracts, while electric forces attract or repel, and the electric force is vastly stronger: about 2.27 × 10³⁹ times gravity for a proton and electron.

What changed about ε₀ in the 2019 SI redefinition?

The elementary charge was fixed at exactly 1.602 176 634 × 10⁻¹⁹ C. As a result the magnetic constant μ₀, which used to be exactly 4π × 10⁻⁷ H/m, became a measured quantity, and so did ε₀ and Coulomb's constant. Their product ε₀μ₀ = 1/c² is still exact.

How does Coulomb's law relate to Maxwell's equations?

Gauss's law for electricity, ∇·E = ρ/ε₀, is Coulomb's law rewritten for fields. For static charges it is equivalent to Coulomb's law. The other three Maxwell equations add magnetism, induction and radiation, which matter once charges move.

Does Coulomb's law work for atoms?

Yes, as the potential energy that goes into quantum mechanics. It sets atomic sizes and energies; what changes is that electrons have probability distributions, not classical paths.

Why does static electricity damage electronics?

A person can charge to tens of thousands of volts in dry air, while many components fail below 100 V. The discharge drives a brief, large current through microscopic structures, melting junctions or breaking thin insulators.

The software that runs the lab: an honest bridge

Safeguard does not do physics. Our models are single purpose security models, and nothing in this post was calculated by them; the numbers come from the published constants and the Python script above. Safeguard also does not analyse hardware, test ESD protection or audit a factory's ESD programme. The link is narrower and practical. Much of the work around electrostatics today is done in software: field solvers, circuit and chip design tools, test equipment control scripts, factory monitoring dashboards for wrist strap and ioniser checks, and the notebooks engineers use to check a calculation. Those programs pull in open source components, and the components need the same dependency hygiene as any other application.

A lab data logger or an ESD monitoring web service is usually a thin layer of your own code on top of open source libraries. When one of them ships a vulnerability, the questions are the usual ones: is there a fixed version, and how quickly can you move to it.

Fix it without waiting for upstream

Whatever the vulnerability, most of the risk sits in two gaps. The first is the time between a fix being published and that fix actually running in your production builds. The second, and worse, is the time before any fix exists at all, while a maintainer works out a patch and a release schedule. Waiting through both gaps is a habit, not a requirement. Safeguard gives teams three ways to close them.

1. Autonomous remediation: pull requests, not tickets

When a fixed version exists, Safeguard's Griffin models open a tested, policy gated pull request that moves the affected dependency to the safe version. You can run it on one repository or in bulk across every project that carries the vulnerable version. Remediation covers eight categories, including dependencies, licenses, security posture and findings. Reachability analysis ranks the vulnerable code paths your application actually calls first, so the first pull requests you merge are the ones that matter most. Autonomous remediation ships switched off, with every category set to Manual, so your team decides how much runs on its own and what still needs a human approval.

2. Gold Open Source: the safe version in seconds, free

gold.safeguard.sh is free and needs no login. Search any CVE or package to see affected and fixed versions, CVSS, EPSS, CISA KEV status and a recommended safe version, or use the "Am I affected?" checker. The same data is available through a free read API, README status badges and RSS threat feeds. The safeguard-sh/gold-scan-action GitHub Action can fail a build on a known exploited vulnerability, on any known vulnerability, or never, so a version you have just removed cannot quietly come back in next week's merge.

3. Gold Registry: start from components that are already fixed

registry.safeguard.sh provides hardened, drop in replacements for upstream packages and container images. Every Gold artifact is held continuously to zero exploitable critical or high CVEs, zero malware and a completed license check, with SLSA L3 or higher provenance, a Sigstore signature and a public evidence page you can hand to an auditor. An artifact loses its Gold badge if it regresses. Teams can point their own package directory at the registry so hardened equivalents resolve without changing how developers install anything.

