Example 1
Two spheres on a torsion balance: 1 µC and −2 µC. The centers are 0.1 m apart, so the charges attract.
F = -1.7975 N
Coulomb's Law Calculator
Enter two point charges and the distance between their centers. The calculator finds the force from Coulomb's law F = k·q₁·q₂/r². Charges of 1 µC and −2 µC at 0.1 m give −1.7975 N. Opposite signs mean attraction. The model does not include a dielectric or a third charge.
Result
Enter both charges and the distance. The force, the entered values, and the |F|(r) chart will appear here.
The field of one charge is on the E = k·Q/r² calculator, and the field from a measured force is on the E = F/q calculator. The force between two masses is on the F = G·m₁·m₂/r² calculator.
How it works
q₁, q₂1 µC, −2 µC
r0.1 m
F-1.7975 N
The force between two point charges in vacuum is F = k·q₁·q₂/r². That is Coulomb's law. The constant k = 8.9875517923×10⁹ N·m²/C² is stored in the calculator and has no field of its own. For q₁ = 1 µC = 1×10⁻⁶ C, q₂ = −2 µC = −2×10⁻⁶ C, and r = 0.1 m, the square of the distance is 0.01 m². The product k·q₁·q₂ is −0.0179751035846 N·m², and the quotient is −1.79751035846 N. The panel rounds the result to four decimal places and shows -1.7975 N. The minus sign means attraction, because the charges have opposite signs.
The calculator uses three numbers you enter: charge q₁ in coulombs, charge q₂ in coulombs, and distance r in meters. The distance must be greater than zero. Enter a microcoulomb as 1e-6, not as 1. A comma, a period, and scientific notation mean the same number. You do not enter k. On the page for the field of one charge, the k field is optional. Here k is always the same number.
What the result panel shows. Besides the force F, the calculator reports the absolute value |F|, the stored constant k, the entered q₁ and q₂, the distance r, the square r², and the force at distance 2r. For the reference values that is -1.7975 N, an absolute value of 1.7975 N, k = 8.9875517923×10⁹ N·m²/C², q₁ = 1.0000e-6 C, q₂ = -2.0000e-6 C, r = 0.1000 m, r² = 0.0100 m², and -0.4494 N at twice the distance. The chart draws |F| against distance at fixed charges, from half of the entered r to three times that distance. The curve falls as 1/r².
Limits of the model. The calculator assumes two charges concentrated at points, vacuum, and a static interaction. It does not divide the result by a relative permittivity, and it does not add a third charge. It does not compute the field E = k·Q/r² or the potential V = k·Q/r. It also does not compute gravitational force. Newton's law has the same inverse-square shape, but it uses the constant G, masses in kilograms, and attraction only. This result does not replace the work W = F·s, because that formula assumes a force that stays constant along the path, while the Coulomb force changes with distance. A radial displacement needs an integral, which this calculator does not evaluate.
The unit switch in the header changes distance to feet and force to pounds-force. The charges stay in coulombs. The square r² on the panel is always in square meters, because the formula uses r in meters. With US units, the distance field is rewritten as 0.3281 ft, and the reference force is shown as -0.4041 lbf (-1.7973 N). The newton figure in parentheses differs in the last digit from the metric result, because the distance returns to the formula after rounding to four decimal places. Numbers in the text stay in SI as the reference. When |F| is at least 1,000,000 N, or smaller than 0.001 N, the panel prints scientific notation with a dot. A charge smaller than 0.001 C uses that notation too, for example 1.0000e-6 C.
The examples on the page report the result of this same formula. The pair 1 µC and −2 µC at 0.1 m gives -1.7975 N. Two charges of +1 µC at 0.05 m give 3.5950 N. The pair 5 nC and −5 nC at 0.01 m gives -0.0022 N. The button “Fill this example” enters the numbers from that card into the fields and refreshes the panel and the chart. The fields are empty when you open the page. A hint in a field is not an entered value.
Formula
F = k · q1 · q2 / r2
k = 1/(4πε₀). The calculator always uses 8.9875517923×10⁹ N·m²/C². The distance satisfies r > 0.
Model: Coulomb's law for the force between two point charges (C.-A. de Coulomb, Academy memoirs, 1785). The calculator finds F with the sign of the product q₁·q₂. It does not compute the electric field, and it does not accept a custom k.
