Coulomb’s Law Explained

What happens when two tiny charges of 1 and 2 microcoulombs sit just 10 centimeters apart? They shove each other with a force of about 1.8 newtons. One short formula predicts that push, and it is called Coulomb’s law. This guide explains each symbol, the sign rule, and why distance matters so much.

What Does the Formula F = k q1 q2 / r^2 Tell You?

It tells you the size of the electric force between two point charges. Multiply the two charges together, multiply by the constant k, then divide by the distance squared. The answer comes out in newtons.

Each symbol has one job. F is the force in newtons, and q1 and q2 are the two charges in coulombs. The letter r is the distance between their centers in meters. OpenStax lists k as 8.988 x 10^9 N m^2/C^2, often rounded to 8.99 x 10^9.

That constant is not random. It equals 1 divided by (4 pi times the vacuum permittivity). NIST lists that permittivity as 8.8541878188 x 10^-12 farads per meter. Plug it in and k comes out at about 8.988 x 10^9, matching the textbook value.

The law is named after the French physicist Charles Coulomb, who lived from 1736 to 1806. He ran careful experiments and first proposed the formula. Modern tests confirm the inverse square to an accuracy of 1 part in 10^16.

When Do Two Charges Push Apart, and When Do They Pull Together?

Charges with the same sign push apart, and charges with opposite signs pull together. Two positives repel, two negatives repel, and a positive and a negative attract. This rule sets the direction of every electric force.

The formula handles direction through signs. Enter each charge with its sign, and the product q1 q2 decides the outcome. A positive result means repulsion, while a negative result means attraction.

Take -2 microcoulombs and +3 microcoulombs placed 5 centimeters apart. The force works out to -21.57 newtons. The minus sign simply says the pair attracts; the strength is still 21.57 newtons.

The force also acts along the straight line joining the two charges. Each charge feels the same size of force, pointed in opposite directions. That matches Newton’s third law, which our guide to Newton’s laws and how forces work covers in depth.

Why Does a Little Extra Distance Weaken the Force So Much?

The distance in Coulomb’s law is squared, so the force drops with the square of the gap. Double the distance and the force falls to one quarter. Triple it and the force falls to one ninth.

Look at the 1 and 2 microcoulomb pair again. At 0.1 meters the force is about 1.80 newtons. At 0.2 meters it drops to 0.449 newtons, and at 0.3 meters it is only 0.200 newtons.

The rule works in reverse too. Move the same pair to 0.05 meters, half the first gap, and the force jumps to about 7.19 newtons. Halving the distance multiplies the force by four.

HyperPhysics notes the electric force has infinite range and follows the inverse square law. It never switches off completely, but it fades fast. This steep falloff is why charges must sit close together to feel a strong push or pull.

How Much Charge Is One Coulomb, Really?

One coulomb is a huge amount of charge, equal to about 6.24 x 10^18 elementary charges. NIST fixes the charge of one proton at exactly 1.602176634 x 10^-19 coulombs. Divide 1 by that value to get the count.

HyperPhysics gives a household picture. A 120 watt bulb on a 120 volt circuit draws 1 ampere. That current moves 1 coulomb of charge through the wire every second.

Now picture two 1 coulomb charges held 1 meter apart. Coulomb’s law gives about 8.99 x 10^9 newtons, an enormous force. Nature rarely piles up that much charge in one spot, because matter stays close to neutral.

That is why textbook problems use smaller units. A microcoulomb (uC) is 10^-6 coulombs, and a nanocoulomb (nC) is 10^-9 coulombs. Always convert to coulombs before you use the formula. Stored charge in circuits follows the same units, as our guide to capacitance and stored charge shows.

What Do Real Coulomb’s Law Numbers Look Like?

Real answers usually land between a fraction of a newton and a few dozen newtons for microcoulomb charges. The three cases below cover a small push, a matched pair, and a close strong pair. Each result was checked with k = 8.9875 x 10^9.

Start with the calculator’s own case. Charges of 1 and 2 microcoulombs sit 0.1 meters apart. The math is 8.9875 x 10^9 times 1 x 10^-6 times 2 x 10^-6, divided by 0.01. That gives about 1.80 newtons, and the positive sign means they repel.

Next, two equal 1 microcoulomb charges at 0.1 meters give 0.899 newtons. Finally, 2 and 3 microcoulombs at 0.05 meters give 21.57 newtons. The closer gap and larger charges multiply the force by 24.

You can also run the law backward. Know the force and both charges, and the distance equals the square root of k q1 q2 divided by F. The Coulomb’s law calculator for force, charge or distance solves for any one missing value from the other three.

How Does the Electric Force Stack Up Against Gravity?

Both forces follow an inverse square law, but the electric force is vastly stronger between particles. OpenStax compares an electron and a proton in a hydrogen atom. The electric pull is about 2.27 x 10^39 times the gravitational pull.

In that OpenStax example, the electric force is 8.19 x 10^-8 newtons. The gravitational force between the same two particles is only 3.61 x 10^-47 newtons. Since both forces scale with 1/r^2, that ratio holds at any distance.

Two big differences separate the laws. Gravity depends on mass and only attracts. The electric force depends on charge and can attract or repel.

So why does gravity rule the scale of planets? Large objects hold nearly equal positive and negative charge. Their electric pushes and pulls cancel out, while their gravity keeps adding up with every kilogram.

FAQs About Coulomb’s Law

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This article explains the physics of Coulomb’s law for two point charges in a vacuum. It is general science education for students and curious readers. Reviewed for accuracy by Prof. Dr. Khalil Mudassar, PhD. Last updated September 27, 2026.