Calculate the electrostatic force between two charges with Coulomb’s law, or solve for either charge or the distance, in common units and media.
Coulomb's Law Formula
F = k * |q1 * q2| / r^2
Where:
- F is the electrostatic force between the two charges, in newtons (N).
- k is Coulomb's constant, equal to 8.9875 x 10^9 N*m^2/C^2, often rounded to 8.99 x 10^9.
- q1 and q2 are the two charges, in coulombs (C).
- r is the distance between the centers of the two charges, in meters (m).
The force points along the line joining the two charges. When q1 and q2 have the same sign, the result is positive and the charges repel. When they have opposite signs, the charges attract. The calculator uses the magnitude of the product, so the number it reports is the strength of the force, and the sign of each charge tells you the direction.
When the charges sit in a material instead of a vacuum, replace k with k divided by the relative permittivity of that material. The relative permittivity (also called the dielectric constant) is written as the Greek letter epsilon with a subscript r. The full constant comes from k = 1 / (4 * pi * epsilon_0), where epsilon_0 is the permittivity of free space, 8.854 x 10^-12 C^2/(N*m^2).
Choose the value to solve for. Inverse charge calculations determine magnitude only; force magnitude does not determine charge sign:
|q1| = F * r^2 * epsilon_r / (k * |q2|)
r = sqrt(k * |q1 * q2| / (epsilon_r * F))
Choose what you want to solve for, set the charge and distance units, and the calculator hides the field it is solving for so only the values you need to enter are shown. Optional settings let you set the relative permittivity of the medium and decimal places. A zero force can give a unique zero charge when the other charge is nonzero, but zero-force distance cases have no unique finite solution.
Charge Units and Dielectric Constants
Charges in physics problems are usually given in microcoulombs or nanocoulombs rather than whole coulombs. Use this table to convert your value to coulombs before checking a by-hand result.
| Unit | Symbol | Value in coulombs |
|---|---|---|
| Coulomb | C | 1 |
| Millicoulomb | mC | 0.001 |
| Microcoulomb | uC | 0.000001 |
| Nanocoulomb | nC | 0.000000001 |
| Picocoulomb | pC | 0.000000000001 |
For ideal point charges in a homogeneous, linear dielectric, divide the vacuum force by its relative permittivity. The values below are illustrative approximations; actual permittivity depends on material and conditions.
| Medium | Relative permittivity (approx.) |
|---|---|
| Vacuum | 1 |
| Air | 1.0006 |
| Paper | 3.5 |
| Glass | 5 to 10 |
| Water | 80 |
Example Problems
Example 1. Two charges of +2 uC and -3 uC sit 5 cm apart in air. Convert the charges to coulombs (2 x 10^-6 C and 3 x 10^-6 C) and the distance to meters (0.05 m). The force is F = (8.99 x 10^9)(2 x 10^-6)(3 x 10^-6) / (0.05)^2 = 21.57 N. Because the charges have opposite signs, the force is attractive.
Example 2. You want the distance at which two +1 uC charges produce a force of 10 N. Rearrange to r = sqrt(k * |q1 * q2| / (epsilon_r * F)) = sqrt((8.99 x 10^9)(1 x 10^-6)(1 x 10^-6) / 10) = 0.03 m, which is about 3 cm.
Frequently Asked Questions
Is the force attractive or repulsive?
Like charges repel and opposite charges attract. The magnitude of the force is the same in both cases and follows the same formula. Only the direction changes, which you read from the signs of the two charges: two positives or two negatives push apart, while a positive and a negative pull together.
What is Coulomb's constant?
Coulomb's constant k is 8.9875 x 10^9 N*m^2/C^2, usually rounded to 8.99 x 10^9. It is defined as 1 / (4 * pi * epsilon_0), where epsilon_0, the permittivity of free space, is 8.854 x 10^-12 C^2/(N*m^2). This constant applies in a vacuum, and air is close enough that you can use the same value for most problems.
How does the surrounding material change the force?
A material between the charges reduces the force by its relative permittivity. If two charges feel a force of 20 N in a vacuum and you place them in water, which has a relative permittivity near 80, the force drops to about 0.25 N. Set the relative permittivity in the advanced options to model this.