Theory, derivation, and domain
Definition of the force between two charges
Coulomb's law gives the force with which two point charges act on each other in vacuum. The magnitude is F = k·q₁·q₂/r², and the direction lies on the line that joins the charges. A positive product q₁·q₂ means repulsion. A negative product means attraction. This calculator reports the number with the sign of the product. It does not draw a vector in space.
The reference values on this page are q₁ = 1 µC, q₂ = −2 µC, and r = 0.1 m. The calculator uses k = 8.9875517923×10⁹ N·m²/C². The panel result is -1.7975 N. That trio of inputs is the reference for the whole text. The unit of force is the newton.
History: Coulomb's torsion balance
In 1785, in the Histoire de l'Académie royale des sciences, Charles-Augustin de Coulomb published measurements of the force between charges on a torsion balance. The force was proportional to the product of the charges and fell as the inverse square of the distance. Henry Cavendish had reached an inverse-square law earlier. James Clerk Maxwell published Cavendish's results only in 1879.
Coulomb measured a force, not a field vector. Michael Faraday later developed the field as a state of space, and Maxwell gave it a mathematical form. Today's formula E = k·Q/r² is Coulomb's force divided by a test charge. This calculator stops at the force between two charges. The field of one charge is on a separate page.
Worked path for the reference values
Step 1. Write the charges in coulombs: 1 µC = 1×10⁻⁶ C, and −2 µC = −2×10⁻⁶ C. Step 2. Square the distance: (0.1 m)² = 0.01 m². Step 3. Multiply the constant by both charges: 8.9875517923×10⁹ × 1×10⁻⁶ × (−2×10⁻⁶) = −0.0179751035846 N·m². Step 4. Divide by r². The quotient is −1.79751035846 N, which the panel writes as -1.7975 N.
School problems often take k = 9×10⁹. For the same pair and the same distance the force is then exactly −1.8 N. This calculator has no field for k and always uses 8.9875517923×10⁹, so the panel shows -1.7975 N. The difference comes only from the constant, not from a different formula. The number in the code is a commonly printed approximation of k = 1/(4πε₀). It is not a live CODATA reading after the 2019 SI redefinition, when ε₀ became a measured quantity.
Assumptions: points, vacuum, and no dielectric
The calculator assumes that both charges are concentrated at points, that the medium is vacuum, and that the charges are at rest. It does not divide k by a relative permittivity εᵣ. In a linear dielectric the school approximation divides the force by εᵣ. That factor has to be applied outside this calculator, because there is no εᵣ field here. The size of a real sphere must be small compared with r. When the distance is comparable to the diameter of the sphere, the point model overstates the force, because the charge is spread out rather than concentrated at one point.
The calculator does not add forces from several pairs. Superposition is a sum of vectors, and the result here is one number for one pair. At r = 0 the formula is singular, and the calculator refuses the division. Moving charges, the magnetic field, and the Lorentz force are outside this model.
Sign of the charge and the inverse-square law
The sign of F is the sign of the product q₁·q₂. For the reference values the product is negative, so the charges attract, and the panel shows -1.7975 N. Two charges of +1 µC at 0.05 m give 3.5950 N and repulsion. The same two charges of +1 µC at 0.1 m give 0.8988 N. Before rounding, the force at 0.05 m is exactly four times the force at 0.1 m, because r² stands in the denominator. The panel shows both numbers after rounding to four decimal places.
Doubling the distance divides the force by four, not by two. For the reference pair, going from 0.1 m to 0.2 m changes the result from -1.7975 N to -0.4494 N. Moving from 0.1 m to 0.05 m, with the same 1 µC and −2 µC, gives -7.1900 N. Half the distance gives four times the force, because r stands in the denominator to the second power.
What this formula is not
The electric field of one charge is E = k·Q/r², and its unit is N/C. The Coulomb force has the unit N and needs two charges. The definition E = F/q uses a measured force and a test charge. This calculator does not accept that pair. The potential of a point charge is V = k·Q/r. The capacitance of a capacitor is C = Q/U. This calculator computes none of those quantities.
Newton's law of gravitation, F = G·m₁·m₂/r², has the same shape, but a different constant, different inputs, and one direction: masses attract. Charges can repel. Replacing q with mass and k with G is not a conversion this calculator performs. Charge from Q = I·t is also on a separate page. Here q₁ and q₂ are already the inputs.
