Hong Life CalculatorsUnderstanding the Coulomb's law calculator — concept and principle
This calculator applies Coulomb's law to two point charges to find the electrostatic force between them, and also gives the electric field and electric potential produced by the first charge at the same distance. It is used in physics courses and for quick electrostatics estimates.
How it is calculated
- Force: F = k · q₁ · q₂ / r², with k = 1/(4πε₀) ≈ 8.98755 × 10⁹ N·m²/C².
- Electric field of charge 1: E = k · q₁ / r², in N/C (equivalent to V/m).
- Electric potential of charge 1: V = k · q₁ / r, in volts, taking zero potential at infinity.
Charges are entered in microcoulombs and converted to coulombs (× 10⁻⁶); the distance is in metres. A negative force means attraction (opposite signs); a positive force means repulsion.
The principle
Charles-Augustin de Coulomb measured in the 1780s, using a torsion balance, that the electric force between two small charged bodies is proportional to the product of their charges and inversely proportional to the square of their separation. Geometrically, the field lines leaving a point charge spread over the surface of a sphere of area 4πr², so their density, and therefore the field strength, falls as 1/r². That is why the constant contains 4π. The field E is the force per unit test charge, and the potential V is the work per unit charge needed to bring a test charge in from infinity, which falls off only as 1/r.
Worked example
- Inputs: q₁ = 5 µC, q₂ = −3 µC, r = 0.1 m.
- Product of charges: 5 × 10⁻⁶ × (−3 × 10⁻⁶) = −1.5 × 10⁻¹¹ C².
- Force: 8.98755 × 10⁹ × (−1.5 × 10⁻¹¹) ÷ 0.01 = −13.481 N (attraction).
- Field from q₁: 8.98755 × 10⁹ × 5 × 10⁻⁶ ÷ 0.01 = 4,493,776 N/C (about 4.49 MV/m).
- Potential from q₁: 8.98755 × 10⁹ × 5 × 10⁻⁶ ÷ 0.1 = 449,378 V.
| Distance (m) | Force (N, − attract) |
|---|---|
| 0.025 | -215.701243 |
| 0.05 | -53.92531075 |
| 0.075 | -23.96680478 |
| 0.1 | -13.48132769 |
| 0.125 | -8.628049721 |
| 0.15 | -5.991701195 |
| 0.2 | -3.370331922 |
How to read the result
Microcoulomb charges are large in everyday terms: two of them 10 cm apart pull with about the weight of a 1.4 kg mass. The computed field of about 4.5 MV/m is above the breakdown strength of dry air (roughly 3 MV/m), so in air such charges would leak away through sparks. Halving the distance multiplies the force by four.
Common mistakes
- Forgetting to square the distance, or entering centimetres in the metre field.
- Dropping the signs and losing the attraction/repulsion information.
- Applying the point-charge formula to large or nearby bodies where the charge is spread out; it holds exactly only for point charges or spherically symmetric distributions.
Frequently asked questions
Why is k sometimes written as 9 × 10⁹?
It is a rounded value of 8.98755 × 10⁹ N·m²/C², accurate to about 0.14%, convenient for quick estimates.
How does gravity compare with the Coulomb force?
Between two protons, the electric repulsion is about 10³⁶ times stronger than their gravitational attraction.
What is the force with three or more charges?
Add the forces from each pair as vectors (superposition principle).
Sources: CODATA recommended values of the fundamental physical constants, NIST (vacuum electric permittivity ε₀) · D. J. Griffiths, Introduction to Electrodynamics, Cambridge University Press, chapter 2