Coulomb's Law Explained: Force Between Two Point Charges

Rub a balloon on your hair and it sticks to the wall. Bring two charged combs close and they shove each other apart. Both moments are the same piece of physics: Coulomb's law, the rule that tells you how hard two electric charges push or pull on each other. It is one of the first equations you meet in electrostatics, and once it clicks, a surprising amount of the electrical world starts to make sense.

This post walks through what the formula means, why distance matters so much, and how the sign of the answer tells you the direction of the force. At the end I work a full example by hand and check it against the Coulomb's law calculator so you can see every step line up.

The formula and what each piece does

Coulomb's law is written:

F = k · q1 · q2 / r²

Four symbols, each with a clear job:

F is the electrostatic force in newtons (N).
q1 and q2 are the two charges in coulombs (C).
r is the distance between them in metres (m).
k is the Coulomb constant, k = 1/(4πε₀) ≈ 8.988 × 10⁹ N·m²/C². Textbooks often round it to 8.99 × 10⁹ or even 9.0 × 10⁹.

Read it as a sentence: the force is bigger when the charges are bigger, and smaller when they sit farther apart. Multiply the two charges together, multiply by that hefty constant, then divide by the square of the gap. The constant k is enormous, which is your first clue that electric forces are ferociously strong compared with gravity. The reason everyday objects do not fling themselves around is that positive and negative charges almost perfectly cancel in ordinary matter.

A quick note on what a coulomb actually is: it is a huge amount of charge. The elementary charge carried by a single proton is only e = 1.602 × 10⁻¹⁹ C. That is why real problems live in microcoulombs (µC, 10⁻⁶ C) and nanocoulombs (nC, 10⁻⁹ C), and why answers usually arrive in scientific notation.

The inverse-square law: distance is the bully

The single most important feature of Coulomb's law is the r² in the denominator. The force does not just fade with distance, it falls off with the square of the distance. This is the inverse-square law, and it is the same shape that governs gravity, sound intensity, and brightness from a light source.

What does squaring do in practice? Double the distance and the force does not halve, it drops to one quarter. Triple the distance and it drops to one ninth. Move ten times farther away and the force becomes one hundredth of what it was.

Here is a concrete chain. Suppose two charges feel 0.9 N of force when they sit 10 cm apart:

At 20 cm (2× the distance): force ≈ 0.9 / 4 ≈ 0.225 N.
At 30 cm (3× the distance): force ≈ 0.9 / 9 ≈ 0.1 N.
At 100 cm (10× the distance): force ≈ 0.9 / 100 ≈ 0.009 N.

This is exactly why the most common mistake in homework is plugging in r where the formula wants r². Forget the square and your close-range forces come out four-plus times too small, and nothing downstream will check out.

Sign of the force: like repels, unlike attracts

Coulomb's law also carries a sign, and the sign is the direction. When you keep the actual signs of q1 and q2 in the calculation, the result tells you which way the charges move:

Positive F = repulsion. Both charges share a sign (both positive, or both negative). They push apart.
Negative F = attraction. The charges have opposite signs. They pull together.

The slogan is short and worth memorizing: like charges repel, unlike charges attract. Two protons shove each other away. A proton and an electron pull together. A balloon that has stolen electrons from your hair (now negative) is drawn to the wall, where it pushes the wall's surface electrons aside and leaves a net positive patch to cling to.

One trap to avoid: a force of −21.6 N is not weaker than +21.6 N. The minus sign is direction, not magnitude. Compare strengths by absolute value, and read the sign only to sort attraction from repulsion.

A worked example, step by step

Let me do one the way I actually do it, then check the machine. Take two charges:

q1 = +3 µC = +3 × 10⁻⁶ C
q2 = −2 µC = −2 × 10⁻⁶ C
separation r = 5 cm = 0.05 m

Plug into F = k · q1 · q2 / r²:

Numerator: k · q1 · q2 = 8.988 × 10⁹ × (3 × 10⁻⁶) × (−2 × 10⁻⁶) = 8.988 × 10⁹ × (−6 × 10⁻¹²) = −0.0539 (units folding in along the way).
Denominator: r² = (0.05)² = 0.0025 m².
Divide: F = −0.0539 / 0.0025 ≈ −21.6 N.

The answer is negative, so the force is attractive, which is exactly what the opposite signs promised. When I first learned this I always double-checked the sign against the like-repels-unlike-attracts rule before trusting the arithmetic, and it caught more of my errors than any other habit.

To verify, I drop the same numbers into the Coulomb's law calculator: charges set to µC, distance set to 5 cm, and it returns F = −21.6 N with the worked steps printed underneath. Because it keeps the sign, it labels the result attractive, so there is nothing left to guess. If I had typed 5 with the unit still reading C instead of µC, I would have computed the force between five whole coulombs and inflated the answer by a factor of a trillion, the classic unit slip.

Working backwards and comparing forces

The same formula rearranges in two useful directions. If you know the force and both charges, solve for distance: r = √(k · |q1 · q2| / |F|). If you know the force, one charge, and the distance, solve for the other charge: q1 = F · r² / (k · q2). This is how you size a charge for a lab demo, or build a quiz where every number lands on a tidy answer.

Coulomb's law is also a twin of Newton's law of gravity, F = G · m1 · m2 / r². Same inverse-square shape, charges and k swapped for masses and G. The difference is scale. For a proton and an electron 1 nm apart, the electric attraction is about −2.3 × 10⁻¹⁰ N, while the gravitational pull between them is roughly 10⁻⁶⁷ N, some 39 orders of magnitude weaker. If you want to make that contrast concrete, run the scientific calculator on both expressions and watch the exponents drift apart. Electricity wins, and it is not even close.

That gap is the whole reason atoms hold together with electric force and planets orbit with gravity. Same equation, wildly different strengths, and Coulomb's law is the half you can feel in a balloon stuck to a wall.

Made by Toolora · Updated 2026-06-13