Coulomb's Law¶
The Formula¶
Scalar Form (Magnitude)¶
Where \(k_e = \frac{1}{4\pi\varepsilon_0} \approx 8.99 \times 10^9 \text{ N·m}^2/\text{C}^2\).
Vector Form (Direction)¶
For rigorous physics, we need to know the direction of the force. If we define \(\hat{r}_{12}\) as the unit vector pointing from charge 1 to charge 2, the force on charge 2 exerted by charge 1 is:
- If \(q_1\) and \(q_2\) have the same sign (both \(+\) or both \(-\)), the product \(q_1 q_2\) is positive. The force points in the direction of \(\hat{r}_{12}\) (away from \(q_1\)): Repulsion.
- If they have opposite signs, \(q_1 q_2\) is negative. The force points opposite to \(\hat{r}_{12}\) (towards \(q_1\)): Attraction.
What It Means¶
Coulomb's Law describes the force between two electric charges. Like charges repel, opposite charges attract — and the force falls off with the square of the distance between them. Double the distance, quarter the force.
It looks almost identical to Newton's law of gravity (\(F = G\frac{m_1 m_2}{r^2}\)), and that's not a coincidence. Both are inverse-square laws. But electricity has one feature gravity doesn't: it can push and pull. Gravity only attracts. Charges can be positive or negative, so the force can go either way.
What is \(\varepsilon_0\) (Epsilon Nought)?¶
You'll often see \(k_e\) written as \(\frac{1}{4\pi\varepsilon_0}\). \(\varepsilon_0\) is the vacuum permittivity. It measures how "permissive" empty space is to electric field lines.
- High permittivity means space "resists" the electric field less (or polarizes more to cancel it out), leading to weaker forces.
- Low permittivity would mean stronger forces.
The fact that it's in the denominator means that the vacuum has a specific, fundamental resistance to the formation of electric fields. It sets the "stiffness" of free space for electromagnetism.
Why It Works — The Story Behind the Formula¶
The Question Nobody Could Answer¶
By the mid-1700s, scientists knew a few things about electricity: rubbing amber on wool makes it attract feathers. Glass rubbed with silk repels other rubbed glass. Benjamin Franklin had shown that lightning was electrical (1752). But nobody could answer the most basic quantitative question: how does electric force depend on distance?
Everyone suspected it was an inverse-square law, like gravity. The reasoning was elegant: if electric force spreads out uniformly in all directions from a charge, then at distance \(r\) it's spread over a sphere of area \(4\pi r^2\). Double the distance → four times the area → one-quarter the force per unit area. So \(F \propto 1/r^2\).
But suspecting and proving are different things. Electric forces are tiny. Charges leak away. You can't just put two charged objects on a scale and measure the attraction. The forces are millions of times weaker than the weight of the objects themselves.
Coulomb's Brilliant Instrument (1785)¶
Charles-Augustin de Coulomb was a French military engineer who specialized in structural mechanics — specifically, the science of torsion (twisting). He knew exactly how fibers and wires behave when twisted, and he realized this knowledge could solve the electricity problem.
He built a torsion balance: one of the most sensitive measuring instruments of the 18th century.
Here's how it worked:
- A thin silver wire hung vertically from a frame, supporting a lightweight horizontal needle
- A small charged metal sphere was mounted on one end of the needle, with a counterweight on the other end
- A second charged sphere (fixed in place) was brought near the first one
The electric repulsion between the two charged spheres pushed the needle, twisting the wire. Coulomb knew exactly how much force corresponded to a given twist angle — he'd calibrated this from his torsion research. The wire acted as a microscopic spring: twist angle proportional to force.
He worked in a sealed, draft-free room. Even the heat from his body could create air currents that disturbed the measurement. So he read the needle position through a telescope aimed through a small window.
The Measurements¶
Coulomb systematically varied the distance between the spheres and recorded the twist angle:
| Distance (\(r\)) | Twist angle (force) |
|---|---|
| \(d\) | \(\theta\) |
| \(d/2\) | \(4\theta\) |
| \(d/3\) | \(9\theta\) |
Halve the distance → force quadruples. Third the distance → force goes up by nine. The pattern was unmistakable:
But What About the Charges?¶
Coulomb also needed to figure out how force depends on the amount of charge. This was trickier — you can't see charge, and there were no units for it yet.
His method was ingenious. He started with a charged sphere carrying some unknown charge \(q\). Then he touched it with an identical uncharged sphere. By symmetry, the charge splits equally: each sphere now carries \(q/2\).
He could now compare the force between two spheres with charge \(q\) at a given distance versus one with \(q\) and one with \(q/2\):
- Force with charges \(q\) and \(q\): \(F_1\)
- Force with charges \(q\) and \(q/2\): \(F_2 = F_1 / 2\)
The force was proportional to each charge individually. Combined with the distance result:
Why It Matters¶
This was the moment electricity stopped being magic and became physics. The same geometric law governing planetary orbits also governed the force between tiny charged particles. The universe was using the same playbook at vastly different scales.
It also opened the door to all of electromagnetism. Maxwell's equations, radio waves, light itself — all of it traces back to this inverse-square law measured by a man with a twisted wire in a sealed room.
