Coulomb's Law Calculator

The Formula

The Coulomb's Law Calculator is built around F = k q1 q2 / r^2, with k = 8.99e9 N m^2/C^2. The exact rearrangement depends on which variable you are solving for; the Coulomb's Law Calculator handles all common arrangements automatically, so you only need to enter the known values and the tool will return the unknown.

How to Use

1. Enter the known values into the input fields.
2. Select the variable you want to solve for, if the Coulomb's Law Calculator offers a reverse mode.
3. Confirm the units of each input match the labels on the fields.
4. Click Calculate to see the result.
5. Read the result in the unit the tool returns; convert manually if you need a different unit.

Frequently Asked Questions

What does this calculator compute? It evaluates F = k q1 q2 / r^2, with k = 8.99e9 N m^2/C^2, and returns the magnitude of the force between two point charges in newtons. Enter any two of charge, charge and separation, and the result follows from the formula directly.

Which units must I enter? Charge in coulombs, separation in metres. A microcoulomb is one millionth of a coulomb, written 1e-6, so 2 microcoulombs is entered as 2e-6 or 0.000002. A separation in centimetres has to be converted to metres before it goes in, because the constant carries metres squared in its units.

Why does the force not have a sign? The formula returns a magnitude. The direction follows from the charge signs you entered: two charges of the same sign push apart, two charges of opposite sign pull together. Read the sign of the product q1 times q2 to decide which case you are in.

Can the calculator handle more than two charges? Not in one step. For three or more charges, work out the force from each pair in turn and add the results as vectors. The superposition principle makes that addition valid, and a worked example below shows the arithmetic.

Why is the answer slightly different from my textbook? Textbooks often use k = 9.0e9 N m^2/C^2 (a round number for teaching) or the full 8.987551792e9 N m^2/C^2 from the defined value of the vacuum permittivity. This page uses 8.99e9. The choice moves the fourth significant figure, so a two-charge example that prints 5.394 N here prints 5.400 N with the rounded teaching value.

Does the law hold at very small separations? The classical inverse-square form is an excellent description at ordinary and atomic distances, but inside the nucleus the strong interaction dominates and this formula alone no longer predicts the behaviour. Treat any result at sub-nuclear separations as an illustration rather than a measurement.

Two point charges worked through

Take a charge of +2.0 microcoulombs and a charge of +3.0 microcoulombs held 0.10 metres apart in air. In SI units that is q1 = 2.0e-6 C, q2 = 3.0e-6 C and r = 0.10 m.

• Multiply the charges: 2.0e-6 times 3.0e-6 = 6.0e-12 C squared.
• Square the separation: 0.10 times 0.10 = 0.01 m squared.
• Multiply the charge product by k: 8.99e9 times 6.0e-12 = 0.05394.
• Divide by the squared separation: 0.05394 / 0.01 = 5.394 N.

The force is 5.394 N, and because both charges are positive the force is repulsive. Each charge pushes the other along the line joining them, away from its partner. Reverse the sign of either charge and the magnitude stays at 5.394 N while the direction flips to attraction.

The same formula runs backwards when the force is known. If the measured force is 5.394 N at 0.10 m and q1 is 2.0e-6 C, then q2 is 5.394 times 0.01 divided by the product of k and q1. That is 0.05394 divided by 17980, which gives 3.0e-6 C, the value that went in.

The inverse square at five separations

Doubling the separation quarters the force, which is the single most useful property of this law. The table holds q1 at 2.0e-6 C and q2 at 3.0e-6 C and varies only the distance.

Halving the separation from 0.10 m to 0.05 m multiplies the force by four, from 5.394 N to 21.576 N. Moving out to 1.00 m cuts the force to 0.054 N, which is one hundredth of the value at 0.10 m, exactly as the ratio of the squared separations predicts. The last column is worth keeping in mind when an answer looks surprising: a factor of two in distance never produces a factor of two in force.

A worked example at the atomic scale

Two protons held 1.0e-15 m apart give the electrostatic repulsion a sense of proportion, because the same pair also attracts by gravity.

