What this calculator does
Find how hard two electric charges push apart or pull together. Coulomb's law describes that push or pull between two charged objects. It works when the objects are small compared with the distance between them. The force grows as you multiply the two charges together. It shrinks with the square of the distance, so halving the gap makes the force four times stronger. Opposite charges (plus and minus) attract each other, and like charges repel.
This calculator works the law three ways. It finds the force from two charges and the distance between them. It finds the distance that gives a certain force. Or it finds the second charge needed for a certain force at a certain distance. You can enter charges in coulombs (the standard unit of charge), or in milli-, micro-, nano- or picocoulombs. You can also enter a count of elementary charges, the charge on one proton. Every conversion to standard (SI) units is shown in the working.
Where it's used
Why this math matters outside the classroom:
- Point charges. Two 1 µC charges 1 m apart push on each other with about 8.99 × 10⁹ × 10⁻¹² ≈ 0.009 N.
- Inverse square. Students see that halving the distance between two charges makes the electric force four times as strong.
- Atoms. Chemistry classes compare the pull between a proton and an electron with gravity to show why electric forces hold atoms together.
- Laser printers. Engineers design laser printers around charged drums whose electric forces pick up toner powder in the shape of the page.
Grade level and standards: Grades 11–12; Common Core State Standards for Mathematics HSA.CED.A.4. © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.
How to use it
- Choose what to solve for: the force, the distance, or the second charge.
- Enter the values you know, with their units. Give a charge a minus sign when it's negative.
- When solving for the second charge, say whether the charges attract or repel. That lets the calculator give it the right sign.
- Pick the unit for the answer. Then read the result, the type of force and the signed force in newtons (the standard unit of force).
What the inputs mean
- Charge 1 (q₁)
- An electron carries −1 elementary charge (−1.602176634 × 10⁻¹⁹ C), and a proton carries +1. Static charges on everyday objects are usually nanocoulombs to microcoulombs.
- Charge 2 (q₂)
- Opposite signs attract; like signs repel.
- Distance between the charges (r)
- Measure between the centers of the charges. For charged balls, Coulomb's law treats each one as a point at its center, as long as the charge is spread evenly over it.
- Size of the force (|F|)
- Enter only the size of the force. The signs of the charges decide whether it attracts or repels.
Formula and method
F = k × q₁ × q₂ ÷ r² k = 1 ÷ (4π × ε₀) ≈ 8.9875517862 × 10⁹ N·m²/C² r = √(k|q₁q₂| ÷ |F|) |q₂| = |F| × r² ÷ (k|q₁|)
Here q₁ and q₂ are the charges in coulombs, r is the distance between them in meters, and F is the force in newtons. The bars | | mean the size of a value without its sign. The constant k comes from ε₀, a fixed number for how electric fields act in empty space (the vacuum permittivity). This calculator uses the CODATA 2022 value, ε₀ = 8.8541878188 × 10⁻¹² F/m.
The sign of q₁ times q₂ tells you the direction. A negative result means the charges attract, and a positive one means they repel. The calculator gives the size of the force as the main answer and shows the signed value separately.
The rearranged forms follow straight from the law. Solving for distance takes a square root. Solving for a charge multiplies the force by r², then divides by k and the other charge.
Worked example
Two small spheres carry +2 µC and −3 µC, and their centers are 5 cm apart. What force acts between them?
- Convert to SI: q₁ = 2 × 10⁻⁶ C, q₂ = −3 × 10⁻⁶ C, r = 0.05 m.
- |q₁q₂| = 6 × 10⁻¹² C², and r² = 0.0025 m².
- |F| = 8.9875517862 × 10⁹ × 6 × 10⁻¹² ÷ 0.0025 = 21.5701 N.
- The charges have opposite signs, so the force is attractive.
Result: The spheres pull on each other with about 21.57 N. That's roughly the weight of a 2.2 kg mass.
Understanding the result
The electric force is remarkably strong. Two charges of one coulomb each, a meter apart, would push on each other with almost nine billion newtons. In practice, everyday objects carry only tiny extra charges. That's why static cling is noticeable but gentle.
The force follows an inverse-square rule, so small changes in distance have big effects. Doubling the distance cuts the force to a quarter. Ten times the distance cuts it to a hundredth.
When several charges are present, the force on any one of them is all the separate Coulomb forces from the others added together, direction included (a vector sum).
Assumptions and limitations
- Both charges are points, or balls with evenly spread charge, and they don't move.
- They sit in a vacuum (empty space). Air changes the force by only about 0.06%.
- Nothing else nearby, such as a metal object or another charge, affects the field between them.
- In a material such as water or oil, the force is weaker by a factor called the material's relative permittivity. This calculator doesn't include it.
- Some objects aren't small compared with the distance between them. For those, the force must be found by adding up (integrating) over how the charge is spread out.
- Moving charges also make magnetic forces, which Coulomb's law alone doesn't describe.
Frequently asked questions
Why is the force sometimes shown as negative?
The signed force, k q₁q₂ ÷ r², is negative when the charges have opposite signs. By custom, a negative value means attraction and a positive one means repulsion. The main result is the size of the force, which is always positive.
How does Coulomb's law compare with Newton's law of gravitation?
Both are inverse-square laws. But gravity always attracts, while electric forces can attract or repel. The electric force between an electron and a proton is about 10³⁹ times stronger than the pull of gravity between them.
What is the Coulomb constant?
It's the scaling number k = 1 ÷ (4πε₀), about 8.99 × 10⁹ N·m²/C². Since the SI units were redefined in 2019, ε₀ is measured rather than exact. But it's known to about one part in ten billion.
Can I use this for charges in water?
Only roughly. Water's relative permittivity is about 80 at room temperature. So the force between charges in water is roughly 80 times smaller than the vacuum value shown here.
How this calculator is checked
Method: Coulomb's law F = q₁q₂ ÷ (4πε₀r²) with ε₀ = 8.8541878188 × 10⁻¹² F/m (CODATA 2022); elementary charge 1.602176634 × 10⁻¹⁹ C (exact, 2019 SI). The calculation is covered by automated tests with known reference values and edge cases. Last updated 2026-10-18. Found a problem? Tell us.
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