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Coulomb's Law Calculator

Coulomb's law quantifies the electrostatic force exerted between two stationary point charges. Whether charges attract or repel depends on their signs—like charges repel, opposite charges attract. Engineers, physicists, and chemists use this principle to model atomic interactions, design capacitors, and predict particle behaviour in electric fields. This calculator applies the inverse-square law to find force magnitude across any separation distance.

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Creators Steven Wooding, BSc Physics
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Understanding Coulomb's Law

Coulomb's law describes the electrostatic interaction between two static point charges. The force acts along the line connecting the charges and follows an inverse-square relationship with distance.

The physical interpretation is straightforward: charges with the same sign experience a repulsive force pushing them apart, while opposite charges experience an attractive force pulling them together. This foundational principle explains atomic bonding, electrostatic shielding, and the behaviour of charged conductors.

The law applies rigorously under three conditions:

  • Both charges must be stationary (non-moving relative to each other)
  • Charges are point-like or spherically symmetric (a metal sphere works; an irregular shape does not)
  • Charges do not occupy the same space—they must maintain measurable separation

When these conditions hold, the force calculation is exact. The constant ke = 8.988 × 109 N⋅m²/C² encodes the permittivity of free space.

Coulomb's Law Formula

The electrostatic force between two charges is proportional to the product of their magnitudes and inversely proportional to the square of the separation distance:

F = ke × (q₁ × q₂) ÷ r²

ke = 1 ÷ (4π × ε₀)

  • F — Electrostatic force in Newtons; positive indicates repulsion, negative indicates attraction
  • k<sub>e</sub> — Coulomb's constant, approximately 8.988 × 10⁹ N⋅m²/C²
  • q₁, q₂ — Charges in Coulombs; sign determines attraction (opposite) or repulsion (same)
  • r — Distance between charge centres in metres
  • ε₀ — Vacuum permittivity, 8.854 × 10⁻¹² F/m

Charge Units and Practical Scale

Electric charge is measured in Coulombs (C), defined as the charge transferred by 1 ampere of current flowing for 1 second. One Coulomb is an enormous charge in everyday contexts—most laboratory and atomic-scale charges range from picocoulombs (10⁻¹² C) to nanocoulombs (10⁻⁹ C).

For perspective:

  • A single electron carries a charge of 1.602 × 10⁻¹⁹ C
  • A typical static shock involves 10⁻⁶ to 10⁻³ Coulombs
  • A phone battery holds roughly 10,000–20,000 Coulombs

This huge range in magnitudes explains why the calculator defaults to nanocoulombs: they align with charge quantities in atoms and small laboratory setups. Always verify your units before comparing results across different problems.

Common Pitfalls and Practical Considerations

When applying Coulomb's law, several subtleties can lead to errors or misinterpretation.

  1. Distance must be centre-to-centre — Always measure separation from the geometric centre of each charge. For non-point charges (e.g., spheres), use the distance between centres, not surface-to-surface. Ignoring this produces large errors at short ranges.
  2. Sign convention determines force direction — The formula returns a signed scalar: negative force means attraction, positive means repulsion. Ensure both charges include their signs. Entering both as positive will give repulsive force; opposite signs give attractive force.
  3. The inverse-square law is exact, not approximate — Distance appears squared in the denominator. Doubling separation reduces force to one-quarter. Halving distance increases force fourfold. Small distance errors amplify dramatically in the result.
  4. Validity breaks down at very short ranges — Quantum effects and electron cloud overlap invalidate Coulomb's law when charges are separated by angstroms or less. The law assumes point charges; extended charge distributions require integration or numerical methods.

Interpreting Results: Attraction and Repulsion

The force value returned carries both magnitude and sign information. A positive result indicates repulsive interaction; the charges push apart. A negative result indicates attractive interaction; the charges pull together.

The sign depends solely on the product of the charge signs:

  • Same-sign charges (both positive or both negative): product is positive → repulsive force
  • Opposite-sign charges (one positive, one negative): product is negative → attractive force

The magnitude tells you the strength of interaction. At the atomic scale, this force is surprisingly strong. For instance, the attraction between a proton and an electron in a hydrogen atom is approximately 1.60 × 10⁻⁸ newtons—tiny in absolute terms, yet enormous relative to the electron's mass, binding it firmly to the nucleus.

Frequently Asked Questions

Why is Coulomb's law called an inverse-square law?

The force is inversely proportional to the square of separation distance. This stems from geometry: an electric charge's field spreads uniformly over a spherical surface, whose area grows as 4πr². As distance doubles, the field intensity drops to one-quarter. Remarkably, experiments confirm this r⁻² dependence to 15 decimal places, validating the point-charge model with extraordinary precision.

Can Coulomb's law apply to moving charges?

No. Coulomb's law strictly describes stationary charges only. When charges move, they generate magnetic fields, and the full electromagnetic interaction requires Maxwell's equations. At low speeds, Coulomb's law remains accurate as an approximation, but relativistic effects and magnetic forces become significant at high velocities. Specialized frameworks like the Lorentz force law are needed for moving charges.

What is the electrostatic force between a proton and electron in a hydrogen atom?

Using Coulomb's law with a separation of approximately 0.120 nm and charge magnitude 1.602 × 10⁻¹⁹ C, the attractive force is roughly 1.60 × 10⁻⁸ N. This enormous force—despite its tiny absolute value—is what binds electrons to nuclei. The force decreases rapidly as distance increases, which is why electron orbitals have sharply defined regions rather than infinite extent.

How do vacuum permittivity and relative permittivity affect the force?

Vacuum permittivity (ε₀) is a fundamental constant that appears in Coulomb's constant. Relative permittivity (εᵣ) is a dimensionless material property: water has εᵣ ≈ 80, glass ≈ 6, and vacuum ≈ 1. When charges are embedded in a dielectric material, the force is reduced by the relative permittivity: F_material = F_vacuum ÷ εᵣ. This reduction occurs because the material's electric field partially cancels the external field.

What happens to Coulomb force if I double the distance between charges?

The force reduces to one-quarter its original value. Since distance appears squared in the denominator, doubling r multiplies the denominator by four. Conversely, halving the distance increases force fourfold. This dramatic distance dependence is why electrostatic effects are strong in compressed geometries—like capacitor plates or nanoscale devices—and weak at large separations.

Does Coulomb's law work for charges of very different magnitudes?

Yes. The force depends on the product of both charges, so a tiny charge in the field of a large charge still experiences a calculable force. However, the principle of superposition becomes important: if multiple charges are present, you must sum forces vectorially. For a tiny test charge in a complex field, you typically calculate the electric field first, then multiply by the test charge.

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