What Is Coulomb's Law? (And What Does k Actually Mean?)

Why Do Charges Push and Pull at All?

Start before the formula. You already know this force, even if you've never written an equation for it.

Rub a balloon on your hair and stick it to a wall — that's this force. Pull a sweater over your head in winter and hear it crackle — same force. Reach for a doorknob and get a shock — same force, just enough of it to jump a gap through the air. Every one of those is charges pushing and pulling on each other.

Two kinds of charge, one simple rule

Matter contains two kinds of electric charge, which we label positive (protons) and negative (electrons). The labels are arbitrary — Benjamin Franklin picked them, and he arguably picked backwards — but the rule they obey is not:

• Opposite charges attract. A positive and a negative pull toward each other.
• Like charges repel. Two positives push apart. So do two negatives.

Most everyday objects are electrically neutral — they hold almost exactly as many protons as electrons, so the pushes and pulls cancel out and you feel nothing. Rubbing a balloon on your hair scrapes some electrons off your hair and onto the balloon. Now the balloon has a slight surplus of negative charge and your hair has a slight deficit, and the balance is broken. Your hair follows the balloon around the room.

What's actually doing the pushing?

Fair question, since the balloon never touches the wall. Nothing physical reaches across the gap.

The modern picture is that a charge alters the space around it — it creates an electric field, an invisible condition filling the surrounding volume. Any other charge that finds itself in that field feels a force. The field is strongest close to the charge and weakens with distance, which is why the balloon has to be close to the wall to stick.

Coulomb's law is the arithmetic of that pushing: given two charges and the gap between them, it tells you precisely how many newtons of force to expect.

Coulomb's Law, Term by Term

In 1785 the French physicist Charles-Augustin de Coulomb measured this force with a delicate instrument called a torsion balance — a needle suspended on a thread so fine that a tiny electric push would visibly twist it. From those measurements he extracted a relationship that has held up ever since:

Read the formula as a sentence and it's almost obvious: the force grows when the charges grow, and shrinks fast when they move apart. The "fast" is the interesting part — distance is squared, so it punishes separation much harder than charge rewards size.

What about the direction?

The formula gives you the strength of the force. For direction, you don't need any maths — just the sign rule from earlier. Opposite signs mean attraction, matching signs mean repulsion.

In introductory problems the cleanest approach is to plug in the sizes of the charges, ignoring their signs, and then state the direction separately in words. If you do plug the signs in, a negative answer simply means "attractive."

What Is k in Coulomb's Law?

This is the question that sends students to search engines at 11pm, and it deserves a real answer rather than "it's a constant."

Why does k need to exist?

Because without it, the two sides of the equation don't agree about what they're describing.

Look at what you feed in: charges measured in coulombs, distance measured in metres. Look at what you want out: a force measured in newtons. Nothing about coulombs and metres automatically produces newtons — those units were defined by different people for different purposes, with no coordination between them. Something has to bridge the gap.

That's k. It's the exchange rate between "coulombs squared per metre squared" and "newtons." Look at its units — N·m²/C² — and you can see it doing exactly that job: it cancels the C² and the 1/m² you fed in, and leaves N behind.

It also encodes something physical, not just bookkeeping: how strongly empty space transmits electric force. Space has a measurable property here, and k is that property in a convenient package.

Where does the number come from?

k isn't arbitrary. It's built out of a more fundamental quantity called the permittivity of free space, written ε₀ ("epsilon nought"), which measures how readily a vacuum permits an electric field to form:

And that stray 4π has a genuinely nice reason for being there: 4πr² is the surface area of a sphere. A charge radiates its influence outward in all directions equally, spreading over an expanding spherical surface. The 4π is the geometry of that sphere, sitting in plain sight in the constant. Writing k as 1/4πε₀ keeps the geometry visible; writing it as 8.99×10⁹ hides it inside a number.

Why is k such an enormous number?

Because the coulomb is an absurdly large unit of charge. k looks dramatic mainly as a comment on our choice of units.

Put 1 coulomb of charge one metre away from another 1 coulomb, and the formula gives 8.99×10⁹ newtons — roughly the weight of 900,000 tonnes. That is not a force anyone has ever produced with two objects on a bench; the charges would tear themselves apart long before you assembled them. You cannot hold a coulomb of net charge in your hand.

For perspective: one coulomb is about 6.24×10¹⁸ elementary charges. Real problems use microcoulombs (μC, millionths of a coulomb), and a good static-electricity shock involves less than that. The enormous k and the tiny charges cancel out into forces of a few newtons — which is why the numbers in the next section come out reasonable.

