Coulomb’s Law – Definition, Formula, and Applications

Coulomb’s law is the fundamental physical law that describes the electrostatic force between two charged objects. It quantifies how the force depends on the magnitudes of the charges and the distance between them, forming the basis of electrostatics in both physics and chemistry.

Key Takeaways: Coulomb’s Law

• Coulomb’s law describes the force between two point charges.
• The force is proportional to the product of the charges and inversely proportional to the square of the distance between them.
• Like charges repel; opposite charges attract.
• The law applies exactly to stationary point charges in a vacuum.
• The Coulomb constant depends on the permittivity of the medium.
• Coulomb’s law underlies electric fields, potentials, and atomic structure.
• It is mathematically analogous to other inverse-square laws, such as gravity.

History of Coulomb’s Law

The understanding of electric forces developed gradually through experimental observations in the 17th and 18th centuries. Early scientists such as William Gilbert and Charles du Fay studied static electricity, identifying attractive and repulsive behaviors between charged objects. Du Fay proposed the existence of two types of electric charge, which we now call positive and negative.

The decisive quantitative breakthrough came from the French physicist Charles-Augustin de Coulomb in the late 18th century. Around 1785, Coulomb performed a series of precise experiments using a torsion balance, an instrument capable of measuring extremely small forces. By observing how charged spheres twisted a suspended fiber, he determined how the force varied with distance.

Coulomb found that the force between charges decreases with the square of the distance between them. This inverse-square relationship mirrored Newton’s law of gravitation, suggesting a deeper mathematical symmetry in nature. His results were published in a series of memoirs to the French Academy of Sciences.

Later developments in the 19th century, particularly by James Clerk Maxwell, placed Coulomb’s law within the broader framework of electromagnetism. Maxwell’s equations generalized electrostatics and showed how electric and magnetic fields arise from charges and currents. Today, Coulomb’s law is understood as a special case of these more general principles.

What Is Coulomb’s Law?

Coulomb’s law describes the electrostatic force between two point charges:

• The force increases as the magnitude of either charge increases.
• The force decreases as the distance between charges increases.
• The force acts along the line connecting the charges.

If the charges have the same sign, the force is repulsive. If they have opposite signs, the force is attractive.

Coulomb’s Law Formula

Coulomb’s law can be expressed in both scalar and vector forms, depending on whether you are interested only in the magnitude of the force or also its direction.

Scalar Form

$F = k_e \frac{|q_1 q_2|}{r^2}$

Where:

• $F$ = magnitude of the force
• $k_e$​ = Coulomb constant
• $q_1, q_2$​ = charges
• $r$ = distance between charges

Vector Form

$\vec{F}_{12} = k_e \frac{q_1 q_2}{r^2} \hat{r}_{12}$​

Where:

• $\vec{F}_{12}$​ is the force on charge 1 due to charge 2
• $\hat{r}_{12}$​ is the unit vector pointing from charge 2 to charge 1

The Coulomb Constant

The Coulomb constant $k_e$​ is given by:$$k_e = \frac{1}{4\pi \varepsilon_0}$$​

Where $\varepsilon_0$​ is the vacuum permittivity.

Numerical value: k_e ​≈ 8.99×10⁹ Nm²/C²

In materials (dielectrics), the effective force decreases because the permittivity is higher than in a vacuum.

Key Quantities in Coulomb’s Law

Worked Example Calculations

Example 1: Force Between Two Charges

Two charges $q_1 = 2.0 \times 10^{-6}$ C and $q_2 = 3.0 \times 10^{-6}$ C are separated by 0.50 m.

Solution:$$F = k_e \frac{q_1 q_2}{r^2}$$$$F = (8.99 \times 10^9) \cdot \frac{(2.0 \times 10^{-6})(3.0 \times 10^{-6})}{(0.50)^2}$$$$F \approx 0.216 \ \text{N}$$

Answer: The force is approximately 0.22 N (repulsive).

Example 2: Force Between Opposite Charges

Two charges $q_1 = -1.0 \times 10^{-6}$ C and $q_2 = 4.0 \times 10^{-6}$ C are 0.20 m apart.

Solution:$$F = (8.99 \times 10^9) \cdot \frac{(1.0 \times 10^{-6})(4.0 \times 10^{-6})}{(0.20)^2}$$​ $$F \approx 0.899 \ \text{N}$$

Answer: The force is approximately 0.90 N (attractive).

Electric Field Form of Coulomb’s Law

Coulomb’s law can be expressed in terms of the electric field produced by a charge rather than the force between two charges.

The electric field $E$ due to a point charge is:$$E = k_e \frac{q}{r^2}$$​

This form describes how a single charge influences the space around it. A second charge placed in this field experiences a force:

$$F = qE$$

This approach is often more useful than directly applying Coulomb’s law because it separates the source of the field from the object experiencing the force. It also provides a bridge to more advanced topics such as field lines, potential, and Maxwell’s equations.

Limitations of Coulomb’s Law

Coulomb’s law is extremely useful but has important limitations:

• It applies strictly to point charges or spherically symmetric charge distributions.
• It assumes charges are stationary (electrostatics).
• It does not account for magnetic effects from moving charges.
• It does not include relativistic effects at high speeds.
• It breaks down at very small (quantum) scales where quantum electrodynamics applies.
• In materials, the simple form must be modified to include permittivity.

