The electric force between two point charges, attractive or repulsive, with the field and potential energy, checked against OpenStax University Physics Volume 2
How Coulomb’s Law Works
The force between two point charges grows with the product of the charges and falls with the square of the distance between them: F = k|q₁q₂| ÷ r². The constant k = 1/(4πε₀) = 8.99 × 10⁹ N·m²/C². The force acts along the line joining the charges, pulling them together if the signs differ and pushing them apart if the signs match.
- Convert charges to coulombs: one elementary charge is 1.602 × 10⁻¹⁹ C; 1 μC is 10⁻⁶ C.
- Convert distance to meters.
- Multiply k by the product of the charge magnitudes and divide by r². For hydrogen: 8.99 × 10⁹ × (1.602 × 10⁻¹⁹)² ÷ (5.29 × 10⁻¹¹)² = 8.24 × 10⁻⁸ N.
- Read the direction from the signs: + and − attract.
Worked Values
Moving two 1 μC charges from 1 m to 10 cm apart multiplies the force by 100, the inverse-square law at work.
More Than Two Charges: Superposition
OpenStax Example 5.2 puts q₂ = −3e at the origin, q₁ = 2e a distance d = 2.0 × 10⁻⁷ m above it, and q₃ = −5e a distance 2d to its right. Use the calculator once per pair: q₁ pulls q₂ upward with 3.46 × 10⁻¹⁴ N, and q₃ pushes it left with 2.16 × 10⁻¹⁴ N. Add the components as vectors: |F| = √(3.46² + 2.16²) × 10⁻¹⁴ = 4.08 × 10⁻¹⁴ N, at tan⁻¹(3.46/2.16) = 58° above the −x axis, matching the textbook.
Common Mistakes
- Forgetting to square the distance. Halving r quadruples the force.
- Leaving charges in μC or nC. k assumes coulombs and meters.
- Adding forces as plain numbers. Forces from several charges add as vectors, component by component.
- Putting the signs into the magnitude. Use |q₁q₂| for the size and the signs only for the direction.
- Using it for spread-out charges. The formula is for point charges or spheres measured center to center.
What This Calculator Does Not Cover
It handles two charges in a vacuum (air is very close). Inside other materials the force is reduced by the dielectric constant, and continuous charge distributions need integration.
Frequently Asked Questions
What is Coulomb’s law formula?
F = k|q₁q₂| ÷ r², where k = 8.99 × 10⁹ N·m²/C², q₁ and q₂ are the charges in coulombs and r is the distance in meters.
What is the value of Coulomb’s constant?
k = 1/(4πε₀) = 8.99 × 10⁹ N·m²/C², the value OpenStax University Physics uses.
What is the force between the proton and electron in hydrogen?
About 8.2 × 10⁻⁸ N, attractive, at the Bohr radius of 5.29 × 10⁻¹¹ m (OpenStax Example 5.1 gives 8.25 × 10⁻⁸ N).
Do like charges attract or repel?
Like charges repel and opposite charges attract.
What happens to the force if the distance doubles?
It drops to one quarter, because force depends on 1/r².
How do you find the net force from several charges?
Find each pair’s force with Coulomb’s law, split them into x and y components, add the components, then combine them with the Pythagorean theorem.
How to Use This Calculator
- Enter both charges — with signs, e.g. 2 and −3, or in scientific notation like 1.5e-6.
- Choose the charge units — elementary charges, coulombs, microcoulombs or nanocoulombs.
- Enter the distance and its units — center to center.
- Read the force — its size in newtons, whether it attracts or repels, plus the field and potential energy.
📚 Sources & References- [1] OpenStax. University Physics Volume 2, 5.3 Coulomb’s Law (2016; accessed October 2026). openstax.org
- [2] OpenStax. University Physics Volume 2, 5.4 Electric Field (2016; accessed October 2026). openstax.org
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