Units and orders of magnitude
The coulomb and the newton are SI units, as in the BIPM brochure. Enter charge in coulombs: 1 µC = 1×10⁻⁶ C, and 1 nC = 1×10⁻⁹ C. Enter distance in meters. A length of 10 cm is 0.1 m. With US units selected in the header, the r field shows feet, and the force shows pounds-force with newtons in parentheses. A comma and a period in the fields are equivalent.
The numbers below are results of the formula for the stated inputs, not a laboratory measurement. The pair 2 µC and −1 µC at 1 m gives -0.0180 N. Two charges of 10 µC at 2 m give 0.2247 N. The pair 100 nC and −50 nC at 0.2 m gives -0.0011 N. Two school elementary charges of 1.6×10⁻¹⁹ C with opposite signs, at 1×10⁻¹⁰ m, give -2.3008e-8 N. That is the classical force between two points, not a model of an atom.
Sources
The definition, the derivation, and the history come from primary papers and textbooks: C.-A. de Coulomb, memoirs in the Histoire de l'Académie royale des sciences, 1785; J. C. Maxwell (ed.), The Electrical Researches of the Honourable Henry Cavendish, Cambridge, 1879; D. Halliday, R. Resnick, and J. Walker, Fundamentals of Physics, the chapter on Coulomb's law; H. D. Young and R. A. Freedman, University Physics, the chapter on the force between point charges; D. J. Griffiths, Introduction to Electrodynamics, chapter 2; R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics, Vol. II, chapter 4; BIPM, SI Brochure, 9th edition; NIST, the CODATA electric constant. The page is educational. The constant in the code is not a live CODATA reading.
How to use
- Enter q₁ in coulombs, for example 1e-6. A minus sign, for example -2e-6, means a negative charge. 1 µC is 1×10⁻⁶ C, not 1.
- Enter q₂ in coulombs. There is no field for k. The calculator always uses 8.9875517923×10⁹ N·m²/C².
- Enter the distance r in meters, for example 0.1. A length of 10 cm is 0.1 m. The distance must be greater than zero.
- Click Calculate, or wait for the live result. For 1 µC, −2 µC, and 0.1 m the calculator shows -1.7975 N.
- Check the panel: the absolute value, the entered charges and distance, the square r², and the force at distance 2r. Clear removes the inputs and hides the panel.
Symbol glossary
The rows below describe the symbols in F = k·q₁·q₂/r² and the form fields where you enter their values. The constant k is stored and fixed. For 1 µC and −2 µC at 0.1 m the panel shows -1.7975 N.
- F
- Force in newtons, with the sign of the product q₁·q₂. For 1 µC and −2 µC at 0.1 m the panel shows -1.7975 N.
- k
- The constant 1/(4πε₀). The calculator always uses 8.9875517923×10⁹ N·m²/C². There is no field in which to change it.
- q₁
- The first charge in coulombs. The reference value is 1×10⁻⁶ C. This is not the single Q from the field formula.
- q₂
- The second charge in coulombs. The reference value is −2×10⁻⁶ C. Its sign relative to q₁ decides attraction or repulsion.
- r
- Distance between the centers, in meters. It must be greater than zero. In the formula this distance is squared.
Worked examples
Example 2
Two positively charged spheres, 1 µC each, 0.05 m apart. That is repulsion. The same pair at 0.1 m gives 0.8988 N, and half that distance gives the force required by the 1/r² law.
F = 3.5950 N
Example 3
Two small spheres carrying 5 nC and −5 nC. The centers are 0.01 m apart. At that short distance the point model is useful only when the spheres are much smaller than r.
F = -0.0022 N
Example 4
Charges of 2 µC and −1 µC whose centers are 1 m apart. The attractive force is less than two hundredths of a newton.
F = -0.0180 N
Example 5
Two charges of 10 µC at 2 m. The signs are the same, so the spheres repel with a force of 0.2247 N.
F = 0.2247 N
Example 6
A model of two rods: 100 nC and −50 nC, with centers 0.2 m apart. A rod is not a point, so the result is a model, not a bench measurement.
F = -0.0011 N
Example 7
Two school elementary charges of 1.6×10⁻¹⁹ C with opposite signs, 1×10⁻¹⁰ m apart. The result is the classical force between two points, not a description of an atom.
F = -2.3008e-8 N
Example 8
Charges of 3 µC and −4 µC at 0.5 m. The product is negative, so the force on the panel has a minus sign.
F = -0.4314 N
Example 9
Two charges of 0.5 µC at 0.15 m. Like signs mean repulsion.