Applications of Coulomb’s Law

Coulomb’s law plays a central role in many fields:

• Atomic structure: Explains attraction between electrons and nuclei.
• Chemistry: Governs ionic bonding and intermolecular forces.
• Electronics: Determines forces in capacitors and circuits.
• Materials science: Describes dielectric behavior.
• Biophysics: Explains interactions between charged biomolecules.
• Particle physics: Forms the classical basis for electromagnetic interactions.

Principle of Superposition

Coulomb’s law applies to systems with more than two charges through the principle of superposition.

The total force on a charge equals the vector sum of the forces from all other charges:

$$\vec{F}_{\text{total}} = \sum \vec{F}_i$$

Each pairwise interaction is calculated using Coulomb’s law, and the results are added vectorially.

This principle is essential for analyzing real systems, including molecules, crystals, and charged conductors.

Coulomb’s Law in Different Media (Dielectrics)

Coulomb’s law changes when charges are not in a vacuum.

The force becomes:$$F = \frac{1}{4\pi \varepsilon} \frac{|q_1 q_2|}{r^2}$$

Where:

• $\varepsilon = \varepsilon_0 \varepsilon_r$r​ is the permittivity of the medium
• $\varepsilon_r$ is the relative permittivity (dielectric constant)

In materials like water, the force between charges is significantly reduced. This effect plays a major role in chemistry and biology, particularly in solutions and biological systems.

Potential Energy of Two Charges

Coulomb’s law also leads to an expression for electric potential energy between two charges:

$$U = k_e \frac{q_1 q_2}{r}$$

Key points:

• Opposite charges have negative potential energy (bound systems).
• Like charges have positive potential energy.
• The energy depends on $1/r$, not $1/r^2$.

This concept is fundamental in atomic physics and chemical bonding.

Coulomb’s Law and Gauss’s Law

Coulomb’s law and Gauss’s law are closely related:

• Coulomb’s law gives the force between individual charges.
• Gauss’s law describes how electric fields relate to charge distributions.

In fact, Coulomb’s law can be derived from Gauss’s law for a point charge, assuming symmetry. Conversely, Gauss’s law can be obtained by generalizing Coulomb’s law.

Coulomb’s Law and Other Inverse-Square Laws

Coulomb’s law shares the same mathematical form as other inverse-square laws, such as Newton’s law of gravitation:

Newton’s law of gravitation: $$F = G \frac{m_1 m_2}{r^2}$$

• Both laws:
  • Depend on $1/r^2$
  • Act along the line between objects
  • Are central forces

Key differences:

• Electric forces can be attractive or repulsive; gravity is always attractive.
• Electric forces are typically much stronger than gravitational forces.

Coulomb’s Law vs Newton’s Law of Gravitation

Does Coulomb’s Law Apply in Relativity?

Coulomb’s law is a non-relativistic approximation:

• It assumes instantaneous interaction between charges.
• In relativity, electromagnetic interactions propagate at the speed of light.
• The full relativistic description comes from Maxwell’s equations and special relativity.

For slowly moving charges, Coulomb’s law remains an excellent approximation.

Common Misconceptions

• “Coulomb’s law applies to all charges.”
  It applies exactly only to point charges at rest.
• “Distance does not matter much.”
  The force decreases rapidly with distance (inverse-square).
• “Only large charges interact.”
  Even tiny charges exert forces, though they may be difficult to measure.
• “Electric force is always stronger than gravity.”
  It is stronger at the particle level, but gravity dominates at large scales due to charge neutrality.

FAQs

What units are used in Coulomb’s law?
The force is in newtons (N), charge in coulombs (C), and distance in meters (m).

What is a coulomb?
A coulomb is the SI unit of electric charge, equal to about $6.24 \times 10^{18}$ elementary charges.

Does Coulomb’s law work in water or other media?
Yes, but the force is reduced due to the medium’s permittivity.

Can Coulomb’s law describe multiple charges?
Yes, using the principle of superposition: add forces vectorially.

Why is it called an inverse-square law?
Because the force depends on $1/r^2$, meaning it decreases with the square of distance.

References and Further Reading

• Coulomb (1785). “Premier mémoire sur l’électricité et le magnétisme” [First dissertation on electricity and magnetism]. Histoire de l’Académie Royale des Sciences [History of the Royal Academy of Sciences] (in French). 569–577.
• Elliott, Robert S. (1999). Electromagnetics: History, Theory, and Applications. Wiley. ISBN 978-0-7803-5384-8.
• Halliday, David; Resnick, Robert; Walker, Jearl (2013). Fundamentals of Physics. John Wiley & Sons. ISBN 9781118230718.
• Purcell, Edward M. (2013). Electricity and Magnetism (3rd ed.). Cambridge. ISBN 9781107014022.
• Tipler, Paul A.; Mosca, Gene (2008). Physics for Scientists and Engineers (6th ed.). New York: W. H. Freeman and Company. ISBN 978-0-7167-8964-2.