Coulomb's Law Calculator

Coulomb’s Law describes the electrostatic force between two electrically charged objects. It is the foundational equation of electrostatics, the electric equivalent of Newton’s gravitational law.

Formula: F = k × (|q₁ × q₂|) ÷ r²

Where:

• F: the electrostatic force in Newtons (N). The absolute-value bars around q₁q₂ mean this formula returns a magnitude, always positive; whether it pulls or pushes comes from the signs of the two charges, not from the sign of F. (Drop the bars and you get a signed result where negative means attractive, but then you must also fix a direction convention along the line joining the charges. The calculator below reports the magnitude and states the direction separately, which is less error-prone.)
• k: Coulomb’s constant = 8.9875 × 10⁹ N·m²/C² (in vacuum)
• q₁, q₂: the electric charges of the two objects, in Coulombs (C). Like charges (both positive or both negative) repel; unlike charges attract.
• r: the distance between the centers of the two charges, in meters.

Alternative form using permittivity: F = (1 ÷ 4πε₀) × (q₁q₂ ÷ r²) where ε₀ = 8.854 × 10⁻¹² C²/(N·m²) is the permittivity of free space.

Key relationships:

• Force doubles if either charge doubles.
• Force quadruples if distance is halved (inverse-square law: same as gravity).
• Force is always along the line connecting the two charges.

Charge reference

One coulomb is the charge of roughly 6.24 × 10¹⁸ electrons, which is an enormous amount to hold in one place. A charged balloon carries maybe a microcoulomb. This is why the calculator defaults to microcoulombs: whole coulombs almost never turn up in a real problem outside a lightning bolt, which moves about 15 C.

The inverse square, and why it matters here

The r² in the denominator means the force falls away fast. Double the separation and you get a quarter of the force; triple it and a ninth. Gravity and light intensity follow the same law, for the same geometric reason: whatever is spreading out from a point is spread over a sphere, and a sphere’s area goes as r².

Run it the other way and the same rule becomes alarming. Halve the distance and the force quadruples. Bring two charges from a centimetre to an angstrom, a factor of 10⁸, and the force rises by 10¹⁶. That is the regime chemical bonds live in, and it is why the calculator offers angstroms as a distance unit.

Worked example: Two charged spheres: q₁ = +3 μC (3 × 10⁻⁶ C), q₂ = −2 μC (2 × 10⁻⁶ C), separated by r = 0.05 m (5 cm). F = 8.9875 × 10⁹ × (3 × 10⁻⁶ × 2 × 10⁻⁶) ÷ (0.05)² F = 8.9875 × 10⁹ × 6 × 10⁻¹² ÷ 0.0025 F = 8.9875 × 10⁹ × 2.4 × 10⁻⁹ F = 21.57 N, an attractive force because the charges are opposite

Real-world applications: DNA binding forces, ionic bond strength, capacitor design, van der Waals interactions, protein folding forces, and electrostatic precipitators in industrial air filtration.

Three things worth keeping in mind. First, the electrostatic force between two protons is about 10³⁶ times stronger than their gravitational pull. The reason gravity wins at planetary and astronomical scales is not strength but neutrality: bulk matter cancels its electric charges, while masses always add. Second, the formula above assumes vacuum. In a medium with relative permittivity ε_r, the force drops by that factor, and water’s ε_r ≈ 80 is why ionic salts dissolve so easily. Third, the superposition principle lets you handle multi-charge systems: the total force on any one charge is the vector sum of the Coulomb forces from each other charge, calculated independently. There is no shielding or interaction term to worry about — each pair simply acts as if the others were not there.