Physics · Electromagnetism · Electrostatics
Coulomb's Law Calculator
Calculates the electrostatic force between two point charges using Coulomb's Law.
Calculator
Formula
F is the magnitude of the electrostatic force in Newtons (N). k_e is Coulomb's constant, approximately 8.9875 \times 10^9 \, \text{N}\cdot\text{m}^2\text{/C}^2. q_1 and q_2 are the magnitudes of the two point charges in Coulombs (C). r is the separation distance between the two charges in meters (m). A positive result indicates a repulsive force (like charges); a negative result indicates attraction (opposite charges), though the magnitude formula uses absolute values.
Source: Coulomb, C.-A. de (1785). Mémoires sur l'électricité et le magnétisme. Histoire de l'Académie Royale des Sciences. Also formalized in: Griffiths, D.J. (2017). Introduction to Electrodynamics, 4th ed., Cambridge University Press.
How it works
Coulomb's Law, published by French physicist Charles-Augustin de Coulomb in 1785, states that the electrostatic force between two point charges is directly proportional to the product of their magnitudes and inversely proportional to the square of the distance separating them. This inverse-square relationship is a direct consequence of the geometry of three-dimensional space and the nature of the electric field emanating isotropically from a point source.
The formula is expressed as F = k_e |q₁ q₂| / r², where k_e ≈ 8.9875 × 10⁹ N·m²/C² is Coulomb's constant (also written as 1/(4πε₀), with ε₀ being the permittivity of free space). q₁ and q₂ are the charges in Coulombs — they can be positive or negative, representing protons and electrons respectively. When both charges share the same sign, the signed force is positive (repulsive); when opposite in sign, it is negative (attractive). The separation r is measured in meters between the centers of the two point charges.
Coulomb's Law has wide-ranging applications: it underpins atomic structure models by describing the attraction between the nucleus and orbital electrons; it is used in capacitor design and electrostatic precipitator engineering; it governs the behavior of ion pairs in electrolyte solutions and ionic crystal lattices; and it forms the basis for more advanced electromagnetic theory including Gauss's Law and the derivation of electric field distributions around charged objects.
Worked example
Suppose two charged particles are separated by r = 0.05 m (5 cm). Particle 1 has a charge of q₁ = +1 × 10⁻⁶ C (1 μC) and Particle 2 has a charge of q₂ = +2 × 10⁻⁶ C (2 μC).
Step 1 — Identify the values: k_e = 8.9875 × 10⁹ N·m²/C², q₁ = 1 × 10⁻⁶ C, q₂ = 2 × 10⁻⁶ C, r = 0.05 m.
Step 2 — Calculate the product of the charges: q₁ × q₂ = (1 × 10⁻⁶) × (2 × 10⁻⁶) = 2 × 10⁻¹² C².
Step 3 — Square the distance: r² = (0.05)² = 0.0025 m².
Step 4 — Apply Coulomb's Law: F = (8.9875 × 10⁹) × (2 × 10⁻¹²) / 0.0025 = (0.017975) / 0.0025 = 7.19 N.
Since both charges are positive (like charges), the force is repulsive with a magnitude of approximately 7.19 Newtons. This is a relatively large force for microcoulomb-scale charges at such close range, illustrating why electrostatic forces dominate at the atomic and molecular scale.
Limitations & notes
Coulomb's Law applies strictly to point charges — idealized objects whose physical size is negligible compared to their separation. For extended charged objects (such as spheres, rods, or surfaces), the law must be integrated over the charge distribution, which requires more advanced techniques. The law is also formulated for static (non-moving) charges; when charges are in motion, magnetic forces arise and the full Lorentz force law or Maxwell's equations must be used. Additionally, Coulomb's Law holds rigorously only in vacuum or free space; in a dielectric medium, the effective force is reduced by the relative permittivity (dielectric constant) of the material, requiring the substitution k_e → k_e / ε_r. At sub-atomic distances (on the order of femtometers), quantum mechanical and nuclear strong-force effects become dominant, rendering the classical Coulomb formulation insufficient. Very high charge densities or relativistic particle speeds also require corrections beyond classical electrostatics.
Frequently asked questions
What is Coulomb's constant and what are its units?
Coulomb's constant k_e equals approximately 8.9875 × 10⁹ N·m²/C². It can also be expressed as 1/(4πε₀), where ε₀ = 8.854 × 10⁻¹² C²/(N·m²) is the permittivity of free space. Its units ensure that when charge is measured in Coulombs and distance in meters, the resulting force is in Newtons.
How does Coulomb's Law differ from Newton's Law of Gravitation?
Both laws follow an inverse-square relationship with distance, but they differ fundamentally in that gravity is always attractive while electrostatic forces can be either attractive or repulsive depending on the signs of the charges. Gravitational force depends on mass and the gravitational constant G, while Coulomb's Law depends on charge and Coulomb's constant k_e. Electrostatic forces between elementary particles are typically many orders of magnitude stronger than gravitational forces at the same distance.
What happens to the electrostatic force if the distance between charges is doubled?
Because force is inversely proportional to the square of the distance, doubling r reduces the force by a factor of four (2² = 4). Similarly, halving the distance increases the force by a factor of four. This inverse-square relationship is characteristic of fields that spread out uniformly in three-dimensional space.
Can Coulomb's Law be used inside a material (not vacuum)?
Yes, but the formula must be modified. Inside a dielectric material with relative permittivity ε_r (also called the dielectric constant), the effective force is F = k_e |q₁ q₂| / (ε_r r²). For example, in water with ε_r ≈ 80, the electrostatic force between two ions is approximately 80 times weaker than in vacuum — which is why ionic compounds readily dissolve in water.
What is the SI unit of electric charge, and how large is one Coulomb?
The SI unit of electric charge is the Coulomb (C). One Coulomb is an extremely large amount of charge in practical terms — it equals the charge of approximately 6.24 × 10¹⁸ protons. In most laboratory and engineering applications, charges are measured in microcoulombs (μC, 10⁻⁶ C), nanocoulombs (nC, 10⁻⁹ C), or picocoulombs (pC, 10⁻¹² C).
Last updated: 2025-01-15 · Formula verified against primary sources.