The Derivation: Why Inverse-Square?¶
The Geometric Argument¶
Imagine a charge \(q\) sitting in empty space. Its electric field radiates outward in all directions, uniformly. At distance \(r\), this field is spread over the surface of a sphere:
If the total "amount of field" (flux) leaving the charge is fixed (this is Gauss's law), then the field strength at distance \(r\) is the total flux divided by the area:
The force on another charge \(q_2\) is \(F = q_2 E\), so:
The inverse-square law is a direct consequence of living in three-dimensional space. If space were two-dimensional, the "sphere" would be a circle (\(C = 2\pi r\)), and the force would be \(\propto 1/r\). If space had four dimensions, it would be \(\propto 1/r^3\). The exponent tells you the dimensionality of space minus one.
Variables Explained¶
| Symbol | Name | Unit | Description |
|---|---|---|---|
| \(F\) | Electric force | Newtons (N) | Force between two charges |
| \(k_e\) | Coulomb's constant | N·m²/C² | \(\approx 8.99 \times 10^9\) |
| \(q_1, q_2\) | Electric charges | Coulombs (C) | Amount of charge on each object |
| \(r\) | Distance | Meters (m) | Distance between the centers of the charges |
| \(\varepsilon_0\) | Vacuum permittivity | C²/(N·m²) | \(\approx 8.85 \times 10^{-12}\) |
Worked Examples¶
Example 1: Two Protons in a Nucleus¶
Two protons (\(q = 1.6 \times 10^{-19}\) C) are separated by \(1 \times 10^{-15}\) m (about the size of a nucleus).
That's about the weight of a 23 kg object — from two particles you can't even see. This is why you need the strong nuclear force to hold nuclei together. Electromagnetism is trying to blow them apart with hundreds of newtons.
Example 2: Coulomb vs. Gravity¶
Compare the electric and gravitational forces between two protons separated by 1 meter.
Electric force:
Gravitational force:
The electric force is \(10^{36}\) times stronger — that's a 1 followed by 36 zeros. Gravity is absurdly weak compared to electromagnetism. The only reason gravity dominates at large scales is that matter is almost perfectly electrically neutral, so the electric forces cancel out. Gravity has no "negative mass" to cancel with.
Example 3: Why Static Electricity Hurts¶
When you shuffle across a carpet and touch a doorknob, you transfer about \(10^{-7}\) C (0.1 microcoulombs). At the moment of the spark, the gap is about 1 mm (\(10^{-3}\) m):
That's like being poked with 9 kg of force — concentrated in a tiny spark. No wonder it stings.
Why Inverse-Square and Not Something Else?¶
Because we live in three-dimensional space.
The \(1/r^2\) law is not a coincidence or an empirical oddity. It's a geometric necessity. Any force that radiates uniformly from a point source in 3D space must follow an inverse-square law, because the surface area of a sphere grows as \(r^2\). Same field, more area, weaker intensity.
If electricity followed a \(1/r\) law, charge conservation would break down. If it followed \(1/r^3\), the math of electrostatics would be inconsistent (Gauss's law would fail). The inverse-square law is the only power law consistent with conservation of charge in three dimensions.
Common Mistakes¶
- Forgetting the sign: When \(q_1\) and \(q_2\) have the same sign, \(F > 0\) (repulsion). Opposite signs give \(F < 0\) (attraction). Many students compute the magnitude but forget the direction.
- Confusing \(k_e\) with \(G\): Coulomb's constant (\(\sim 10^{9}\)) and gravitational constant (\(\sim 10^{-11}\)) differ by a factor of \(10^{20}\). Electric forces are enormously stronger than gravity.
- Using this for moving charges: Coulomb's Law is only valid for static (or slowly moving) charges. For charges in motion, you need the full machinery of Maxwell's equations, which adds magnetic forces.
- Assuming it works inside conductors: Inside a conductor, free charges rearrange themselves until the electric field is zero. Coulomb's Law still applies to individual charges, but you can't ignore the redistribution.
Related Formulas¶
- Newton's Laws — the force framework underlying all of classical physics
- Measuring the Constants — how Coulomb's torsion balance and similar devices measured fundamental constants
- Gravitational Potential Energy — another inverse-square force law, but for gravity
History¶
- 1600 — William Gilbert publishes De Magnete, distinguishing electricity from magnetism for the first time
- 1733 — Charles François de Cisternay du Fay discovers two types of electricity ("vitreous" and "resinous" — what we now call positive and negative)
- 1752 — Benjamin Franklin's kite experiment proves lightning is electrical
- 1767 — Joseph Priestley suspects an inverse-square law for electricity, based on an observation by Franklin that charge resides on the outside of a conductor (same reasoning Newton used for gravity inside a shell)
- 1785 — Coulomb publishes his torsion balance measurements, establishing \(F \propto q_1 q_2 / r^2\)
- 1865 — Maxwell publishes his equations, showing that Coulomb's Law is just the static case of a much deeper electromagnetic theory
References¶
- Coulomb, C. A. (1785). "Premier mémoire sur l'électricité et le magnétisme." Mémoires de l'Académie Royale des Sciences
- Priestley, J. (1767). The History and Present State of Electricity
- Griffiths, D. J. Introduction to Electrodynamics, 4th ed., Ch. 2.
- Feynman, R. P. The Feynman Lectures on Physics, Vol. 2, Ch. 4-5.