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Coulomb's Law Calculator - Electric Force Between Charges

Enter any three known values to solve Coulomb's law for electric force, charge q1, charge q2, or distance r. The calculator also labels attraction or repulsion, shows the force on q2 due to q1, and keeps all calculations local in your browser.

F = k q1 q2 / (εr r^2) | enter any 3 known values to solve the missing one

Inputs

Choose the missing variable. Fill the other three values below.
Used for force output, and as the input unit when force is known.
Use positive or negative values. Example: 1 µC.
Opposite signs attract. Like signs repel.
Center-to-center distance. Must be greater than zero.
A larger εr weakens the force in this idealized model.
Used for displayed output only.

Tip: Press Ctrl/Cmd + Enter to calculate. The URL updates with your values, but no inputs are sent to a server.

Live recalculation is on after each valid input change.

Results

Enter values and calculate to see the interaction summary.
Force Magnitude - Primary answer
Direction - Attractive or repulsive
Electric Field from q1 at q2 - Magnitude only; sign noted below
Potential Energy - Electrostatic potential energy
    Equation used-
    Signed force on q2-
    q1 in coulombs-
    q2 in coulombs-
    Distance in meters-
    Relative permittivity-
    Inverse-square factor-

    Step-by-step solution

    Enter values and calculate to see the substitutions.

    Coulomb's Law Formula, Solve Modes, and Sign Convention

    Coulomb's law describes the electrostatic force between two stationary charges. In SI units the magnitude is F = k |q1 q2| / (εr r2), where k is Coulomb's constant, q1 and q2 are the charges in coulombs, r is their center-to-center separation in meters, and εr is the relative permittivity of a uniform medium. The inverse-square structure is the key scaling rule: double the separation and the force falls to one quarter; cut the separation in half and the force becomes four times larger. The interaction acts along the line joining the charges.

    You can solve the same equation four ways: force from two charges and distance, q1 from force, q2 from force, or distance from force and both charges. When solving for an unknown charge, the force magnitude alone does not determine the sign, so the calculator asks whether the intended interaction is attractive or repulsive.

    The sign of q1 q2 determines whether the force is attractive or repulsive. The reported magnitude is always positive. The signed scalar uses a specific diagram convention: q1 is on the left, q2 is on the right, and positive force on q2 points to the right in the +x direction. Opposite charges usually give a negative signed force on q2 in that diagram because q2 is pulled back toward q1. The calculator also reports the electric field due to q1 at the location of q2, using E = k q1 / (εr r2), and the electrostatic potential energy using U = k q1 q2 / (εr r).

    Those extra outputs matter because force alone does not tell the whole story. The electric field describes how strongly space around a charge would push on a positive test charge, while potential energy tells you whether the charge configuration is energetically bound or resistant to compression. Negative potential energy corresponds to an attractive configuration; positive potential energy corresponds to a repulsive one. If you are comparing Coulomb's law with the inverse-square ideas in our Gravity & Newton's Second Law tool or force scaling in the Centripetal Force Calculator, the mathematical pattern will look familiar even though the physical source differs.

    This page uses the standard idealization of point charges in electrostatic equilibrium. That is appropriate for homework checks, back-of-the-envelope reasoning, and many clean lab examples. It is not a full simulator for finite-sized conductors, corona discharge, dielectric breakdown, polarization effects, or moving charges. Very small separations, very large charges, or real engineered assemblies can violate the assumptions behind the simple formula. Treat the result as an ideal electrostatics estimate rather than design approval or safety guidance.

    F = k q1 q2 / (εr r^2) |F| = k |q1 q2| / (εr r^2) |q1| = |F| εr r^2 / (k |q2|) |q2| = |F| εr r^2 / (k |q1|) r = sqrt(k |q1 q2| / (εr |F|)) E(q1 at q2) = k q1 / (εr r^2) U = k q1 q2 / (εr r) k = 8.9875517862 x 10^9 N·m^2/C^2

    Practical examples

    Two microcoulomb charges, 10 cm apart

    +1 µC and -1 µC at 0.10 m in air attract with |F| about 0.899 N. The signed force on q2 is negative in the page diagram because q2 is pulled left toward q1.

    Nanocoulomb static charges

    +25 nC and +40 nC at 5 cm repel with |F| about 0.00359 N, or about 359 dyn. Small static charges can still matter at short range.

    Proton-electron scale

    At a Bohr-radius distance of 0.529 angstrom, a proton and electron attract with |F| about 8.24 x 10-8 N. This is an ideal classical comparison, not a full quantum atom model.

