Coulomb’s Law & Electric Field Calculator
Work out the electric force between two charges. Solve F = kq₁q₂/r² for the force, a charge or the distance, find the field and potential of a point charge, and add up the field of two charges at any point.
How to Use
- Pick what to find: the Force between two charges, the second Charge, the Distance, the Field and potential of one charge, or the field of Two charges at a point.
- Enter each charge with its sign and choose a unit: coulombs, mC, µC, nC, pC or e (elementary charges). A minus sign makes it negative.
- Enter the distance or force with their units. To find a charge, say whether the two charges attract or repel.
- For two charges, type the x and y position of each charge and of the point you are interested in, all in one length unit.
- Leave the relative permittivity at 1 for a vacuum (air is 1.0006), or enter a material’s value to weaken the force inside it.
- Press a preset to load an example, and read Show Work for every conversion and step.
Worked Example
Two equal charges. Two +1 µC charges sit 10 cm (0.1 m) apart. F = k × q₁ × q₂ ÷ r² = 8.98755 × 10⁹ × (1 × 10⁻⁶)² ÷ 0.1² = 0.8988 N. Both charges are positive, so they repel, and the stored energy is U = kq₁q₂ ÷ r = 0.08988 J.
The hydrogen atom. A proton (+e) and an electron (−e) at the Bohr radius, 5.29177 × 10⁻¹¹ m: F = 8.98755 × 10⁹ × (1.602177 × 10⁻¹⁹)² ÷ (5.29177 × 10⁻¹¹)² = 8.239 × 10⁻⁸ N of attraction. The proton’s field there is 5.142 × 10¹¹ V/m and the potential energy is −27.21 eV.
The common mistake: leaving the distance in centimetres. For the first pair, typing r = 10 instead of 0.1 m gives 8.988 × 10⁻⁵ N, ten thousand times too small, because the error is squared. Forgetting to square r at all gives 0.08988 N, ten times too small. Convert to metres first, then square: the right answer is 0.8988 N.
Show Work
Formulas
Coulomb’s Torsion Balance
Charles-Augustin de Coulomb, a French military engineer, measured the electric force directly in 1785. His torsion balance hung a light needle with a small charged ball from a fine wire; a second charged ball pushed it round, and the twist of the wire, which he had already shown is proportional to the angle, measured the force. Halving the distance between the balls made the twist about four times larger: an inverse-square law, like Newton’s gravity.
He was not the first to suspect it. Joseph Priestley guessed the inverse square in 1767 from the fact that there is no electric force inside a charged hollow conductor, and Henry Cavendish confirmed it with a cleverer experiment in the 1770s but never published; his notes were edited by James Clerk Maxwell in 1879. Michael Faraday later pictured the force as lines of field filling the space between charges, the idea behind the field lines drawn above.
The SI unit of charge is named after Coulomb. Since the 2019 redefinition of the SI the elementary charge is exactly 1.602176634 × 10⁻¹⁹ C, and the vacuum permittivity ε₀, once fixed by definition, is now a measured constant: this calculator uses the NIST CODATA 2022 value.
About This Tool
This calculator solves Coulomb’s law for the force, the second charge or the distance, keeping the signs so it can tell you whether the charges attract or repel. It also gives the field and potential of a single point charge, and for two charges anywhere in a plane it adds their fields as vectors, adds their potentials, and traces the field lines so you can see where they cancel. Charges can be entered in coulombs, mC, µC, nC, pC or elementary charges, and a relative permittivity lets you put the charges in a material such as water or oil.
The formulas are for point charges (or uniformly charged spheres, measured from their centres) that are not moving. Everything runs in your browser; nothing you enter is sent anywhere.
Related tools: Gravity Calculator, Inverse-Square Law Calculator, and Photon Energy & Wavelength Calculator.
Frequently Asked Questions
What is Coulomb’s law?
The force between two point charges is F = k × q₁ × q₂ ÷ r², with k = 8.988 × 10⁹ N·m²/C². Two +1 µC charges 10 cm apart push each other away with 0.8988 N. Like charges repel and unlike charges attract, and because of the r² the force at 20 cm is a quarter as big: 0.2247 N.
Where does k = 8.988 × 10⁹ come from?
k = 1 ÷ (4π ε₀), where ε₀ = 8.8541878188 × 10⁻¹² F/m is the vacuum permittivity (NIST CODATA 2022). That gives k = 8.9875517862 × 10⁹ N·m²/C²: two charges of 1 C each, 1 m apart, would push with almost 9 billion newtons, which is why everyday charges are measured in µC and nC.
What is the difference between electric field and electric potential?
The field E = kq ÷ r² is the force per coulomb on a test charge and has a direction; the potential V = kq ÷ r is the energy per coulomb and is just a number. 10 cm from a 1 nC charge, E = 898.8 N/C (the same as V/m) pointing away from the charge, and V = 89.88 V.
How much stronger is the electric force than gravity?
Much stronger. In a hydrogen atom, the proton and electron 0.0529 nm apart (the Bohr radius) attract electrically with 8.239 × 10⁻⁸ N, but their gravity is only 3.632 × 10⁻⁴⁷ N, a ratio of 2.27 × 10³⁹. Gravity only wins for big objects because their positive and negative charges almost exactly cancel.
How many electrons make one microcoulomb?
One electron carries 1.602176634 × 10⁻¹⁹ C (exact in the SI since 2019), so 1 µC is 1 × 10⁻⁶ ÷ 1.602176634 × 10⁻¹⁹ = 6.242 × 10¹² extra or missing electrons. That sounds like a lot, but it is a tiny fraction of the electrons in any object you can hold.
How do I use the Coulomb’s Law & Electric Field Calculator?
Just type your numbers. The answer shows up right away — there is no button to press. Change anything and it updates by itself.
Do I need to install or sign up for anything?
Not at all — it runs in the browser with nothing to install and no account. After it loads once, it even works without an internet connection.
Is my information private?
Yes. Everything happens in your browser. Nothing you type is sent to a server or saved anywhere.
Common Use Cases
Physics homework
Find how far apart two 1 µC charges must be to push with 1 N: r = √(k q₁q₂ ÷ F) = 9.48 cm, with every step and unit conversion shown.
Atomic physics
Proton and electron at the Bohr radius: 8.239 × 10⁻⁸ N, a field of 5.142 × 10¹¹ V/m and a potential energy of −27.21 eV, one hartree.
Dipole fields
+1 nC and −1 nC 10 cm apart give 2,542 N/C at a point 5 cm above the middle, pointing from the + side to the − side, while the potential there is exactly 0 V.
Finding a null point
+4 nC at x = 0 and −1 nC at x = 3 cm: the two fields cancel at x = 6 cm, where E = 0 but V is still 299.6 V.
Charges in water
Water at 20 °C has a relative permittivity of about 80, so the 0.8988 N between two 1 µC charges at 10 cm falls to 0.0112 N, which is why salts dissolve.
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