A Worked Example, Step by Step

Two small charged spheres sit on a bench. One carries +3 μC, the other −5 μC. They're 20 cm apart. What force do they feel, and which way?

Step 1 — Convert everything to SI units

This is where most lost marks happen, so do it before anything else. The formula only works in coulombs and metres. If the prefixes trip you up, our unit converter handles the micro and nano steps in one go.

Step 2 — Put the numbers in

Step 3 — Work through it

1. Multiply the charges: 3×10⁻⁶ × 5×10⁻⁶ = 15×10⁻¹² = 1.5×10⁻¹¹ C²
2. Square the distance: (0.20)² = 0.04 m²  — note this is smaller than 0.20, a classic place to slip up
3. Multiply by k: 8.99×10⁹ × 1.5×10⁻¹¹ = 0.1348
4. Divide by r²: 0.1348 ÷ 0.04 = 3.37

Is that a sensible answer? 3.37 N is roughly the weight of a full soda can — a real, easily felt force, from two charges far too small to see. That's a good sanity check to run on every answer: if you get 10⁻²⁰ N or 10¹² N from bench-scale numbers, you've dropped a power of ten somewhere.

Once you've done a few of these by hand, checking your work is faster than repeating it. You can verify any setup — and see the field around the charges drawn out — with our Coulomb's law calculator.

What if you double the distance?

Ask most students and they'll say the force halves. It doesn't — it drops to a quarter. Move the spheres from 20 cm to 40 cm and the force falls from 3.37 N to 0.84 N. Move them to 10 cm instead and it jumps to 13.5 N, four times stronger.

That's the inverse-square law, and it has a beautifully simple geometric reason behind it.

That's the whole story behind the exponent. Nothing is being absorbed or lost — the same influence is simply spread thinner across a bigger surface, and surfaces grow with the square of distance. Which brings us to a force you already know that behaves the exact same way.

Coulomb's Law vs. Newton's Gravity

Put the two laws side by side and the resemblance is almost suspicious:

Same structure, for the same reason: both forces radiate outward from a point into three-dimensional space, so both get diluted across the surface of a sphere. Swap charge for mass and k for G and you have swapped one law for the other — you can watch the gravitational version play out in our gravity simulator.

That last row is worth pausing on. Take the electron and proton in a hydrogen atom. The electric attraction holding them together is about 8.2×10⁻⁸ N. The gravitational attraction between the same two particles is about 3.6×10⁻⁴⁷ N. The electric force is roughly 2×10³⁹ times stronger — a 2 followed by 39 zeros. Gravity is, by an almost unimaginable margin, the weakest force in physics.

So why does gravity run the universe?

Here's the resolution, and it's the most satisfying idea in this article.

Charge comes in two signs and cancels. Mass comes in one sign and accumulates.

The Earth contains something like 10⁵¹ electrons and just about exactly as many protons. All that colossal electric force is almost perfectly balanced — every proton's pull answered by an electron's push — leaving a net electric force of essentially zero. Meanwhile every single kilogram of the Earth contributes its gravity in the same direction, with nothing to oppose it, and it all adds up.

So the strongest force cancels itself into irrelevance at large scales, and the feeblest one accumulates until it holds galaxies together. Electricity dominates the small world — atoms, molecules, chemistry, your entire body's structure. Gravity dominates the big one. Not because it's strong, but because nothing ever cancels it.

Quick Checklist Before You Submit an Answer

• Charges converted to coulombs? (μC = 10⁻⁶, nC = 10⁻⁹)
• Distance converted to metres, and then squared?
• Did you state the direction — attractive or repulsive — not just the number?
• Is the answer plausible? Bench-scale charges should give you something in the range of fractions of a newton up to a few newtons.
• Are the units on your answer newtons? If not, something went wrong upstream.

The Takeaway

Coulomb's law is one short formula doing a lot of work: multiply the two charges, divide by the distance squared, scale by k. The squared distance comes from the geometry of a sphere. The k comes from the fact that coulombs, metres and newtons were invented independently and need a translator — and it's a big number only because the coulomb is a big unit.

Get comfortable with this one and a lot of physics opens up behind it. Electric fields, voltage, capacitance and the structure of the atom itself are all built on top of what you have just read — and you can see that structure directly in our atomic structure simulator.

Check your work

Enter two charges and their separation to get the force, with the electric field drawn around them. Free, no sign-up, runs in your browser.