Coulomb's Law — The Complete Physics Guide
Coulomb's Law describes the force between any two electrically charged objects. Formulated by French physicist Charles-Augustin de Coulomb in 1785 using a torsion balance of his own design, it states that the force between two point charges is directly proportional to the product of their charges and inversely proportional to the square of the distance between them. In equation form: F = kQ₁Q₂/r². This single relationship underlies everything from the structure of the atom to the operation of every capacitor and every static-electricity spark.
Coulomb's Law is structurally identical to Newton's Law of Universal Gravitation — both are inverse-square laws — but electrostatic force is vastly stronger. The electrostatic force between a proton and an electron in a hydrogen atom is about 10³⁹ times stronger than the gravitational force between them, which is why electromagnetism dominates atomic and molecular physics while gravity dominates on astronomical scales.
What is Coulomb's Law?
Coulomb's Law quantifies the electrostatic force — the push or pull between charged particles due to their electric charge alone, independent of any motion. Two charges with the same sign (both positive or both negative) repel one another; two charges with opposite signs attract. The magnitude of this force depends only on the size of the two charges and the distance separating them.
The law is a cornerstone of electrostatics — the study of charges at rest — and forms the foundation from which the entire concept of the electric field is built. In fact, the electric field at a point is simply defined as the Coulomb force per unit test charge placed at that point: E = F/q.
The Formula Explained
Each symbol carries specific physical meaning. F is the electrostatic force in newtons (N) — the magnitude of the push or pull each charge experiences (by Newton's third law, both charges feel an equal and opposite force). Q₁ and Q₂ are the two point charges, measured in coulombs (C) — note that the elementary charge on a single electron or proton is tiny, just 1.602 × 10⁻¹⁹ C, so everyday charges of microcoulombs (10⁻⁶ C) already represent trillions of electrons. r is the distance between the charge centres, in metres. k is Coulomb's constant, equal to 8.988 × 10⁹ N·m²/C² — sometimes written as 1/(4πε₀), where ε₀ is the permittivity of free space (8.854 × 10⁻¹² C²/N·m²).
The inverse-square relationship means distance dominates the result: doubling the separation reduces the force to a quarter of its original value; tripling the distance reduces it to a ninth. This is why static shocks are sharp and localised — the force falls away extremely quickly as you move apart.
How to Use This Calculator
Choose which quantity you want to find using the mode selector. If you know both charges and the distance between them, use "Q₁, Q₂ & r" to find the force directly. If you know the force you want to achieve along with both charges, use "F, Q₁ & Q₂" to find the required distance. If you know the force and distance and are looking for two identical charges that would produce it, use "F & r". Enter charges as positive or negative numbers (using a minus sign for negative charges) — the calculator automatically determines whether the resulting force is attractive or repulsive and displays the reasoning in the step-by-step solution.
Worked Example 1 — Two Point Charges
Problem: Two charges of +2 μC and +3 μC are placed 0.5 m apart. Find the force between them.
F = kQ₁Q₂/r² = (8.988 × 10⁹)(2 × 10⁻⁶)(3 × 10⁻⁶) / (0.5)²
F = 0.05393 / 0.25 = 0.216 N (repulsive — both charges positive)
Worked Example 2 — The Hydrogen Atom
Problem: Find the electrostatic force between the proton and electron in a hydrogen atom, separated by the Bohr radius of 5.29 × 10⁻¹¹ m.
F = k|Q₁Q₂|/r² = (8.988 × 10⁹)(1.602 × 10⁻¹⁹)² / (5.29 × 10⁻¹¹)²
F = 2.307 × 10⁻⁴⁷ / 2.798 × 10⁻²¹ = 8.24 × 10⁻⁸ N (attractive) — tiny in absolute terms, but this force is what holds the atom together, since it acts on an electron of mass just 9.11 × 10⁻³¹ kg.
Worked Example 3 — Finding the Required Distance
Problem: Two charges of 5 μC and 8 μC must be positioned to produce exactly 1 N of force. How far apart should they be placed?
r = √(kQ₁Q₂/F) = √[(8.988 × 10⁹)(5 × 10⁻⁶)(8 × 10⁻⁶) / 1]
r = √(0.3595) = 0.600 m
Worked Example 4 — Two Identical Charged Spheres
Problem: Two identical charged spheres 0.3 m apart repel each other with a force of 0.6 N. Find the charge on each sphere.
Q = √(Fr²/k) = √[(0.6)(0.3)² / (8.988 × 10⁹)]
Q = √(6.006 × 10⁻¹²) = 2.45 × 10⁻⁶ C = 2.45 μC (on each sphere)
Common Mistakes
Forgetting to square the distance: the most frequent error — Coulomb's Law is an inverse-square law, not an inverse law. Halving r quadruples F, not doubles it.
