Coulomb’s law calculator: force between two charges
This Coulomb’s law calculator uses F = k·|q₁·q₂|/r² from OpenStax College Physics with k = 1/(4π·ε₀) = 8.988 × 10⁹ N·m²/C². Example: 2 µC and 3 µC, 10 cm apart, push each other away with about 5.39 N.
Example: Calculate: force F from the distance · Charge q₁ 2 · Charge q₂ 3 · Charge unit: µC · Distance r 10 · Distance unit: cm → Result: F = 5.3925 N (repulsive). Source: OpenStax College Physics 2e, 18.3 Coulomb’s Law: F = k·|q1·q2|/r², k = 8.988 × 10⁹ N·m²/C²; like charges repel, unlike charges attract; Example 18.1 electron–proton at 0.530 × 10⁻¹⁰ m → 8.19 × 10⁻⁸ N (read 10.10.2026). Updated: .
How it is calculated
Coulomb’s law: F = k·|q₁·q₂| / r². The electrostatic force between two point charges grows with the product of the charges and falls with the square of their distance. OpenStax gives k = 8.988 × 10⁹ N·m²/C² ≈ 8.99 × 10⁹; the calculator uses k = 1/(4π·ε₀) with ε₀ = 8.854 187 8188 × 10⁻¹² F/m from NIST, which is 8.988 × 10⁹ as well. The force acts along the line joining the charges: like charges repel, unlike charges attract.
Worked examples
| q₁ | q₂ | r | F |
|---|---|---|---|
| 2 µC | 3 µC | 10 cm | 5.39 N, repulsive |
| 2 µC | 3 µC | 20 cm | 1.35 N (one quarter) |
| 1 µC | −1 µC | 9.48 cm | 1 N, attractive |
| e | −e | 0.530 × 10⁻¹⁰ m | 8.21 × 10⁻⁸ N |
The last row is OpenStax Example 18.1, an electron and a proton at their average distance in a hydrogen atom. With the rounded values e = 1.60 × 10⁻¹⁹ C and k = 8.99 × 10⁹ the book gets 8.19 × 10⁻⁸ N; with the exact elementary charge 1.602 176 634 × 10⁻¹⁹ C the result is 8.21 × 10⁻⁸ N. OpenStax compares this with gravity between the two particles: the electric force is 2.27 × 10³⁹ times stronger.
Inverse square: double the distance, a quarter of the force
Because r is squared, doubling the distance divides the force by 4, and ten times the distance divides it by 100. To find the distance for a given force, solve for r: r = √(k·|q₁·q₂| / F). For gravity between masses the same 1/r² rule holds, see the gravitational force calculator; the potential V = k·q/r around a single charge is in the electric potential calculator.
Frequently asked questions
What is the formula of Coulomb’s law?
F = k·|q₁·q₂| / r² with k ≈ 8.99 × 10⁹ N·m²/C². Charges in coulomb, distance in metres, force in newtons.
How strong is the force between 2 µC and 3 µC at 10 cm?
About 5.39 N, and repulsive because both charges are positive.
What happens if the distance doubles?
The force drops to a quarter: 5.39 N becomes 1.35 N.
How do I know if the force is attractive or repulsive?
Look at the signs. Same sign (+ + or − −) means repulsion, opposite signs mean attraction.
Why does OpenStax get 8.19 × 10⁻⁸ N for the hydrogen atom?
The book rounds e to 1.60 × 10⁻¹⁹ C and k to 8.99 × 10⁹. With the unrounded NIST values (exact e, CODATA k) the result is 8.21 × 10⁻⁸ N.
What units does the calculator use?
Charges in C, mC, µC, nC, pC or elementary charges e; distances in m, cm, mm or nm. Internally everything is converted to coulomb and metre, the force comes out in newtons.
Sources and legal basis
- OpenStax College Physics 2e, 18.3 Coulomb’s Law: F = k·|q1·q2|/r², k = 8.988 × 10⁹ N·m²/C²; like charges repel, unlike charges attract; Example 18.1 electron–proton at 0.530 × 10⁻¹⁰ m → 8.19 × 10⁻⁸ N (read 10.10.2026)
- NIST CODATA: vacuum electric permittivity ε0 = 8.854 187 8188 × 10⁻¹² F/m (k = 1/(4π·ε0)) and elementary charge e = 1.602 176 634 × 10⁻¹⁹ C, exact (read 10.10.2026)
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