F = 0.0999 N
Example 10
Charges of 1 µC and −1 µC at 0.2 m give -0.2247 N. The same pair at 0.4 m gives -0.0562 N, one quarter of that force. The button fills the distance 0.2 m.
F = -0.2247 N
Common questions
What does this calculator compute, and from which formula?
The calculator finds the force between two point charges in vacuum from F = k·q₁·q₂/r². For 1 µC and −2 µC at 0.1 m, the panel shows -1.7975 N.
Can the Coulomb constant be changed?
No. The calculator always uses 8.9875517923×10⁹ N·m²/C². The page for the field of one charge has an optional k field. This page does not.
Which units should q₁, q₂, and r use?
Enter both charges in coulombs and the distance in meters. The result is in newtons. 1 µC is 1×10⁻⁶ C. With US units selected in the header, distance is in feet and force is in pounds-force, with newtons in parentheses. The charges stay in coulombs.
What happens when the distance is zero or negative?
The formula is singular at r = 0, so the calculator does not divide. The distance must be greater than zero. Both charges must be numbers. A charge may be negative.
How is this formula different from E = k·Q/r²?
Here the result is the force between two charges, in newtons. The field E = k·Q/r² belongs to one charge and has the unit N/C. A separate calculator evaluates that formula, and k can be entered there.
How is this formula different from E = F/q?
Here the force comes from two charges and a distance. The definition E = F/q uses a measured force and a test charge. This calculator does not accept that pair.
What does the sign of the force mean?
The sign of F is the sign of the product q₁·q₂. Opposite signs mean attraction, and like signs mean repulsion. For 1 µC and −2 µC at 0.1 m the panel shows -1.7975 N.
Why does half the distance not double the force?
Because the distance stands in the denominator as a square. Half of r multiplies the force by four. For the reference pair, going from 0.1 m to 0.05 m changes the result from -1.7975 N to -7.1900 N. Doubling r, from 0.1 m to 0.2 m, gives -0.4494 N.
Does the calculator include a dielectric or a third charge?
No. The model is one pair of point charges in vacuum. The calculator does not ask for a relative permittivity εᵣ. Superposition of several charges needs a sum of vectors, not one product.
Are a comma and a period equivalent?
Yes. The calculator reads 0.1 and 0,1 as the same distance, and it reads 1e-6 as 1×10⁻⁶. The fields are empty when you open the page. A hint in a field is not an entered value.
Why is the result for 1 µC and −2 µC at 0.1 m not 1.8 N?
Because the stored constant k is 8.9875517923×10⁹, not the school value 9×10⁹. The quotient is -1.7975 N. The value −1.8 N appears only with k = 9×10⁹, and this calculator does not accept that k.
Is gravitational force computed with this same formula?
No. Newton's law has the shape F = G·m₁·m₂/r², but it uses the constant G and masses in kilograms, and the direction is always attractive. This calculator accepts coulombs and the Coulomb constant.
How do I calculate the force with F = k·q₁·q₂/r²?
Multiply the constant k by the product of the charges and divide by the square of the distance. At 1×10⁻⁶ C, −2×10⁻⁶ C, and r = 0.1 m the result is about −1.7975 N.
How do I enter 1 µC and −2 µC in coulombs?
1 µC is 1×10⁻⁶ C, and −2 µC is −2×10⁻⁶ C. In the fields use 0.000001 and -0.000002, or 1e-6 and -2e-6.
What does a negative result of −1.7975 N mean?
Opposite charge signs mean attraction. The panel also shows the absolute value |F| ≈ 1.7975 N.
How is this page different from the point-charge field calculator?
Here the result is the force F between two charges. The companion page computes E = k·Q/r² at a point without a second charge in the fields.
Does the calculator divide the force by permittivity εᵣ?
No. The constant k is built in as for vacuum. Water and other dielectrics have no εᵣ field on this page.
Sources
The definition, the unit, the history, and the derivation come from established sources in physics and metrology.
- Wikipedia — Coulomb's law
- Feynman Lectures on Physics, Vol. II, chapter 4 — Electrostatics
- BIPM — The International System of Units (SI), 9th edition
- NIST — CODATA electric constant ε₀
- Coulomb, C.-A., memoirs on electricity and magnetism, Histoire de l'Académie royale des sciences, 1785.
- Griffiths, D. J., Introduction to Electrodynamics, chapter 2.
Page updated November 3, 2026.