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How Coulomb's law works
The force between two electric charges is:
\[F = k \frac{|q_1 q_2|}{r^2}\]Where F is the force in newtons, k is Coulomb’s constant, q₁ and q₂ are the charges in coulombs, and r is the distance between them in meters. This one equation governs static electricity, the attraction that holds electrons in atoms, and the repulsion that keeps like-charged objects apart.
The constants and units
Coulomb’s constant (k):
\[k = 8.9875 \times 10^9 \text{ N·m}^2\text{/C}^2\]This is sometimes written as:
\[k = \frac{1}{4\pi\varepsilon_0}\]Where ε₀ = 8.854 × 10⁻¹² F/m is the permittivity of free space. Both forms are equivalent; the k notation is more common in introductory physics.
The coulomb (C) is the SI unit of electric charge. One coulomb is an enormous charge in practical terms. Most electrostatic interactions involve charges measured in microcoulombs (μC, 10⁻⁶ C) or nanocoulombs (nC, 10⁻⁹ C). The charge of a single electron is −1.602 × 10⁻¹⁹ C.
| Charge source | Charge value |
|---|---|
| Electron | −1.602 × 10⁻¹⁹ C |
| Proton | +1.602 × 10⁻¹⁹ C |
| Neutron | 0 C |
| Typical static shock | ±0.1 to 1 μC |
| Lightning bolt | ±5 C (approximately) |
Worked example: two microcoulomb charges
Two charges of +1 μC each, placed 1 meter apart:
\[q_1 = q_2 = 1 \times 10^{-6} \text{ C}\] \[F = (8.9875 \times 10^9) \times \frac{(1 \times 10^{-6})^2}{1^2}\] \[F = (8.9875 \times 10^9) \times (1 \times 10^{-12})\] \[F = 8.9875 \times 10^{-3} \text{ N} \approx 0.009 \text{ N}\]That is approximately 9 millinewtons, or about the weight of a 0.9-gram object. Small in everyday terms, but significant at the atomic scale.
| Because both charges are positive, the force is repulsive: each charge pushes the other away. The absolute value in the formula | q₁q₂ | means the equation gives you the magnitude of the force. The direction (attractive or repulsive) is determined by the signs of the charges separately. |
Attractive vs repulsive forces
The rule is simple: opposite signs attract, same signs repel.
- +q₁ and +q₂: repulsive
- −q₁ and −q₂: repulsive
- +q₁ and −q₂: attractive
For a proton (+e) and an electron (−e) separated by 1 nm (1 × 10⁻⁹ m):
\[F = (8.9875 \times 10^9) \times \frac{(1.602 \times 10^{-19})^2}{(1 \times 10^{-9})^2}\] \[F = (8.9875 \times 10^9) \times \frac{2.566 \times 10^{-38}}{10^{-18}}\] \[F = (8.9875 \times 10^9) \times (2.566 \times 10^{-20})\] \[F \approx 2.307 \times 10^{-10} \text{ N} \approx 0.23 \text{ nN}\]The force is attractive (proton pulls electron toward it, electron pulls proton toward it). At 1 nm separation this is about 0.23 nanonewtons, which is small in absolute terms but enormous relative to the masses involved.
The inverse square law
The r² in the denominator makes Coulomb’s law an inverse square law: doubling the distance reduces the force to one quarter. Tripling the distance reduces the force to one ninth.
\[\text{If } r \to 2r: \quad F \to \frac{F}{4}\] \[\text{If } r \to 3r: \quad F \to \frac{F}{9}\]This relationship has a geometric explanation. Electric field lines from a point charge spread outward in all directions. As distance doubles, the same number of field lines spread over four times the area (since the surface area of a sphere scales as r²). The field strength, and therefore the force on a second charge, drops by the same factor.
For the 1 μC pair from the earlier example at different distances:
| Distance (m) | Force (N) | Relative to 1 m |
|---|---|---|
| 0.5 | 0.0360 | 4× |
| 1.0 | 0.0090 | 1× (baseline) |
| 2.0 | 0.00225 | 1/4 |
| 3.0 | 0.00100 | 1/9 |
| 10.0 | 0.000009 | 1/100 |
Comparison with gravity
Coulomb’s law and Newton’s law of gravitation have the same mathematical form:
\[F_E = k \frac{|q_1 q_2|}{r^2} \qquad F_G = G \frac{m_1 m_2}{r^2}\]Both are inverse square laws. Both depend on a product of two quantities (charges vs masses) and a constant. The differences are:
Magnitude of the constant. k = 8.99 × 10⁹ N·m²/C², while G = 6.67 × 10⁻¹¹ N·m²/kg². Electrostatic forces are far stronger than gravitational forces at short distances.
Sign of the force. Mass is always positive, so gravity is always attractive. Charge can be positive or negative, so electrostatic force can be either attractive or repulsive.
Dominance at different scales. At the atomic scale, electrostatics dominate completely. The electrostatic attraction between a proton and electron is approximately 10³⁹ times stronger than their gravitational attraction. At planetary and stellar scales, gravity dominates because matter is electrically neutral on average (positive and negative charges cancel), while mass has no equivalent cancellation.
Worked example: proton-electron at Bohr radius
The Bohr radius is the average distance between the proton and electron in a hydrogen atom’s ground state: a₀ = 5.292 × 10⁻¹¹ m.
\[F = (8.9875 \times 10^9) \times \frac{(1.602 \times 10^{-19})^2}{(5.292 \times 10^{-11})^2}\] \[F = (8.9875 \times 10^9) \times \frac{2.566 \times 10^{-38}}{2.800 \times 10^{-21}}\] \[F = (8.9875 \times 10^9) \times (9.165 \times 10^{-18})\] \[F \approx 8.24 \times 10^{-8} \text{ N} = 82.4 \text{ nN}\]For comparison, the gravitational force between a proton and electron at the same distance:
\[F_G = (6.674 \times 10^{-11}) \times \frac{(1.673 \times 10^{-27})(9.109 \times 10^{-31})}{(5.292 \times 10^{-11})^2}\] \[F_G \approx 3.6 \times 10^{-47} \text{ N}\]The electrostatic force is approximately 2.3 × 10³⁹ times larger. Gravity is negligible at the atomic scale.
Real-world applications
Static electricity is the most familiar application. When you walk across carpet and touch a metal doorknob, charge that built up on your body (typically in the range of 0.1 to 1 μC) discharges rapidly. The spark you see and feel results from the sudden current as that charge flows through a conductive path.
Lightning is the same phenomenon at much larger scale. Charge separation in storm clouds (primarily through ice crystal collisions) builds up to hundreds of coulombs across voltage differences of millions of volts. The discharge channel reaches temperatures around 30,000 K.
At the molecular level, Coulomb’s law governs ionic bonding (the attraction between Na⁺ and Cl⁻ in table salt), intermolecular forces, and the behavior of proteins and DNA in solution. The force responsible for holding together the nucleus works differently (the strong nuclear force), but the repulsion between protons that the strong force must overcome is electrostatic in nature.
Use the Coulomb’s law calculator to compute the force between any two charges at any distance, with automatic unit conversion for microcoulombs and nanometers.