Coulomb's Law Calculator

The Coulomb's Law Calculator computes the electrostatic force between two point charges from \( F = k_e \dfrac{q_{1} q_{2}}{\varepsilon_{r}\, r^{2}} \). Pick the unknown — force F, either charge, or the separation r — and type the other three quantities in any common unit (coulombs, microcoulombs, picocoulombs, elementary charges e, or even CGS statcoulombs). The calculator returns the force magnitude, the attract-or-repel direction (with arrows that flip in the live SVG), the electric field at the second charge's position, the electrostatic potential energy, the dramatic Coulomb-to-gravity ratio that explains why chemistry is electric, and a step-by-step LaTeX derivation. A dielectric-medium selector handles vacuum, air, water, glass, silicon, and a free-form custom εᵣ, so you can model how a surrounding material screens the force.

How to Use This Coulomb's Law Calculator

1. Select the unknown in the Solve for dropdown — F, q₁, q₂, or r. The matching input field hides itself automatically, and the remaining three become required.
2. Enter the two charges with their signs. Positive and negative numbers are both accepted, and you can mix units (e.g., q₁ in nanocoulombs and q₂ in elementary charges).
3. Enter the separation r in any of the supported units, from picometers and angstroms for atomic problems to kilometers for storm-cloud examples.
4. Pick the surrounding medium. Vacuum and air are nearly identical (εᵣ ≈ 1); water at εᵣ ≈ 80 cuts the force by almost two orders of magnitude. For unusual dielectrics, choose Custom εᵣ and type the value.
5. Press Calculate and read the result, the attract-or-repel visualization, the F_electric ⁄ F_gravity ratio, the step-by-step derivation, and any contextual notes.

What Makes This Calculator Different

Coulomb's Law in One Line

Two point charges q₁ and q₂ separated by r in a medium of relative permittivity εᵣ exert a force on each other given by

\[ F \;=\; k_{e}\,\dfrac{q_{1}\,q_{2}}{\varepsilon_{r}\,r^{2}} \]

where Coulomb's constant \(k_{e} = 1/(4\pi\varepsilon_{0}) \approx 8.9875 \times 10^{9}\) N·m²/C². If the product \(q_{1}\,q_{2}\) is positive the force is repulsive (pushing the charges apart along the line that joins them); if the product is negative, the force is attractive. The force on each charge has the same magnitude — Newton's third law.

The corresponding electric field of q₁ at q₂'s location is

\[ E \;=\; k_{e}\,\dfrac{q_{1}}{\varepsilon_{r}\,r^{2}} \]

and the electrostatic potential energy stored in the configuration is

\[ U \;=\; k_{e}\,\dfrac{q_{1}\,q_{2}}{\varepsilon_{r}\,r} \]

U is positive for same-sign pairs (energy must be supplied to bring them together) and negative for opposite-sign pairs (energy is released as they approach).

Worked Example: Hydrogen Atom

Consider the electron–proton pair inside a hydrogen atom in its ground state, separated by the Bohr radius \(r \approx 5.29 \times 10^{-11}\) m.

• \( F = (8.9875 \times 10^{9})(1.6 \times 10^{-19})(1.6 \times 10^{-19}) / (5.29 \times 10^{-11})^{2} \approx 8.24 \times 10^{-8}\) N — about 82 nanonewtons.
• Gravitational pull on the same pair: \( F_{g} = G\,m_{e}\,m_{p}/r^{2} \approx 3.6 \times 10^{-47}\) N.
• Ratio: \( F/F_{g} \approx 2.3 \times 10^{39} \). The electromagnetic force is ~10³⁹ times stronger than gravity at every scale where both act — which is why atoms exist and stones do not fly apart.

Worked Example: Two Charged Spheres

Two small conducting spheres each carry +5 µC and sit 1 m apart in air.

