Like charges repel and opposite charges attract. Charles-Augustin de Coulomb measured exactly how strongly in 1785 with a delicate torsion balance, and found the same inverse-square pattern Newton had found for gravity. This calculator applies Coulomb’s law to two point charges. It finds the force from the charges and their separation, or works back to either charge or the distance, keeps track of signs so you can see whether the force pulls or pushes, and lets you place the charges in air, oil, ethanol, water or any material with a known dielectric constant.
How to use the Coulomb’s law calculator
- Pick what to Solve for: Force, Charge 1, Charge 2 or Distance.
- Choose what lies Between the charges: vacuum, dry air, transformer oil, ethanol, water or a custom relative permittivity.
- Enter Charge 1 and Charge 2 with their signs, in µC, nC, pC, mC, C or elementary charges (e).
- Enter the Distance between charges, center to center, in cm, m, mm, µm, nm or ångströms.
- When solving for a charge or a distance, enter the Electric force with its sign: positive for repulsion, negative for attraction.
- Read the result with the force magnitude, whether it is attractive or repulsive, the electric field of charge 1 at charge 2, the potential energy and charge 1 expressed in electrons.
Coulomb’s law formula
εr is the relative permittivity of the medium (1 for vacuum). The field of charge 1 at the position of charge 2 is E = kq1 ÷ (εrr2), and the potential energy of the pair is U = kq1q2 ÷ (εrr).
Worked example
Two small charged spheres
A sphere carrying +3 µC sits 12 cm from one carrying −5 µC, in vacuum.
F = 8.9875517923 × 10⁹ × (3 × 10⁻⁶) × (−5 × 10⁻⁶) ÷ 0.12² = −9.362 N
The negative sign means attraction, with a magnitude of 9.36 N (about 2.1 lbf). The first sphere's field at the second is 1.87 × 10⁶ N/C, and charging it took removing about 1.87 × 10¹³ electrons.
Inside an atom. In the Bohr model of hydrogen, a proton (+1 e) and an electron (−1 e) are 52.9 pm apart. The calculator gives an attractive force of 8.24 × 10⁻⁸ N and a potential energy of −27.2 eV, twice the 13.6 eV needed to ionize hydrogen, as the model predicts.
In water. Put the same two spheres in water (εr ≈ 80.1) and the force drops to −0.117 N, eighty times weaker.
Relative permittivity of common media
| Medium | εr |
|---|---|
| Vacuum | 1 (exact) |
| Dry air | 1.00059 |
| Transformer oil | about 2.2 |
| Glass | 4–10 |
| Ethanol (25 °C) | about 24.3 |
| Water (20 °C) | about 80.1 |
When Coulomb’s law applies
The formula is exact for point charges at rest and for uniformly charged spheres measured from their centers. For charges spread over irregular shapes, the force must be summed over small pieces. Moving charges also feel magnetic forces, and close to conductors induced charges change the picture. Treating a material with a single εr is an approximation that works well for large-scale fields but not at the molecular scale.
The gravitational force calculator uses the same inverse-square form for masses. For circuits rather than isolated charges, see the Ohm’s law calculator, and to handle the very large and very small numbers involved, the scientific notation calculator.
Frequently asked questions
What is Coulomb's constant?
k = 1/(4πε₀) = 8.9875517923 × 10⁹ N·m²/C², from the CODATA value of the vacuum permittivity ε₀. It is the force in newtons between two 1-coulomb charges 1 meter apart, which shows how enormous a coulomb is.
How do I tell whether the force attracts or repels?
Enter the charges with their signs. If the product q₁q₂ is positive (both positive or both negative), the force is positive and repulsive. If the charges have opposite signs, the force is negative and attractive. The calculator states which in words.
Why does the force get weaker in water?
Water molecules are polar and turn to partly cancel the field of each charge. Water's relative permittivity is about 80, so ions in water attract each other about 80 times more weakly than in a vacuum. That is a big part of why salt dissolves.
What units can I use for charge?
Coulombs, millicoulombs, microcoulombs, nanocoulombs, picocoulombs and elementary charges (e = 1.602176634 × 10⁻¹⁹ C exactly). Static electricity on everyday objects is usually nanocoulombs to microcoulombs.
How does Coulomb's law compare with gravity?
Both fall off with the square of distance, but the electric force is vastly stronger. Between a proton and an electron it is about 2 × 10³⁹ times their gravitational attraction. Gravity wins on large scales only because matter is almost exactly neutral.