Coulomb's Law Calculator Guide: Electric Force Between Charges

What Is Coulomb's Law?

Coulomb's law describes the electrostatic force between two electrically charged particles. Published by French physicist Charles-Augustin de Coulomb in 1785, it was one of the first quantitative laws of electromagnetism and remains fundamental to physics, chemistry, and electrical engineering.

The law states that the magnitude of the electrostatic force between two point charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. This inverse-square relationship mirrors Newton's law of universal gravitation, published a century earlier in 1687.

The Formula

F = k × |q&sub1; × q&sub2;| / r²

Where:

• F = electrostatic force in newtons (N)
• k = Coulomb's constant ≈ 8.9875 × 10&sup9; N·m²/C²
• q&sub1;, q&sub2; = electric charges in coulombs (C)
• r = distance between charge centers in meters (m)

Coulomb's constant can also be written as k = 1 / (4πε&sub0;), where ε&sub0; is the permittivity of free space (8.854 × 10−¹² C²/(N·m²)). This form appears frequently in university-level electromagnetism courses and in Maxwell's equations.

Worked Example 1: Two Point Charges

Two charges of +3 μC and −5 μC are separated by 0.2 meters. What is the force between them?

Convert to SI: q&sub1; = 3 × 10−&sup6; C, q&sub2; = 5 × 10−&sup6; C, r = 0.2 m

F = (8.988 × 10&sup9;) × (3 × 10−&sup6;) × (5 × 10−&sup6;) / (0.2)²
F = (8.988 × 10&sup9;) × (15 × 10−¹²) / 0.04
F = 0.13482 / 0.04
F ≈ 3.37 N (attractive, since charges are opposite)

For perspective, 3.37 N is roughly the weight of a medium apple. Two charges measured in millionths of a coulomb, separated by just 20 cm, produce a force you could easily feel. This illustrates just how powerful the electromagnetic force is.

Worked Example 2: The Inverse-Square Relationship

If the distance in the above example doubles from 0.2 m to 0.4 m:

F = 3.37 / (2)² = 3.37 / 4 ≈ 0.84 N

Doubling the distance reduced the force to exactly one-quarter. Triple the distance and the force drops to one-ninth. This rapid falloff explains why electrostatic effects dominate at small scales but become negligible at large distances.

Electric Force vs Gravitational Force

Coulomb's law and Newton's law of gravitation share the same mathematical structure:

The ratio of electromagnetic to gravitational force between a proton and electron is approximately 2.27 × 10³&sup9;(Griffiths, Introduction to Electrodynamics, 4th ed.). Gravity only dominates at astronomical scales because most matter is electrically neutral — positive and negative charges cancel out.

The Superposition Principle

When more than two charges are present, the net force on any charge is the vector sum of the forces from every other charge. This is the superposition principle, and it holds exactly in classical electrostatics.

For example, if charge A experiences forces from charges B, C, and D:

F&sub(net) = F&sub(AB) + F&sub(AC) + F&sub(AD)

Each force is calculated independently using Coulomb's law, then the vectors are added. This is how molecular dynamics simulations compute electrostatic interactions — every pair of charges contributes to the total force. Modern computational chemistry tools process billions of these pairwise calculations per second (Journal of Chemical Theory and Computation, 2023).

Coulomb's Law in Dielectric Media

In a material medium (not vacuum), the electrostatic force is reduced by the dielectric constant (κ, also written as ε&sub(r)):

F = k × |q&sub1; × q&sub2;| / (κ × r²)

Water's high dielectric constant is the reason ionic compounds like NaCl dissolve in it. The electrostatic attraction between Na&sup+ and Cl− ions is reduced by a factor of 80 in water, allowing thermal energy to pull the ions apart. This insight, first quantified by Peter Debye in the 1920s, earned foundational importance in physical chemistry.

Real-World Applications

Electrostatic Precipitators

Coal-fired power plants use Coulomb's law to remove particulates from exhaust. Charged plates attract and capture soot particles, removing over 99% of particulate matter (EPA, 2024). The technology handles over 2 billion tons of flue gas annually in the United States alone.

Inkjet Printing

Inkjet printers use electric fields to steer charged ink droplets toward precise locations on paper. The deflection is governed by Coulomb's law. Modern inkjet heads place drops with accuracy of 5 micrometers— roughly one-tenth the width of a human hair (HP Labs, 2023).

Molecular Biology

DNA's double helix is held together partly by electrostatic attraction between hydrogen bond donors and acceptors. Gel electrophoresis separates DNA fragments by size using electric fields — smaller fragments experience less drag relative to the Coulombic driving force and migrate faster through the gel matrix.

Frequently Asked Questions

What is Coulomb's law formula?

Coulomb's law states that the electric force between two point charges is F = k × |q&sub1; × q&sub2;| / r², where k is Coulomb's constant (8.9875 × 10&sup9; N·m²/C²), q&sub1; and q&sub2; are the charges in coulombs, and r is the distance between them in meters. The force is attractive for opposite charges and repulsive for like charges.

What is Coulomb's constant?

Coulomb's constant (k) equals approximately 8.9875 × 10&sup9; N·m²/C². It can also be expressed as k = 1/(4πε&sub0;), where ε&sub0; is the permittivity of free space (8.854 × 10−¹² C²/(N·m²)). The constant quantifies how strongly charges interact in a vacuum.

How does distance affect electric force?

Electric force follows an inverse-square law. Doubling the distance between charges reduces the force to one-quarter. Tripling the distance reduces it to one-ninth. This is the same relationship that governs gravitational force, but electric forces are vastly stronger — about 10³&sup6; times stronger than gravity for elementary particles.

What is the difference between Coulomb's law and gravitational force?

Both follow inverse-square laws (F ∝ 1/r²), but Coulomb's law uses charges and the electric constant, while Newton's gravitational law uses masses and the gravitational constant G. The key difference: electric force can be attractive or repulsive, while gravity is always attractive. The electric force is roughly 10³&sup6; times stronger than gravity at the atomic scale.

Does Coulomb's law work inside materials?

In a dielectric medium (like water or glass), the force is reduced by the material's dielectric constant (κ). The modified formula becomes F = k × |q&sub1; × q&sub2;| / (κ × r²). Water has κ ≈ 80, meaning electric forces in water are roughly 80 times weaker than in a vacuum. This is why ionic compounds dissolve in water — the reduced electrostatic attraction allows ions to separate.