Coulomb's law is one of the fundamental laws of electrostatics, describing the force of interaction between two electric charges. Formulated by the French physicist Charles-Augustin de Coulomb in 1785, it remains today the basis for understanding electrical phenomena — from the structure of the atom to how capacitors work.
The Coulomb's law formula
Coulomb's law states that the force between two point charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them:
F = k × |q₁ × q₂| / r²
- F — the force of interaction (in newtons, N),
- k — the electrostatic constant (Coulomb's constant), in vacuum approx. 8.99 × 10⁹ N·m²/C²,
- q₁, q₂ — the charge values (in coulombs, C),
- r — the distance between the charges (in metres, m).
The direction of the force depends on the signs of the charges: like charges (two positive or two negative) repel each other, while opposite charges attract.
Where does the constant k come from
The electrostatic constant k is related to the permittivity of free space ε₀ by the formula k = 1 / (4πε₀). The value of ε₀ is approximately 8.854 × 10⁻¹² F/m, which after substitution gives the familiar value k ≈ 8.99 × 10⁹ N·m²/C². In media other than vacuum (for example water), the force is weaker because the permittivity of the medium is higher.
Worked example
Let's calculate the force between two charges: q₁ = 2 × 10⁻⁶ C, q₂ = 3 × 10⁻⁶ C, separated by a distance of r = 0.1 m.
F = 8.99 × 10⁹ × (2 × 10⁻⁶ × 3 × 10⁻⁶) / 0.1² =
F = 8.99 × 10⁹ × 6 × 10⁻¹² / 0.01 =
F = 8.99 × 10⁹ × 6 × 10⁻¹⁰ =
F ≈ 5.39 N
Since both charges are positive, the force is repulsive.
Coulomb's law vs Newton's law of gravitation
The formula for Coulomb's law is mathematically analogous to Newton's law of universal gravitation — in both cases the force decreases with the square of the distance. The key difference: the gravitational force is always attractive, while the electrostatic force can be attractive or repulsive depending on the signs of the charges. Electrostatic forces are also typically many orders of magnitude stronger than gravitational ones for comparable masses and charges of elementary particles.
Applications of Coulomb's law
- Atomic structure — the Coulomb force keeps electrons in orbitals around the positively charged nucleus.
- Capacitors — the design and operation of components that store electric charge.
- Applied electrostatics — powder coating, electrostatic filters, laser printers.
- Chemistry — ionic bonds arise directly from Coulomb attraction between ions of opposite charge.
Common calculation mistakes
Students most often go wrong by: forgetting the absolute value of the charges (force as a scalar quantity is always positive; direction is determined separately), incorrectly converting units (microcoulombs, nanocoulombs to coulombs), and confusing r with r² in the denominator.
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Frequently asked questions about Coulomb's law
What is Coulomb's law in simple terms?
Coulomb's law describes how strongly two electrically charged bodies attract or repel each other. The larger the charges and the closer they are to each other, the greater the force of interaction.
What is the value of Coulomb's constant?
Coulomb's constant in vacuum is approximately k ≈ 8.99 × 10⁹ N·m²/C², often rounded to 9 × 10⁹ N·m²/C² in school problems.
Is the Coulomb force always attractive?
No. Charges of the same sign (two positive or two negative) repel each other, while charges of opposite signs attract. The formula itself gives the magnitude of the force — the direction must be determined from the signs of the charges.
How does the Coulomb force change when the distance is doubled?
Because the force is inversely proportional to the square of the distance, doubling the distance reduces the force to one quarter of its original value.
How does Coulomb's law differ from the law of gravitation?
Both laws have a similar mathematical structure (the force decreases with the square of the distance), but gravity is always attractive, while the electric force can be attractive or repulsive depending on the signs of the charges.
What is the unit of electric charge?
The SI unit of electric charge is the coulomb (C). One coulomb is a very large charge — in practice microcoulombs (μC) or nanocoulombs (nC) are more commonly encountered.
Does Coulomb's law only work in a vacuum?
The basic formula applies to a vacuum. In other media (such as water or air), the force is weaker because the relative permittivity of the medium must be included in the denominator of the formula.
Does Coulomb's law apply to extended charges?
Strictly speaking, Coulomb's law applies to point charges. For bodies with an extended charge distribution, integration over the entire volume or surface is used, which is mathematically more complex.
Who formulated Coulomb's law and when?
The law was formulated by the French physicist Charles-Augustin de Coulomb in 1785, based on experiments with a torsion balance that he built himself.
Is Coulomb's law part of high-school physics exams?
Yes, electrostatics and Coulomb's law appear in the extended physics curriculum in many countries and commonly show up in final school-leaving exams.