Coulomb's Law Calculator — force between two charges

Coulomb's law gives the force between two stationary point charges: proportional to the product of the charges, inversely proportional to the square of the distance, and directed along the line joining them. This Coulomb's law calculator applies it with the charge units you actually have, accounts for the medium the charges sit in, and reports not just the force but the electric field, the stored potential energy and whether the interaction attracts or repels.

Arb Digital builds free calculators that treat units as part of the problem rather than an afterthought. Charge spans an enormous range — from the elementary charge of 1.602 × 10⁻¹⁹ C up to coulombs in a lightning stroke — so the tool accepts coulombs, millicoulombs, microcoulombs, nanocoulombs and multiples of the elementary charge directly. This page derives a force from the physics; if all you need is to rescale a charge value between units, the electric charge converter does that job and nothing else.

What This Coulomb's Law Calculator Does

Enter both charges with their signs, the separation between them and the relative permittivity of whatever lies between. The result hero gives the magnitude of the force in newtons. The grid states the direction as attraction or repulsion, gives the electric field at the position of the second charge, the electrostatic potential energy of the pair, and — because newtons at these scales are hard to picture — the mass whose weight would produce the same force.

Potential energy is worth attention because it behaves differently from the force. It falls as 1/r rather than 1/r², and it carries a sign: negative for opposite charges, meaning the pair is bound and work must be done to separate them, positive for like charges, meaning energy was invested to bring them together and would be released if they were let go.

The tool guards a separation of zero, which would give infinite force. That is not a numerical inconvenience but a genuine limit of the point-charge idealisation: real charges occupy volume, and once separation approaches their size, the law no longer describes the situation.

How to Use It

1. Include the signs. Coulomb's law is a vector statement, and the sign of the product determines whether the force pulls the charges together or pushes them apart.
2. Match the unit selector to your data. A charge quoted in nanocoulombs entered as microcoulombs is a factor of a thousand out, which becomes a factor of a million in the force.
3. Measure separation centre to centre. For uniformly charged spheres this is exact outside the surfaces; for irregular conductors it is an approximation that degrades as they get close.
4. Set the medium. Vacuum and air are effectively identical, but oil, glass and especially water reduce the force substantially.
5. Sanity-check with the equivalent mass. If the force corresponds to hanging several kilograms off a microscopic charge, a unit is wrong somewhere.

The Formula: How the Force Is Calculated

The law is F = k |q₁q₂| / (ε_rr²), where k = 1/(4πε₀) is Coulomb's constant, about 8.9876 × 10⁹ N·m²/C². That constant is derived from the vacuum electric permittivity, whose 2022 CODATA value is 8.8541878188 × 10⁻¹² F/m as published by NIST. The relative permittivity of the medium divides the result, because a dielectric polarises in response to the field and partly screens the charges from each other.

Working the defaults: 1 µC and −2 µC separated by 0.05 m in vacuum. The product of magnitudes is 2 × 10⁻¹² C², the separation squared is 2.5 × 10⁻³ m², so F = 8.9876 × 10⁹ × 2 × 10⁻¹² ÷ 2.5 × 10⁻³ = 7.19 N. The signs are opposite, so it is an attraction. Seven newtons is the weight of about 733 grams — a substantial pull between two charges most people would describe as small, which is the first thing this law teaches.

The electric field at the second charge is the force per unit charge there: E = kq₁/(ε_rr²) = 3.60 × 10⁶ V/m. The potential energy is U = kq₁q₂/(ε_rr), keeping the signs, which gives −0.360 J. Note the difference between these two: the field is a property of the space regardless of what sits there, and our electric field calculator deals with it directly. The OpenStax University Physics section on Coulomb's law sets out the vector form and the superposition principle for more than two charges.

Just How Strong the Electrostatic Force Is

The default case gives 7 N between two charges of a millionth of a coulomb. Scale that up and the numbers become absurd. Two charges of one coulomb each, one metre apart in vacuum, would push each other apart with about 9 × 10⁹ N — roughly the weight of a billion kilograms. That is why a coulomb of separated static charge does not exist in any ordinary setting: the force required to assemble it exceeds anything a laboratory can hold together.