When upstream has no fix yet, or will not ship one, Safeguard first works to get the fix accepted upstream. If the issue is severe or upstream will not move, Safeguard builds a fork, fixes the vulnerability there and publishes a hardened Gold build, so you are not stuck on a maintainer's release schedule. You can also request a hardened, compatibility tested build of a specific package. This fork and patch flow is in early access today, so ask which packages it covers before you plan around it.

One honest limit: when the vulnerable code lives in a vendor appliance, a hosted service or an operating system runtime you do not build yourself, the fix still has to come from that vendor or distribution. The steps above apply to the open source dependencies and container images you build and ship, which is where most teams carry the largest and least tracked share of their exposure.

See how Safeguard would help

None of these demos involve electrostatics calculations; they show the software side described in the bridge above.

The first demo is a real Safeguard CLI session: it installs the CLI, opens the interactive terminal and asks it to fix a lodash CVE in a repository. It reads the finding and advisory, asks before each command, edits package.json to lodash 4.17.21, runs npm install and npm test with approval, prints a summary and pushes nothing. The same flow applies to a lab dashboard or an ESD monitoring web app your team maintains: scan the repository, then let the assistant propose the dependency upgrade under your approval.

Click through it at your own pace, or open it full screen.

The second demo sets up the Safeguard runner on your own machine, approves it with its device code and then scans from the terminal. That suits teams who keep simulation scripts, instrument control code or test bench tools on lab machines and want the scan to run where that code lives.

Click through it at your own pace, or open it full screen.

The third demo connects Claude to the Safeguard MCP server with OAuth and lets you choose which tools Claude can use. For an engineer who already works with Claude while writing analysis or test code, it brings dependency findings into the same conversation, limited to the tools you allow.

Click through it at your own pace, or open it full screen.

To try it on your own repositories, download Safeguard and follow the CLI documentation.

Quick check

Sources

• NIST CODATA 2022, elementary charge
• NIST CODATA 2022, vacuum electric permittivity
• NIST CODATA 2022, vacuum magnetic permeability
• NIST CODATA 2022, speed of light in vacuum
• NIST CODATA 2022, Bohr radius
• NIST CODATA 2022, electron mass
• NIST CODATA 2022, proton mass
• NIST CODATA 2022, Newtonian constant of gravitation
• NIST CODATA 2022, fine structure constant
• NIST CODATA 2022, Hartree energy
• NIST, SI redefinition
• NIST, SI redefinition: the ampere
• BIPM, The International System of Units (SI Brochure), 9th edition
• BIPM, SI Brochure 9th edition, English PDF
• OpenStax University Physics Volume 2, 5.1 Electric Charge
• OpenStax University Physics Volume 2, 5.3 Coulomb's Law
• OpenStax University Physics Volume 2, 6.3 Applying Gauss's Law
• OpenStax University Physics Volume 2, 8.1 Capacitors and Capacitance
• OpenStax University Physics Volume 2, 8.5 Molecular Model of a Dielectric
• OpenStax University Physics Volume 2, 16.1 Maxwell's Equations and Electromagnetic Waves
• OpenStax College Physics 2e, 18.8 Applications of Electrostatics
• MacTutor History of Mathematics, Charles Augustin de Coulomb
• Histoire de l'Académie royale des sciences, Année 1786 (Paris, 1788), with Coulomb's fourth memoir on electricity, via Internet Archive
• The Electrical Researches of the Honourable Henry Cavendish, written between 1771 and 1781, edited by J. Clerk Maxwell (Cambridge, 1879), via Internet Archive
• Plimpton and Lawton, "A Very Accurate Test of Coulomb's Law of Force Between Charges", Physical Review 50, 1066 (1936)
• Williams, Faller and Hill, "New Experimental Test of Coulomb's Law: A Laboratory Upper Limit on the Photon Rest Mass", Physical Review Letters 26, 721 (1971)
• EOS/ESD Association, ESD Fundamentals Part 1: An Introduction to ESD
• EOS/ESD Association, Standards

Last reviewed: 24 October 2026.