Coulomb's law

Coulomb's law
Name	Coulomb's law
Caption	Charles-Augustin de Coulomb, whose torsion balance experiments established the law.
Fields	Electromagnetism, Classical physics
Discovered by	Charles-Augustin de Coulomb
Year	1785
Related laws	Gauss's law, Newton's law of universal gravitation

Coulomb's law is a fundamental principle of electromagnetism that quantifies the electrostatic force between two stationary, point charges. Formulated by Charles-Augustin de Coulomb in the late 18th century, it states that the force is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. This inverse-square law is analogous in form to Newton's law of universal gravitation and serves as a cornerstone for understanding electrostatics within the framework of classical physics.

Statement of the law

The scalar magnitude of the electrostatic force \( F \) between two point charges is given by the equation \( F = k_e \frac{|q_1 q_2|}{r^2} \), where \( q_1 \) and \( q_2 \) are the signed magnitudes of the charges, and \( r \) is the distance separating them. The law explicitly describes a force that is repulsive for like charges, such as two protons, and attractive for opposite charges, like a proton and an electron. This behavior was rigorously confirmed through Coulomb's experiments using a torsion balance, building upon earlier qualitative observations by other scientists like Joseph Priestley and Henry Cavendish. The mathematical form solidified the concept of action at a distance for electric phenomena, paralleling the structure of gravitational interaction described by Isaac Newton.

Vector form

To account for both the magnitude and direction of the force, Coulomb's law is expressed in vector form as \( \vec{F}_{12} = k_e \frac{q_1 q_2}{{|\vec{r}_{12}|}^2} \hat{r}_{12} \), where \( \vec{F}_{12} \) is the force on charge \( q_1 \) due to \( q_2 \), and \( \hat{r}_{12} \) is the unit vector pointing from \( q_2 \) to \( q_1 \). This formulation is essential for applying the superposition principle, which allows the net force on any charge to be calculated as the vector sum of forces exerted by all other charges in a system, such as in complex arrangements studied in electrostatics. The vector nature underscores that the force acts along the line joining the two charges, a concept central to the work of Siméon Denis Poisson and Carl Friedrich Gauss in developing the field theory of electromagnetism.

Electrostatic constant

The proportionality constant \( k_e \) in Coulomb's law is known as the Coulomb constant or the electrostatic constant. In the International System of Units (SI), its value is approximately \( 8.9875517873681764 \times 10^9 \ \mathrm{N \cdot m^2 / C^2} \). This constant is often expressed in terms of more fundamental parameters as \( k_e = \frac{1}{4 \pi \varepsilon_0} \), where \( \varepsilon_0 \) is the vacuum permittivity or the electric constant. The value of \( \varepsilon_0 \) is defined in relation to the defined speed of light in vacuum, \( c \), and the magnetic constant, \( \mu_0 \), through the equation \( c^2 = 1/(\mu_0 \varepsilon_0) \), a relationship arising from James Clerk Maxwell's equations.

Relation to Gauss's law

Coulomb's law is mathematically equivalent to Gauss's law in its integral form for static electric fields in a vacuum. Gauss's law, one of Maxwell's equations, states that the flux of the electric field through any closed surface is proportional to the enclosed electric charge. This equivalence can be derived by applying Gauss's law to a point charge, which yields the electric field described by Coulomb's law. This deep connection demonstrates that the inverse-square law nature of the force is intrinsically linked to the three-dimensional geometry of space, a principle also explored in the context of gravitational potential in Newtonian mechanics.

Limitations and applicability

Coulomb's law is strictly valid only for stationary point charges in a vacuum. Its applicability is limited in several key scenarios: it fails at very small distances comparable to the Compton wavelength, where quantum electrodynamics and effects like vacuum polarization become significant. It also does not account for the magnetic forces that arise between moving charges, which are described by the Lorentz force law and Ampère's force law. In material media, the presence of dielectric substances alters the effective force, often incorporated by using the permittivity of the medium rather than \( \varepsilon_0 \). Furthermore, at relativistic speeds, the effects of special relativity must be considered, requiring a more complete description from classical electromagnetism.

Comparison with Newton's law of gravitation

Both Coulomb's law and Newton's law of universal gravitation are inverse-square laws describing central forces, but they govern fundamentally different interactions. The gravitational force between masses, as formulated by Isaac Newton, is always attractive and proportional to the product of the masses, with a proportionality constant \( G \) (the gravitational constant) that is exceedingly small compared to \( k_e \). In contrast, the electrostatic force can be either attractive or repulsive and is vastly stronger; for instance, the force between an electron and a proton in a hydrogen atom is about \( 10^{39} \) times stronger than their gravitational attraction. This disparity in strength explains why electromagnetic forces dominate atomic and molecular structure, while gravity governs large-scale astrophysical phenomena like the orbits of planets in the Solar System.

Category:Electromagnetism Category:Physics laws Category:Electrostatics