Coulomb's Law Calculator

Background.

The Coulomb's law calculator computes the electrostatic force between two stationary point charges, or solves for any one variable—charge magnitude or separation distance—when the other three are known. It is the foundational quantitative relationship in electrostatics, preceding field concepts and Gauss's law in most physics curricula. Students in Advanced Placement Physics C, introductory university physics, and electrical engineering courses encounter this calculation repeatedly, both in conceptual problem sets and in laboratory exercises involving Coulomb balances or charged pith balls. Beyond the classroom, the law governs the design of electrostatic precipitators, the modeling of ionic bonding potentials, and the baseline force estimates in vacuum microelectromechanical systems where gravitational and inertial effects are negligible compared with electric interactions.

Search demand for Coulomb's law calculators is sustained by the sheer volume of STEM enrollments globally. Every academic year, millions of students complete problem sets that require evaluating the force between two charges separated by a given distance, or inferring an unknown charge from a measured force and known geometry. Unlike more specialized engineering calculators, the audience here is broad: high school seniors, undergraduates, teaching assistants creating answer keys, and hobbyists building high-voltage demonstrations. The calculation is simple in principle—an inverse-square law with a proportionality constant—but students frequently struggle with unit conversions, scientific notation, and the sign conventions that determine attractive versus repulsive forces.

The historical context matters because Coulomb's law is not merely an approximation; it is the electrostatic limit of quantum electrodynamics, validated to extraordinary precision. Charles-Augustin de Coulomb published his findings in 1785 after experiments with a torsion balance, measuring the torque on charged spheres as a function of their separation. His result—that force varies as the inverse square of distance—was anticipated by Joseph Priestley and Daniel Bernoulli, but Coulomb provided the first direct quantitative evidence. The proportionality constant, now written k_e, was not isolated as a distinct fundamental constant until the development of the SI system and the explicit definition of the ampere. Modern values derive from CODATA recommended values, where k_e is exact by definition because it equals 1/(4πε₀) and ε₀ is fixed by the defined speed of light and the defined magnetic constant. This definitional exactness means that, unlike the gravitational constant G, k_e carries no measurement uncertainty in SI.

When using the calculator, it is important to recognize the domain of validity. Coulomb's law applies exactly to point charges in a vacuum. For extended conductors or dielectric media, the net force can be computed by integration or by introducing a relative permittivity factor, but the point-charge formula itself is no longer exact. At atomic scales, the law breaks down because charge is quantized and quantum effects dominate; at relativistic velocities, magnetic fields and retardation effects enter. Nevertheless, for the macroscopic, static, vacuum-scale problems that dominate homework and introductory design, Coulomb's law remains the correct starting point. The calculator therefore serves both as a computational shortcut and as a pedagogical reinforcement of the inverse-square relationship that underlies much of classical electromagnetism. Modern Coulomb balances used in national metrology institutes to realize the ampere rely on this same force law, achieving relative uncertainties below one part in 10⁸.

What is coulomb's law calculator?

Coulomb's law states that the magnitude of the electrostatic force between two point charges is directly proportional to the product of the charge magnitudes and inversely proportional to the square of the distance between them. The force acts along the straight line joining the charges. If the charges have the same sign, the force is repulsive; if opposite, attractive. The scalar magnitude is expressed as F = k_e |q₁ q₂| / r², where q₁ and q₂ are the charge values in coulombs, r is the center-to-center separation in meters, and k_e is Coulomb's constant. In SI units, k_e equals 8.9875517923 × 10⁹ newton meter squared per coulomb squared, exact by definition.

The coulomb is the SI unit of electric charge, defined by the fixed numerical value of the elementary charge e = 1.602176634 × 10⁻¹⁹ C. Consequently, one coulomb corresponds to roughly 6.242 × 10¹⁸ elementary charges. The law assumes charges are stationary or moving slowly enough that magnetic effects are negligible. It also assumes the medium is vacuum or air; for other materials, the force is reduced by a factor of the relative permittivity κ. The range of validity extends from submillimeter charged-particle traps to planetary-scale charge distributions, provided the problem remains in the electrostatic regime and quantum corrections are unnecessary.

How to use this calculator.

1. 01Enter the magnitude of the first point charge in coulombs, using scientific notation if needed.
2. 02Enter the magnitude of the second point charge in coulombs.
3. 03Input the center-to-center separation distance in meters.
4. 04Select the quantity to solve for: force, first charge, second charge, or distance.
5. 05Review the computed result, which uses the exact CODATA value of Coulomb's constant.
6. 06Interpret the sign: like charges yield a positive (repulsive) force magnitude; opposite charges yield attraction.

The formula.

The modern form of Coulomb's law, F = k_e |q₁ q₂| / r², emerged from decades of refinement after Coulomb's 1785 torsion-balance experiments. The inverse-square dependence on distance was the critical discovery, consistent with the geometric spreading of electric flux that later enabled Gauss's law. The proportionality to the product of charges reflects the superposition principle: the force exerted by a collection of charges on a test charge is the vector sum of the individual pairwise forces, each scaling linearly with the test charge.

The constant k_e is not an independently measured empirical quantity in the current SI; it is exact because the vacuum electric permittivity ε₀ is fixed by definition. Since k_e = 1/(4πε₀) and ε₀ is defined through the magnetic constant μ₀ and the speed of light c (with c exact and μ₀ = 4π × 10⁻⁷ N/A² exact), k_e inherits exactness. This definitional architecture, adopted in the 2019 SI revision, means that a Coulomb's law calculation in SI units carries no uncertainty from k_e itself; any uncertainty enters only through the measured charges and distances.

The formula can be algebraically rearranged to solve for any isolated variable. Solving for separation gives r = √(k_e |q₁ q₂| / F), which is useful when a known force is measured between calibrated charges. Solving for an unknown charge gives |q₁| = F r² / (k_e |q₂|), the basis of historical charge measurement before the electron's charge was known. The vector form introduces unit vectors: F₁₂ = k_e q₁ q₂ / r² r̂₁₂, where r̂₁₂ points from charge 1 to charge 2. This vector formulation is essential when combining Coulomb forces with other vectors, but the scalar magnitude suffices for collinear problems.

An important subtlety is that Coulomb's law describes the force between point charges. For continuous charge distributions, the total force requires integrating the pairwise force over both distributions, which reduces to the point-charge result when the separation is large compared with the charge dimensions. Inside a uniform spherical shell, the enclosed charge contributes nothing by symmetry, a result that generalizes to Newton's shell theorem and its electrostatic analogue.

A worked example.

Frequently asked questions.

References& sources.

1. [1]CODATA (2018). "CODATA Recommended Values of the Fundamental Physical Constants: 2018." Journal of Physical and Chemical Reference Data, 50(3), 033105.
2. [2]Coulomb, C.A. (1785). "Premier mémoire sur l'électricité et le magnétisme." Histoire de l'Académie Royale des Sciences, 569-577.
3. [3]Halliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of Physics, 10th ed. Wiley. Ch. 21.
4. [4]BIPM (2019). The International System of Units (SI Brochure), 9th ed.
5. [5]Jackson, J.D. (1999). Classical Electrodynamics, 3rd ed. Wiley. Ch. 1.