Coulomb’s law calculator
Enter two charges and a distance, get the force.
What is Coulomb’s law?
The force between two point charges is F = k |q₁ q₂| ÷ r², where r is the distance between them and k = 1 ÷ (4πε₀) ≈ 8.988 × 10⁹ N·m²/C². Like charges repel; opposite charges attract.
What value of k does the calculator use?
k = 8.9875517862 × 10⁹ N·m²/C², from the CODATA 2022 value of the vacuum permittivity ε₀ = 8.8541878188 × 10⁻¹² F/m. Textbooks often round it to 8.99 × 10⁹.
Example: the hydrogen atom
A proton and an electron 5.29 × 10⁻¹¹ m apart attract with a force of about 8.24 × 10⁻⁸ N, using the exact elementary charge 1.602176634 × 10⁻¹⁹ C.
How does distance change the force?
It follows an inverse square: twice as far, one-quarter of the force; ten times as far, one-hundredth.
Does this work in water or other materials?
The formula here is for charges in a vacuum (air is very close). In other materials the force is weaker, divided by the material’s dielectric constant.
How do I tell whether the force attracts or repels?
Like charges repel and opposite charges attract, and the force lies along the line joining them (OpenStax University Physics, Volume 2, section 5.3). Give each charge its sign: a negative product means attraction. The calculator reports the direction for you. The two charges feel equal and opposite forces, whatever their sizes.
How do I find the force when there are three or more charges?
Add the forces from the other charges one by one as vectors; each pair obeys Coulomb’s law independently (superposition, Eq. 5.2). For charges on one line, give attractive and repulsive forces opposite signs and add. OpenStax Example 5.2 combines the forces from three charges, 2e, −3e and −5e, and gets a net force of 4.08 × 10⁻¹⁴ N. This page computes one pair at a time.
How many elementary charges are in one microcoulomb?
One µC is 10⁻⁶ C and the elementary charge is exactly 1.602176634 × 10⁻¹⁹ C, so 1 µC is 6.24 × 10¹² elementary charges. Choose “elementary charges (e)” in the unit menu to enter charges in multiples of e.
The formula and what it assumes
The force between two point charges is F = k·|q₁q₂| ÷ r², with k = 1 ÷ (4πε₀) (OpenStax University Physics, Volume 2, section 5.3, Eq. 5.1). The calculator uses ε₀ = 8.8541878188 × 10⁻¹² F/m, which gives k = 8.9876 × 10⁹ N·m²/C²; OpenStax rounds these to 8.85 × 10⁻¹² and 8.99 × 10⁹. The law is stated for stationary point charges. It does not depend on the masses. For charged bodies that are not small compared with their separation, the formula is only an approximation.
Worked example: 2.0 µC and −3.0 µC, 5.0 cm apart
Convert to SI first: q₁ = 2.0 × 10⁻⁶ C, q₂ = −3.0 × 10⁻⁶ C, r = 0.050 m. Then F = 8.9876 × 10⁹ × (2.0 × 10⁻⁶ × 3.0 × 10⁻⁶) ÷ 0.050² = 21.57 N, and because the charges have opposite signs the force is attractive. Halve the distance to 2.5 cm and the force rises fourfold to 86.3 N. For the electron in hydrogen (r = 5.29 × 10⁻¹¹ m), OpenStax Example 5.1 finds 8.25 × 10⁻⁸ N; with the exact elementary charge and ε₀ the calculator gives 8.24 × 10⁻⁸ N.
Common mistakes
The most common error is the prefix: entering 2 mC instead of 2 µC makes each charge 1,000 times larger and the force a million times larger, 2.16 × 10⁷ N in the example above. Next is the distance, which is squared: measuring to the wrong point, or forgetting that cm and inches need converting (the unit menu does it), changes the answer by the square of the error. Finally, keep the signs: the direction comes from the sign of the product, so dropping a minus sign turns an attraction into a repulsion.
Sources: OpenStax University Physics, Volume 2, section 5.3 (Coulomb’s law, Eq. 5.1–5.2; Examples 5.1 and 5.2); ε₀ from CODATA 2022 (NIST) and the elementary charge, exact by the 2019 SI definition. Related: gravitational force calculator, magnetic field calculator and Ohm’s law calculator.