Coulomb's Law Calculator
Calculate electrostatic force between charges using F = kq₁q₂/r². Solve for force, charge, or separation distance.
What Is Coulomb's Law?
Coulomb's Law describes the electrostatic force between two point charges. The formula is F = k·q₁·q₂/r², where k = 8.99 × 10⁹ N·m²/C² (Coulomb's constant), q₁ and q₂ are the charges (coulombs), and r is the distance between them (meters). Like charges (both positive or both negative) repel; opposite charges attract.
The inverse-square dependence (1/r²) is the same mathematical form as Newton's Law of Gravitation. Doubling the distance reduces the force by a factor of 4; halving it quadruples the force. This rapid falloff explains why electrostatic forces dominate at atomic scales but gravity dominates at astronomical scales — gravity has no repulsive form to cancel out.
Coulomb's constant k = 1/(4πε₀) = 8.99 × 10⁹ N·m²/C², where ε₀ = 8.854 × 10⁻¹² F/m is the permittivity of free space. In a material with relative permittivity εᵣ, the force is reduced by εᵣ: F = kq₁q₂/(εᵣr²). Water (εᵣ ≈ 80) reduces electrostatic forces dramatically, explaining why ionic compounds dissolve easily in it.
Coulomb's Law is the foundation of classical electrostatics. It allows calculation of electric fields (E = F/q = kq/r²), potentials, capacitance, and forms the basis for understanding chemical bonds, crystal structures, and electronic devices. For more than two charges, the principle of superposition applies: the total force on any charge is the vector sum of Coulomb forces from all other charges.
Formula Reference Table
| Quantity | Formula | Notes |
|---|---|---|
| Force (F) | F = k·q₁·q₂/r² | k = 8.99×10⁹ N·m²/C²; positive = repulsion |
| Charge (q₁) | q₁ = F·r²/(k·q₂) | Coulombs; 1 μC = 1×10⁻⁶ C |
| Distance (r) | r = √(k·q₁·q₂/F) | Separation between charges |
| Electric field (E) | E = F/q = k·q/r² | Field due to charge q; N/C |
| Coulomb's constant k | 8.99×10⁹ N·m²/C² | = 1/(4πε₀) |
| In medium | F = kq₁q₂/(εᵣr²) | εᵣ = relative permittivity |
3 Worked Examples
Find the electrostatic force between two protons (q = 1.602×10⁻¹⁹ C) separated by 1×10⁻¹⁰ m (atomic scale).
- F = k·q₁·q₂/r²
- F = 8.99×10⁹ × (1.602×10⁻¹⁹)² / (1×10⁻¹⁰)²
- F = 8.99×10⁹ × 2.566×10⁻³⁸ / 10⁻²⁰
- F = 23.1 N — a large force at atomic scale!
Two spheres carry q₁ = 3 μC and q₂ = −5 μC, separated by 0.2 m. Find the force.
- q₁ = 3×10⁻⁶ C; q₂ = −5×10⁻⁶ C; r = 0.2 m
- F = 8.99×10⁹ × 3×10⁻⁶ × 5×10⁻⁶ / 0.04
- F = 8.99×10⁹ × 15×10⁻¹² / 0.04 = 3.37 N
- Negative product = attraction (opposite charges)
Two 1 μC charges must repel with exactly 0.1 N. How far apart should they be?
- r² = k·q₁·q₂/F = 8.99×10⁹ × (1×10⁻⁶)² / 0.1
- r² = 8.99×10⁹ × 10⁻¹² / 0.1 = 0.08991
- r = √0.08991 = 0.2998 m ≈ 0.3 m
Real-World Applications
Common Mistakes to Avoid
F ∝ 1/r², not 1/r. If distance doubles, force drops by a factor of 4, not 2. This is the most common error in Coulomb's Law calculations.
1 μC = 1×10⁻⁶ C. Entering "3" instead of "3×10⁻⁶" gives a result 10¹² times too large. Always convert to coulombs before calculating.
When q₁q₂ > 0, charges are same-sign → repulsion. When q₁q₂ < 0, charges are opposite-sign → attraction. The sign of the product tells you the direction; always note this.
F = kq₁q₂/r² applies to point charges. For charged spheres, rods, or planes, integration is needed (or simplified forms like Gauss's Law). This formula only works exactly for spherically symmetric charge distributions at distances > sphere radius.
In a dielectric material, k_eff = k/εᵣ. In water (εᵣ ≈ 80), forces are 80× weaker. Forgetting this when analyzing forces in biological or chemical systems gives vastly incorrect results.
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Interpretation: This relationship connects motion, force, momentum, work or energy in a mechanical system. Assumption: Choose a consistent reference direction and unit system. The model may assume constant acceleration, rigid bodies, negligible losses or an isolated system.