    Solve distance from a target force

    If q1 = +2 µC, q2 = -3 µC, and |F| = 1 N in air, the center-to-center distance is about 0.232 m. Use distance mode for this rearrangement.

    Solve an unknown charge

    If q2 = -1 µC, r = 10 cm, and |F| = 0.899 N, an attractive interaction gives q1 about +1 µC. A repulsive target would choose the opposite sign.

    Assumption note: The dielectric presets are simple relative-permittivity approximations. They are useful for educational comparison, but real materials can be frequency-dependent, temperature-dependent, and geometry-dependent.

    Source and Review

    Last updated: June 23, 2026.

    Reviewed by: Starlight Robotics for formula use, unit conversions, and calculator behavior.

    Constant source: Coulomb's constant is derived from k = 1/(4πε0) using CODATA/NIST fundamental constants. See the NIST Reference on Constants, Units, and Uncertainty.

    Assumptions: point charges, electrostatic conditions, center-to-center distance, and a uniform isotropic medium represented by εr.

    Coulomb's Law Notes and Common Mistakes

    Convert charge units first

    Microcoulombs and nanocoulombs must become coulombs before the formula is evaluated. Forgetting that 1 µC = 1 x 10-6 C changes the answer by a factor of one million.

    Units

    Use center-to-center distance

    The r in Coulomb's law is the distance between charge centers or idealized point locations. Surface-to-surface gaps are usually not the right distance for spheres or electrodes.

    Distance

    Separate magnitude from sign

    A negative signed scalar is not a negative magnitude. It means the force points opposite the chosen positive direction in the diagram.

    Direction

    Distance dominates changes

    If distance doubles, force drops to one quarter. If distance is cut in half, force becomes four times larger. Many answer surprises come from the r2 term.

    Inverse square

    Moving charges need more physics

    Coulomb's law is an electrostatic model. Moving charges, currents, radiation, magnetic forces, discharge, and complex dielectrics can require a broader electromagnetic model.

    Limits

    Frequently Asked Questions

    How do I calculate force between two charges?

    Convert q1 and q2 to coulombs, convert r to meters, then use |F| = k|q1q2| / (εrr2). The calculator does the unit conversions and shows the substituted steps.

    Can Coulomb force be negative?

    The force magnitude is not negative. The signed scalar can be negative after a direction convention is chosen. Here q1 is on the left, q2 is on the right, and positive force on q2 points right.

    What is Coulomb's constant?

    Coulomb's constant is approximately 8.9875517862 x 109 N·m2/C2. It is the vacuum proportionality constant in Coulomb's law and equals 1/(4πε0).

    What does r mean?

    r is the center-to-center separation between the two charges. In a point-charge problem, it is the distance between the two point locations.

    How do I solve for distance?

    Use r = sqrt(k|q1q2| / (εr|F|)). Choose distance mode, enter both charges and the target force magnitude, then choose the distance unit you want for the answer.

    How do I solve for an unknown charge?

    Use |q1| = |F|εrr2 / (k|q2|) or the matching q2 form. Because force magnitude does not determine sign by itself, choose whether the interaction should be attractive or repulsive.

    What units should q1 and q2 use?

    You can enter C, mC, µC, nC, pC, fC, elementary charge, or statC/esu. Internally, the calculator converts charge to coulombs before applying the equation.

    When is Coulomb's law invalid?

    The simple formula can fail for moving charges, extended conductive shapes, strong fields that cause discharge, nonuniform media, and quantum or relativistic situations. It is an ideal electrostatics model.

    Does this calculator show attraction or repulsion?

    Yes. It uses the signs of q1 and q2 to label the interaction. Opposite signs attract; like signs repel.

    What unit should I use for charge?

    Use the unit your problem gives, then let the calculator convert it. For homework problems, microcoulombs and nanocoulombs are common; for electrostatic cgs problems, statC/esu may appear.

    Why is the force so small for tiny charges?

    In everyday static-electricity problems the charges are often in the nanocoulomb or picocoulomb range. Those are extremely small amounts of charge, so even with a large constant k, the final force may still be modest unless the distance is also very small.

    Can I use this for real hardware or safety limits?

    Only as an ideal estimate. Real hardware may involve distributed charge, conductive geometry, field concentration, discharge paths, humidity, insulation failure, and moving charges. Use a detailed engineering model when the consequences matter.

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