The comparison with gravity makes the point sharper. Two protons attract each other gravitationally and repel each other electrostatically, and the electrostatic repulsion is stronger by a factor of about 10³⁶. Gravity dominates the universe at large scales only because matter is electrically neutral to an extraordinary precision — positive and negative charges cancel so nearly that the vastly weaker gravitational attraction is what remains. Our gravitational force calculator lets you compute the other side of that comparison.

Neutrality is also why static electricity is a matter of nanocoulombs. Rubbing a balloon on hair transfers perhaps a few hundred nanocoulombs, enough to lift the balloon's own weight against gravity, and that transfer involves only about a trillionth of the electrons present. Static effects are the visible residue of a nearly perfect cancellation.

What the Medium Does to the Force

Relative permittivity — the dielectric constant — is not a small correction. Water's value near 80 means two ions in water attract each other with one-eightieth of the force they would exert in vacuum at the same separation. That single fact explains why water dissolves salts: the electrostatic bonds holding an ionic crystal together are weakened enough that thermal motion can break them apart, while the same crystal is entirely stable in air.

Air is close enough to vacuum for most purposes, differing by less than a tenth of a percent. Insulating oils sit around 2 to 3, and their role in high-voltage equipment is partly this screening effect and partly a much higher breakdown strength than air. Solid dielectrics in capacitors work on the same principle from the opposite direction: a higher permittivity between the plates means more charge stored for the same voltage.

The model does have limits. Treating a medium as a single permittivity number assumes it is uniform, isotropic and linear, and that it responds instantly. Real dielectrics are frequency-dependent — water's effective permittivity falls dramatically at microwave frequencies, which is exactly what a microwave oven exploits — so a static value is only appropriate for a static problem.

Where the Point-Charge Assumption Breaks

Coulomb's law describes point charges. It applies exactly to uniformly charged spheres as long as you are outside them, which is a result worth knowing because it means two charged balls can be treated as points at their centres. It stops applying once they are close enough that each one's field redistributes the charge on the other, a process called induction. Two conducting spheres brought near each other will polarise, concentrating opposite charge on their facing surfaces, and the true force becomes larger than the simple formula predicts.

The law also assumes the charges are stationary. Moving charges produce magnetic fields as well, and at high speeds the electrostatic picture is incomplete. For anything static or slow-moving, which covers most laboratory and everyday situations, the simple form is entirely adequate.

Finally, superposition applies. For more than two charges, calculate each pairwise force with this tool and add them as vectors. There is no shortcut: the forces do not simply add as numbers unless the charges happen to lie on a line. When the numbers get awkward to read, the scientific notation converter makes the exponents easier to handle, and the force converter takes the result into other force units.

Common Mistakes to Avoid

• Dropping the signs — the sign of the product is what tells you whether the charges attract or repel, and the sign of the potential energy tells you whether the pair is bound.
• Mixing charge units — nanocoulombs entered as microcoulombs is a factor of a thousand per charge, and a factor of a million in the force.
• Forgetting to square the separation — force falls as the inverse square, so halving the distance quadruples it. This is the most common arithmetic slip in the whole calculation.
• Leaving permittivity at 1 for a charge in water — water's value near 80 reduces the force by almost two orders of magnitude, which is the difference between a salt dissolving and not.
• Applying it to very close conductors — induction redistributes charge on each body when they are near, and the simple point-charge result understates the true force.

Related Free Tools From Arb Digital

Electrostatics runs from charge to field to circuits. Use the electric field calculator for the field a single charge produces, the electric charge converter for unit work on charge alone, and the electrical power calculator once you move from static charge to current. For comparison with the other inverse-square force, try the gravitational force calculator, and for handling very large or very small numbers the scientific notation converter. Everything else is in the free online tools hub.

Frequently Asked Questions

This tool is provided for educational and study use. It applies the idealised electrostatic law and is not high-voltage, electrical safety or equipment design guidance.