Coulomb's Law Calculator
Calculate the electrostatic force between two charged particles using Coulomb's Law. The formula F = kq₁q₂/r² gives the force magnitude, where k is Coulomb's constant, q₁ and q₂ are the charges, and r is the distance between them.
Coulomb's Law describes the electrostatic force between two charged particles. Discovered in 1785 by Charles-Augustin de Coulomb using a torsion balance, it is the foundational equation of electrostatics — the equivalent of Newton's law of gravitation, but for electric charge instead of mass. The mathematical form is strikingly similar: force varies inversely with the square of distance, scales with the product of the source quantities, and has a universal proportionality constant.
The formula F = kq₁q₂/r² tells you the magnitude of force between two point charges. Sign matters: positive force means repulsion (like charges), negative means attraction (opposite charges). Coulomb's constant k ≈ 8.988 × 10⁹ N·m²/C² is enormous — even tiny charges produce significant forces. One coulomb on each particle at one meter would generate roughly 9 billion newtons, more than the thrust of a Saturn V rocket. Real-world charges are microcoulombs or smaller, scaling forces down dramatically.
Electrostatic forces hold atoms together (electrons bound to nuclei), make tape stick, cause static shocks, drive printer toner to paper, and underlie all of chemistry. They are vastly stronger than gravity — the electrostatic force between two protons is about 10³⁶ times their gravitational attraction. The only reason large-scale matter is gravity-dominated is that positive and negative charges nearly cancel everywhere.
Common applications: introductory physics problems, electrostatic motor design, electrostatic precipitators (air filters), laser printer/copier physics, semiconductor device physics, and any analysis of charged-particle behavior.
Inputs
Results
Force
8.987e-1 N
Direction
Repulsive
E-field (q₁)
8.987e+5 N/C
Coulomb's Law Results
| Parameter | Value |
|---|---|
| Charge q₁ | 1 μC (1.0000e-6 C) |
| Charge q₂ | 1 μC (1.0000e-6 C) |
| Distance | 0.1 m (10.00 cm) |
| Force | 8.9875e-1 N |
| Force (absolute) | 8.9875e-1 N |
| Direction | Repulsive (like charges) |
| E-field from q₁ | 8.9875e+5 N/C |
| E-field from q₂ | 8.9875e+5 N/C |
| Formula | F = kq₁q₂/r² |
Formula
How to use this calculator
- Enter the first charge in microcoulombs (μC). Positive or negative.
- Enter the second charge in microcoulombs.
- Enter the separation distance in meters.
- Calculator returns force magnitude and direction (attractive/repulsive).
- For multiple charges, calculate pairwise and add vectorially.
- In dielectric media, divide by relative permittivity εᵣ.
Worked examples
Two charged balloons
**Scenario:** Two balloons rubbed against hair each acquire 5 nC of negative charge. They hover 5 cm apart. Force on each? **Calculation:** F = 8.988e9 × (5e-9)² / (0.05)² = 8.988e9 × 2.5e-17 / 0.0025 = 8.99 × 10⁻⁵ N. **Result:** ~90 μN repulsive force. Each balloon has mass ~3 g (weight 30 mN). So the electrostatic force is ~0.3% of weight — barely perceptible. Increasing charge or decreasing distance dramatically increases force (squared dependence on both).
Electron-proton attraction
**Scenario:** Force between an electron and proton at the Bohr radius (5.29 × 10⁻¹¹ m) in a hydrogen atom. **Calculation:** F = 8.988e9 × (1.6e-19)² / (5.29e-11)² ≈ 8.2 × 10⁻⁸ N. **Result:** ~82 nN attractive force. Tiny in absolute terms but huge for particles with mass ~10⁻³¹ kg. The resulting electron orbital speed is ~2.2 × 10⁶ m/s (~1% of light speed). This force defines all of chemistry — without it, atoms wouldn't hold together.