• Charge product: the elementary charge is 1.602176634e-19 C, so e squared is 2.566970e-38 C squared.
• Squared separation: 1.0e-15 times 1.0e-15 = 1.0e-30 m squared.
• Electrostatic force: 8.99e9 times 2.566970e-38 divided by 1.0e-30 = 230.77 N.
• Gravitational force: with the gravitational constant at 6.674e-11 and the proton mass at 1.672621924e-27 kg, 6.674e-11 times 2.797664e-54 divided by 1.0e-30 = 1.867e-34 N.

Both protons carry the same sign, so the electric force pushes them apart with 230.77 N of force at that separation. The gravitational attraction between them is 1.867e-34 N, smaller by a factor of about 1.24e36. The comparison is the point of the example: for charged particles at short range, gravity is irrelevant, and any calculation that includes it while ignoring the electric force is modelling the wrong interaction.

Three charges and the net force

Superposition means the force on any one charge is the vector sum of the forces from each other charge, taken one pair at a time. Put +2.0e-6 C at x = 0, +3.0e-6 C at x = 0.10 m, and +1.0e-6 C at x = 0.30 m, and find the net force on the middle charge.

• From the charge at x = 0, separated by 0.10 m: 8.99e9 times 2.0e-6 times 3.0e-6 divided by 0.01 = 5.394 N, directed towards positive x, away from the like charge behind it.
• From the charge at x = 0.30 m, separated by 0.20 m: 8.99e9 times 1.0e-6 times 3.0e-6 divided by 0.04 = 0.674 N, directed towards negative x, away from the like charge ahead of it.
• Net force: 5.394 minus 0.674 = 4.720 N towards positive x.

The two forces oppose each other, so the net result is smaller than either contribution in this layout. Work the pairs in a fixed order, give each force a direction before adding anything, and keep the signs consistent. Adding magnitudes without directions is the most common error in a multi-charge problem, and it produces an answer that is too large rather than one that is merely mis-signed.

Conventions behind the numbers

The formula and the numbers above follow four conventions.

• The charges are treated as points. A charged sphere behaves like a point charge as long as the separation is measured between centres and the spheres do not overlap.
• The separation is measured centre to centre and entered in metres. A value entered in centimetres makes the force one hundred times too small, because the separation is squared.
• The medium is a vacuum or air. Air has a relative permittivity within about 0.06 percent of a vacuum at ordinary conditions, so the difference is invisible in the third significant figure.
• The charges are static. Moving charges also produce magnetic effects that this law does not include, so the formula describes the electric force alone.

The constant is worth stating precisely. The vacuum electric permittivity is 8.8541878128e-12 F/m in the CODATA 2018 adjustment, and k is 1 divided by 4 pi times that value, which is 8.987551792e9 N m^2/C^2. The value used on this page, 8.99e9, is that number rounded to three significant figures and 0.027 percent larger. Any answer computed here will therefore sit a little below the answer from the full-precision constant, in the fourth significant figure.

Comparing three values of the constant

The constant is quoted at different precisions in different places, and the choice is visible in the fourth significant figure of a result. The table reruns the two-charge example above, q1 = 2.0e-6 C, q2 = 3.0e-6 C and r = 0.10 m, with each value of k.

All three answers agree to three significant figures, which is why the rounded figures are safe for coursework. Reach for the full constant when the result feeds another calculation, because the 0.027 percent difference accumulates rather than cancelling.

Where the constant and the law are defined

The numerical values of the elementary charge and the vacuum electric permittivity used above are the CODATA 2018 recommended values published by the National Institute of Standards and Technology. The inverse-square form of the electrostatic force was established experimentally by Charles-Augustin de Coulomb, whose 1785 memoir Premier mémoire sur l'électricité et le magnétisme reported the torsion-balance measurements behind it.

References

• HyperPhysics, Coulomb's Law, Georgia State University, the standard treatment of the electrostatic force between point charges. http://hyperphysics.phy-astr.gsu.edu/hbase/electric/elefor.html
• Coulomb, C.-A. (1785), Premier mémoire sur l'électricité et le magnétisme, the original formulation of the inverse-square electrostatic force law.