Pull a sweater over your head on a dry winter day and your hair lifts toward the fabric. Rub a balloon on your sleeve and it clings to a wall. Both are the same effect: separated electric charge exerting a force across empty space. The pull feels faint, almost playful — yet the law that governs it is the same one that holds every atom together and sets the scale of all of chemistry. Coulomb's law is where the study of electricity begins.
This article explains what Coulomb's law states, how the inverse-square distance dependence shapes the force, and how to compute the force between two charges from first principles.
Why this calculation matters
Coulomb's law is the foundation that everything electrical is built on. Voltage, capacitance, the binding of electrons to nuclei, the behavior of insulators and dielectrics — all of it traces back to the force between charges. Before you can reason about fields, potentials, or circuits, you need the force law underneath them.
It matters in practical engineering, too. The electrostatic force sets the breakdown limit of an insulator, drives the attraction in electrostatic precipitators and laser printers, and explains why high-voltage equipment needs generous clearances. It governs the behavior of charged particles in detectors and accelerators. Whenever charge sits still or moves slowly, Coulomb's law is the tool that tells you how hard it pushes. It is also the cleanest possible example of an inverse-square law, which makes it the natural place to build intuition that later transfers to gravity and to radiated fields.
The core formula
Coulomb's law gives the magnitude of the electrostatic force between two point charges:
F = k * q1 * q2 / r^2
Here q1 and q2 are the two charges in coulombs, r is the distance between them in meters, and k is Coulomb's constant:
k = 8.99e9 N.m^2/C^2
Two features carry most of the physics. First, the force is proportional to the product of the charges — double either charge and the force doubles. Second, and more striking, the force falls off with the square of the distance. Move the charges twice as far apart and the force drops to a quarter; move them ten times as far and it drops to a hundredth. This inverse-square behavior is why electric forces are intense at short range and fade quickly at a distance.
Direction follows a simple rule. The force acts along the line joining the two charges. Like charges — two positives or two negatives — repel each other. Opposite charges attract. In a medium other than vacuum, the force is reduced by the relative permittivity of that medium, so the same charges separated by the same distance push less hard in water or in a dielectric than they do in air.
The same law generalizes neatly. Treat one charge as the source of an electric field, and the force on the other charge is just that field multiplied by the charge it acts on — the bridge from Coulomb's law to the broader language of fields.
A worked example
Take two equal point charges, each q = 1 microcoulomb, held a distance r = 0.1 m apart in vacuum. Find the force between them.
Step 1 — convert to base units. Each charge is 1 microcoulomb = 1e-6 C. The separation is r = 0.1 m.
Step 2 — substitute into Coulomb's law.
F = k * q1 * q2 / r^2
F = 8.99e9 * (1e-6 * 1e-6) / (0.1)^2
Step 3 — evaluate term by term.
q1 * q2 = 1e-6 * 1e-6 = 1e-12 C^2
r^2 = (0.1)^2 = 0.01 m^2
F = 8.99e9 * 1e-12 / 0.01
F = 8.99e-3 / 0.01 = 0.899 N
Result. The two charges push apart with a force of about 0.899 N — roughly the weight of a 90-gram object. Because both charges are positive, the force is repulsive; had one been negative, the same magnitude would act as an attraction.
It is worth feeling the sensitivity to distance here. Halve the separation to 0.05 m and the force quadruples to about 3.6 N. Double it to 0.2 m and the force collapses to about 0.225 N. A microcoulomb, incidentally, is a substantial amount of static charge — which is why these everyday-scale charges still produce a force you could measure on a kitchen scale.
Common mistakes
Leaving charges in microcoulombs. The constant k expects charges in coulombs. Plugging in "1" for a microcoulomb instead of 1e-6 inflates the force by twelve orders of magnitude. Convert every charge to coulombs before substituting.
Forgetting to square the distance. The r in the denominator is squared. Using r instead of r^2 turns an inverse-square law into an inverse-first-power law and badly overestimates the force at any real separation.
Applying the point-charge form to large bodies. Coulomb's law in this form is exact for point charges, and a good approximation only when the bodies are small compared with their separation. For two charged spheres almost touching, the charge distributions shift and the simple formula no longer holds.
Ignoring the medium. The plain F = k*q1*q2/r^2 is the vacuum (or close-enough air) result. Immersed in water or a dielectric, the force is reduced by the relative permittivity — a large factor for water — and skipping it overstates the force.
Treating the force as one-sided. By Newton's third law, each charge feels a force of equal magnitude and opposite direction. There is no "stronger" charge in the pair; a tiny charge near a huge one feels exactly the force the huge one feels back.
Try the interactive NovaSolver calculator
Running the arithmetic once fixes the method; seeing the force respond as you drag the charges and the separation is what builds intuition. The Coulomb Law Simulator — Electric Force Between Point Charges on NovaSolver lets you set the two charges, the separation, and the relative permittivity of the medium, then returns the Coulomb force, the electric field, the potential energy, and whether the interaction is attractive or repulsive — with field lines and a log-log plot of force against distance.
Related calculators
- Electromagnetic Induction (Faraday's Law) Simulator — the next step beyond static charge, where a changing magnetic flux drives an EMF.
- Coulomb Force Visualizer — a focused visual companion for seeing how the force vector grows and shrinks with separation.
- Electromagnetic Induction & Faraday's Law Simulator — extends the same charge-and-field thinking into induced EMF, mutual inductance, and transformers.
You can browse the full set in the electromagnetics tools hub.
Closing note
Coulomb's law is a short equation that carries an enormous amount of physics. The force between two charges scales with the product of the charges and falls off with the square of the distance — like repels like, opposites attract, and the medium in between softens the effect. Convert your charges to coulombs, square the separation, and respect the inverse-square sensitivity, and you have the starting point for fields, potentials, and ultimately every electrical system built on top of them.
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