Mixing up units: charge must be in coulombs, not microcoulombs or nanocoulombs, before substituting into the formula. A charge of "5 μC" must first be converted to 5 × 10⁻⁶ C.
Ignoring the sign for direction (but not magnitude): the magnitude of the force always uses the absolute value of Q₁Q₂ — the sign of the product only tells you whether the interaction is attractive or repulsive, it does not make the force itself "negative."
Confusing Coulomb's Law with the electric field formula: F = kQ₁Q₂/r² gives the force between two charges. E = kQ/r² gives the field from a single charge. The force on a test charge in that field is then F = qE — a two-step calculation students often try to shortcut incorrectly.
Real-World Applications
Atomic and molecular structure: Coulomb's Law governs how electrons are bound to nuclei and how atoms bond into molecules — ionic bonds are a direct consequence of electrostatic attraction between oppositely charged ions.
Photocopiers and laser printers: use electrostatic attraction to transfer toner particles onto paper, exploiting precisely controlled charge distributions.
Electrostatic precipitators: remove pollutant particles from industrial exhaust gas by charging them and attracting them to oppositely charged collector plates — a major air-pollution control technology.
Van de Graaff generators and capacitors: both rely directly on Coulomb's Law to predict how charge distributes itself and how much force (and therefore energy) is stored in a charge separation.
From Coulomb's Law to the Electric Field
Coulomb's Law describes the force between two specific charges, but physicists often prefer to describe the effect of a single charge on the space around it — the electric field. The field at a distance r from a charge Q is E = kQ/r², measured in newtons per coulomb (N/C) or equivalently volts per metre (V/m). Any other charge q placed at that point then experiences a force F = qE, which is mathematically identical to Coulomb's Law but conceptually separates "what one charge does to space" from "how a second charge responds."
This field-based view becomes essential for more complex charge distributions — continuous lines, sheets, or spheres of charge — where the field can be calculated once and then applied to any test charge, rather than recomputing pairwise forces for every possible charge placement.
Historical Context — Coulomb's Torsion Balance
Charles-Augustin de Coulomb measured the electrostatic force using an extraordinarily sensitive torsion balance: a horizontal rod suspended by a thin silver wire, with a charged sphere at one end balanced by a counterweight at the other. A second charged sphere, brought near the first, caused the rod to twist — and the angle of twist, calibrated against the known stiffness of the wire, gave a direct measurement of the force. This apparatus could detect forces as small as a hundred-thousandth of a newton, remarkable precision for 1785.
Coulomb's experiments confirmed both the inverse-square dependence on distance and the direct proportionality to charge, and his results have been refined but never overturned. Modern precision tests — using charged spherical shells rather than point charges — have confirmed the inverse-square exponent is 2 to within one part in 10¹⁶, making Coulomb's Law one of the most stringently verified relationships in all of physics.
Coulomb's Law and Electric Potential Energy
Coulomb's Law gives the force between two charges at a single instant, but it also determines how much energy is stored when charges are moved relative to each other. The electric potential energy of two point charges separated by distance r is U = kQ₁Q₂/r — notice this has one fewer power of r than the force formula, since energy is the integral of force with respect to distance. Bringing two like charges closer together requires doing work against their mutual repulsion, storing that work as potential energy; releasing them lets that energy convert back into kinetic energy as they fly apart.
This relationship between force and potential energy is exactly analogous to gravity: just as raising a mass against gravity stores gravitational potential energy that converts to kinetic energy on release, separating (or bringing together) opposite charges stores electric potential energy. This is the same physics that powers a capacitor, a particle accelerator, and — on a cosmic scale — the electrostatic repulsion that must be overcome to fuse nuclei together in stars.
Superposition — Forces from Multiple Charges
Coulomb's Law as stated applies to exactly two point charges. When more than two charges are present, the total force on any one charge is found using the principle of superposition: calculate the Coulomb force from each other charge individually, treating each pair in isolation, then add all the resulting force vectors together. Because force is a vector, this addition must account for both magnitude and direction — forces from charges in different positions can partially or fully cancel.
This principle is what makes Coulomb's Law so powerful despite its simplicity: however complex a real charge distribution — a charged rod, a charged sphere, or a crystal lattice of ions — the total force can always be built up, in principle, from the pairwise interactions of Coulomb's Law. In practice, continuous distributions are handled with calculus (integrating over infinitesimal charge elements), but the underlying physics is unchanged.