Coulomb's Law Calculator

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How to Use the Coulomb's Law Calculator

1. 1. Pick the mode. Force between two charges solves Coulomb's law for the force, either charge or the distance. Net force and field from several charges sums up to six point charges at positions you give. Electric field of one charge solves E = kq/r².
2. 2. Enter charges with their signs. A negative charge takes a minus sign. Each box has a unit menu, from coulombs down to picocoulombs and elementary charges (e), and distances run from kilometres to picometres and ångströms. Scientific notation such as 1.6e-19 works.
3. 3. Choose the medium. Vacuum and air are almost identical; water, with a relative permittivity near 80, cuts the force by that factor. You can also type your own εᵣ.
4. 4. Read the force and the working. A positive force is repulsive and a negative one attractive. The calculator shows every term with your numbers, and a diagram of the charges.

Coulomb's Law

F = kₑ q₁ q₂ / (εᵣ r²)

E = kₑ q / (εᵣ r²)

V = kₑ q / (εᵣ r)

kₑ = 1 / (4π ε₀) = 8.9876 × 10⁹ N·m²/C²

Charles-Augustin de Coulomb measured this force with a torsion balance in 1785. Two point charges exert equal and opposite forces on each other along the line joining them, proportional to the product of the charges and inversely proportional to the square of the distance. Like charges repel and opposite charges attract. The constant kₑ comes from the vacuum permittivity, ε₀ = 8.8541878188 × 10⁻¹² F/m (CODATA 2022). The elementary charge is exactly 1.602176634 × 10⁻¹⁹ C (values from the NIST CODATA tables). OpenStax University Physics covers the law in Volume 2, section 5.3.

The coulomb is a very large charge: two charges of 1 C placed 1 m apart would repel with 8.988 × 10⁹ N. Everyday static electricity involves nanocoulombs to microcoulombs, so the default examples use those sizes.

Worked Examples

1. Two +1 µC charges, 1 m apart. F = 8.9876 × 10⁹ × (10⁻⁶)² / 1² = 0.008988 N: a repulsion of about 9 mN.
2. +2 nC and −3 nC, 5 cm apart. F = −2.157 × 10⁻⁵ N. The minus sign means attraction, with a magnitude of about 22 µN.
3. The hydrogen atom. A proton and an electron separated by the Bohr radius (5.29177 × 10⁻¹¹ m) attract each other with 8.239 × 10⁻⁸ N, or about 82 nN. At that distance the proton's field is 5.142 × 10¹¹ N/C and its potential is 27.21 V.
4. In water. Two 1 µC charges 1 m apart in water (εᵣ = 80.2) repel with 1.121 × 10⁻⁴ N, 80 times weaker than in vacuum. In air it is 0.008982 N, a difference of 0.06%.
5. A charge between two others. A +1 µC test charge at the origin, with +3 µC at (0.3, 0) m and −2 µC at (0, 0.4) m. The +3 µC charge pushes it with 0.2996 N in the −x direction, and the −2 µC charge pulls it with 0.1123 N in the +y direction. The net force is 0.3200 N at 159.4° from the +x axis.
6. Field above two equal charges. Two +1 µC charges at (±0.5, 0) m give a field at (0, 0.5) m of 25,420 N/C pointing straight up: the horizontal parts cancel and the vertical parts add.

Adding Forces From Several Charges

Electric forces superpose: the force on a charge from several others is the vector sum of the forces from each alone. The multi-charge mode adds each charge's field at a point as a vector, kₑ q r̂ / r², then multiplies by a test charge q₀ to get the force. It also gives the potential at that point, which adds as a plain number.

Take a dipole of +1 nC at (−5, 0) cm and −1 nC at (5, 0) cm. At (20, 0) cm the exact field is 255.6 N/C, pointing toward the negative charge (180°). The textbook dipole formula 2kp/r³ gives 224.7 N/C, 12% too low, because 20 cm is not far from a 10 cm dipole. At 2 m the two agree to within 0.2%.

Assumptions and Limits

• Point charges at rest. Spheres with spread-out charge act as point charges when measured from their centres. Moving charges also produce magnetic forces.
• Uniform medium. The factor εᵣ models a medium that fills all the space; real dielectrics near a boundary are more complex.
• No induced charge. Conductors and polarisable objects rearrange their charge in a field, so a charged ball near a metal sheet behaves differently.
• Not for atomic nuclei. At the smallest distances quantum effects take over.

Frequently Asked Questions

What is Coulomb's law?

Coulomb's law gives the electrostatic force between two point charges: F = kₑ q₁ q₂ / r², where kₑ = 8.9876 × 10⁹ N·m²/C², q₁ and q₂ are the charges in coulombs and r is the distance between them in metres. Like charges repel and opposite charges attract.

How do you find the net force on a charge from several others?

Work out the force from each other charge separately, as a vector pointing along the line between them, then add the x and y components. Equivalently, find the electric field at the charge's position by adding each source's field as a vector and multiply by the charge. The several-charges mode does this for you.

What is the difference between electric force and electric field?

The field E at a point is the force per unit of positive test charge, E = F/q₀, in N/C or V/m. It exists whether or not a charge is there. The force on a charge q placed in it is F = qE.

Why is the force negative for opposite charges?

The sign carries the direction. With q₁q₂ positive (like charges) the force is positive and repulsive; with q₁q₂ negative it is negative and attractive. The magnitude is the same either way, which is why the calculator reports both.

What is relative permittivity and how does it change the force?

Relative permittivity εᵣ, the dielectric constant, tells you how much a medium reduces the electric force: F = kₑ q₁ q₂ / (εᵣ r²). It is 1 in vacuum, about 1.0006 in air and about 80 in water at 20 °C, which is why salt dissolves in water: the ions' attraction is 80 times weaker.