Lightning charge
**Scenario:** A typical thunderstorm cell holds ~15 C separated between top and bottom of a 5 km tall cloud. **Calculation:** F = 8.988e9 × 15² / 5000² = 8.988e9 × 225 / 2.5e7 = 80,900 N. **Result:** ~81 kN attractive force between the cloud top and bottom — about 8 tonnes. When the air's dielectric strength is exceeded (~3 MV/m), the charge dumps in a millisecond as lightning, releasing ~5 × 10⁹ J — equivalent to 100 kg of TNT.
When to use this calculator
**Use Coulomb's Law for:**
- **Point-charge problems**: introductory physics and quantum mechanics. - **Atomic physics**: electron orbits, ionization energies. - **Molecular physics**: ionic bonds, intermolecular forces. - **Electrostatic devices**: precipitators, paint sprayers, copiers/printers. - **Capacitor physics**: charge separation forces. - **Electrostatic motors**: small-scale rotation from charge. - **Space charge effects**: in vacuum tubes, plasmas.
**Limitations:**
- **Point charges only**: real objects have charge distributions; use integration or Gauss's law. - **Vacuum form**: in dielectric media, divide by εᵣ. - **Static only**: moving charges produce magnetic fields too (use Maxwell's equations). - **Relativistic**: at very high speeds, retarded potentials matter (Liénard-Wiechert). - **Classical**: at atomic scale, quantum mechanics modifies things (orbitals, exchange).
**Beyond point charges:**
For continuous charge distributions: - **Line charge** (infinite): E = λ/(2πε₀r). - **Surface charge** (infinite plane): E = σ/(2ε₀). - **Sphere** (outside): acts like point charge at center. - **Sphere** (inside, uniform): E proportional to r.
Gauss's law (∮E·dA = Q_enc/ε₀) often simpler for symmetric problems.
**Electric field perspective:**
Coulomb's law can be split into two steps: 1. A charge creates an electric field: E = kq/r². 2. Other charges feel a force: F = qE.
This field picture generalizes better to moving charges and electromagnetic waves.
**Common applications:**
- **Inkjet printing**: charged droplets steered by electric fields. - **Photocopiers/laser printers**: charged toner adheres to charged drum. - **Electrostatic precipitators**: charge dust, attract to collection plate. - **Mass spectrometry**: ion deflection in electric fields. - **Ion implantation**: semiconductor doping. - **Particle accelerators**: charged particles accelerated by E fields. - **Capacitors**: stored energy comes from electrostatic forces.
**Dielectric effects:**
A dielectric medium reduces effective forces: - Air: εᵣ ≈ 1 (essentially vacuum). - Water: εᵣ ≈ 80 (dissolves ionic compounds). - Glass: εᵣ ≈ 4-10. - Most plastics: εᵣ ≈ 2-4.
This is why salt dissolves in water — ions feel much weaker electrostatic attraction in water than in air.
**Software:**
- **Comsol, ANSYS**: full electrostatic finite element analysis. - **Python (NumPy + matplotlib)**: educational visualization. - **SPICE**: circuit-level analysis (capacitances, not raw forces).
**Pitfalls:**
- **Wrong units**: μC vs nC (factor 1,000); cm vs m (factor 100 → factor 10,000 in force). - **Sign errors**: products of signed charges give correct force sign. - **Point-charge assumption**: real objects need integration. - **Ignoring dielectric**: water dramatically reduces forces. - **Confusing F and E**: force on charge vs field from charge. - **Squared distance**: tiny distance changes matter hugely.
Common mistakes to avoid
- Mixing charge units (μC vs nC vs C).
- Mixing distance units (cm vs m — distance squared amplifies the error).
- Forgetting the inverse square law (halving distance quadruples force).
- Ignoring dielectric medium (water reduces force by factor 80).
- Treating extended objects as point charges when they're not far apart.
- Forgetting sign convention (same signs repel, opposite attract).
- Confusing force with field or energy.
- Using the formula for moving charges (need full electromagnetics).