Coulomb's Law Calculator

What the Coulomb's law calculator does

Two electric charges placed near each other push or pull on one another. This calculator works out the strength of that electrostatic force from the two charges and the distance between them, using Coulomb's law. You can run it simply, in vacuum, or in advanced mode set the Coulomb constant, the permittivity, and the medium yourself.

Below is what Coulomb's law is, the equation behind it, how attraction and repulsion arise, and a worked example.

How to use it

1. Enter the two charges and the distance between them, each with its unit. Charges can be given in coulombs and its fractions, in elementary charges, or in ampere-hours.
2. Choose simple or advanced mode. Advanced mode exposes the Coulomb constant, the vacuum permittivity, and the relative permittivity of the medium.
3. Press Calculate for the electrostatic force, or Reset to clear it.

What Coulomb's law is

Coulomb's law is the rule for the force between electric charges, the electrical counterpart of Newton's law for gravity. It says that two charges exert a force on each other that grows with the size of each charge and weakens with the square of the distance between them. Charles-Augustin de Coulomb established it in the 1780s, and it is the foundation of electrostatics.

The force it describes is one of the most important in nature. It holds electrons to atomic nuclei, binds atoms into molecules, and underlies all of chemistry and electricity. Where gravity depends on mass, this force depends on charge, and unlike gravity it comes in two flavours, because charge can be positive or negative. That single difference gives the electrostatic force a richness gravity does not have.

The equation it uses

Coulomb's law gives the force F between two charges q₁ and q₂ separated by a distance r:

F = kₑ × q₁ × q₂ ÷ r²

The constant kₑ is Coulomb's constant, which sets the strength of the force, with a value of about 8.99 × 10⁹ in SI units. It is tied to a more fundamental quantity, the permittivity of free space ε₀, through the relation kₑ = 1 ÷ (4πε₀), and the calculator keeps the two linked, updating one when you change the other in advanced mode. The force is proportional to each charge and falls off with the square of the distance, exactly mirroring the form of the gravitational law.

Attraction, repulsion, and magnitude

What makes the electrostatic force distinctive is that it can both pull and push. Two charges of the same sign, both positive or both negative, repel each other, while two charges of opposite sign attract. This is the familiar rule that like charges repel and opposites attract, and it has no parallel in gravity, which only ever attracts.

The calculator works with the sizes of the charges and returns the magnitude of the force, how strong the push or pull is. Which of the two it is, attraction or repulsion, you read from the signs of the charges themselves: matching signs mean the force pushes the charges apart, opposite signs mean it draws them together. The number tells you how hard; the signs tell you which way.

The effect of the medium

The force between charges also depends on what surrounds them. In empty space, or near enough in air, the force is at its full strength. But place the same charges in another material, such as water, oil, or glass, and the force weakens, because the material responds to the charges and partly screens them.

This screening is captured by the relative permittivity of the medium, sometimes called its dielectric constant. The calculator divides the force by this factor, so a higher relative permittivity means a weaker force. Water, for instance, has a high value and dramatically reduces the force between charges immersed in it, which is part of why so many salts dissolve readily in water: it loosens the electrical grip holding their ions together. In vacuum the relative permittivity is one, and the force takes its full value.

How it compares to gravity

Coulomb's law and Newton's law of gravitation share the same shape, both proportional to the product of two source quantities and both falling off as the inverse square of distance. But the forces they describe differ enormously in strength. The electrostatic force is staggeringly stronger than gravity for the same objects.

Compare the two forces between a pair of protons, and the electrical repulsion outweighs the gravitational attraction by a factor of around 10³⁶, a one followed by thirty-six zeros. Gravity wins out over cosmic distances only because matter is, on the whole, electrically neutral, with positive and negative charges balancing so their forces cancel, leaving gravity's feeble but uncancelled pull to dominate at large scales. Up close, where charges are not balanced, electricity is overwhelmingly the stronger force, and this calculator works in that regime.

Units and precision

The calculator works in SI units underneath, taking charges in coulombs and the distance in metres and returning the force in newtons, while the menus let you enter charges in fractions of a coulomb, in elementary charges, or in ampere-hours, and distances from nanometres upward. The Coulomb constant and permittivity carry their standard measured values by default. The relationship is exact; in practice the accuracy depends on how well the charges and distance are known. Results are shown to many significant figures.

A worked example

Take two charges of 1 microcoulomb each, placed 1 metre apart in vacuum.

The force is F = kₑq₁q₂/r² = (8.99 × 10⁹ × 10⁻⁶ × 10⁻⁶) ÷ 1² ≈ 9 × 10⁻³ newtons, about 9 millinewtons. Since both charges have the same sign, this force pushes them apart. That may sound small, but a microcoulomb is itself a modest charge, and the gravitational attraction between two one-gram masses at the same separation would be smaller by a factor of trillions, showing how much stronger electricity is than gravity.

Questions people ask

How do you calculate the force between two charges?

Multiply Coulomb's constant by both charges and divide by the square of the distance, F = kₑq₁q₂/r². The constant is about 8.99 × 10⁹ in SI units.

When do charges attract and when do they repel?

Like charges repel and opposite charges attract. Two positives or two negatives push apart; a positive and a negative pull together. The calculator gives the magnitude, and the signs tell you the direction.

Does the surrounding material change the force?

Yes. A medium weakens the force according to its relative permittivity, or dielectric constant. Water, with a high value, greatly reduces the force; in vacuum the relative permittivity is one and the force is at full strength.

Why is the electrostatic force so much stronger than gravity?

For the same objects it is vastly stronger, by around 10³⁶ for two protons. Gravity dominates at large scales only because matter is mostly electrically neutral, so its charges cancel while gravity's pull never does.

References

A quick note on where the physics comes from. Coulomb's law, its inverse-square form, and the role of permittivity are standard electromagnetism, set out in OpenStax's University Physics and in Georgia State University's HyperPhysics. The values of Coulomb's constant and the permittivity of free space follow the US National Institute of Standards and Technology. The HyperPhysics link is worth a quick click to confirm it lands where you expect.

1. OpenStax, University Physics Volume 2, Section 5.3, Coulomb's Law. https://openstax.org/books/university-physics-volume-2/pages/5-3-coulombs-law
2. HyperPhysics, Coulomb's Law. http://hyperphysics.phy-astr.gsu.edu/hbase/electric/elefor.html
3. National Institute of Standards and Technology (NIST), Fundamental Physical Constants (Coulomb constant and vacuum electric permittivity). https://physics.nist.gov/cuu/Constants/