• \( F = k\,q_{1}\,q_{2}/r^{2} = (8.9875 \times 10^{9})(5 \times 10^{-6})^{2} / 1^{2} \approx 0.225\) N — roughly the weight of a paperclip.
• The force is repulsive because both charges are positive, so the spheres push apart along the line that joins them.
• The electric field one sphere creates at the other's centre is \( E = kq/r^{2} \approx 44 950\) V/m — strong but well below the dry-air breakdown of about 3 × 10⁶ V/m.

Same Charges, Different Medium: Ionic Bond in Water

A Na⁺ and a Cl⁻ ion sit at the typical NaCl bond length \(r \approx 2.82\) Å.

• In vacuum: \( F \approx 2.9 \times 10^{-9}\) N — a strong atomic-scale attraction worth several electronvolts of potential energy.
• In water (εᵣ ≈ 80.4): the same geometry gives \( F \approx 3.6 \times 10^{-11}\) N — about 80× weaker. The dielectric screening is large enough that thermal motion (kT ≈ 25 meV at 25 °C) can break the bond, which is exactly why ionic salts dissolve so readily in water.

Centripetal Force vs Centrifugal Force vs Coulomb Force

Coulomb force is one of the four real inward (or outward) forces nature offers. When you put a charged particle on a circular path (a particle accelerator, an electron in an atomic orbit in the semi-classical picture), the Coulomb force becomes the centripetal force that bends the trajectory into a circle. By contrast, the "centrifugal" feeling is a fictitious outward push that only exists in a rotating reference frame — the real inward pull is still Coulomb's.

Where the Force Actually Comes From: Physical Examples

Why εᵣ < 1 Is Not Allowed

Vacuum has the smallest possible permittivity. A material can only weaken the Coulomb force by aligning its bound charges so they partially cancel the source field — it can never strengthen the force at static frequencies. The calculator therefore requires εᵣ ≥ 1; entering a smaller value raises a validation error. For high-frequency or anomalous-dispersion problems where εᵣ < 1 can appear, Coulomb's law in this simple form no longer applies.

Frequently Asked Questions

What is Coulomb's law formula?
F = k · q₁ · q₂ / r², where k ≈ 8.9875 × 10⁹ N·m²/C² is Coulomb's constant, q₁ and q₂ are the charges in coulombs, and r is the separation in meters. In a non-vacuum medium, divide by the relative permittivity εᵣ.

How do I know if the force is attractive or repulsive?
Multiply the two charge signs. Like signs (both + or both −) repel; unlike signs attract. The calculator shows the direction directly with arrows that flip in the live SVG.

What is Coulomb's constant?
k = 1 / (4π ε₀) ≈ 8.9875517873681764 × 10⁹ N·m²/C². ε₀ is the vacuum permittivity, 8.8541878128 × 10⁻¹² F/m.

What is one elementary charge in coulombs?
e = 1.602176634 × 10⁻¹⁹ C — exact since the 2019 SI redefinition. Protons carry +1 e and electrons carry −1 e.

Does the medium between the charges change the force?
Yes. The force is divided by the relative permittivity εᵣ of the medium. Vacuum has εᵣ = 1, water has εᵣ ≈ 80 — so ionic forces in water are about 80× weaker than in vacuum at the same separation.

Why is electrostatic force so much stronger than gravity?
For a proton–electron pair the Coulomb attraction is roughly 2.3 × 10³⁹ times stronger than their mutual gravity at any separation — because the electromagnetic coupling constant is vastly larger than the gravitational one. The calculator reports the ratio explicitly.

Can I solve for the separation r instead of the force?
Yes. Set Solve for to "Separation distance r" and the calculator rearranges to r = √( k · q₁ · q₂ / (εᵣ · F) ). The r input then hides itself automatically.

Can I enter charges in elementary charges e or in CGS statcoulombs?
Yes. The charge-unit dropdown includes coulombs, milli- through femto-coulombs, elementary charges e, and statcoulombs (esu). The calculator converts everything to SI internally.