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Coulomb's law

Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law[1] of physics that calculates the amount of force between two electrically charged particles at rest. This electric force is conventionally called the electrostatic force or Coulomb force.[2] Although the law was known earlier, it was first published in 1785 by French physicist Charles-Augustin de Coulomb. Coulomb's law was essential to the development of the theory of electromagnetism and maybe even its starting point,[1] as it allowed meaningful discussions of the amount of electric charge in a particle.[3]

The law states that the magnitude, or absolute value, of the attractive or repulsive electrostatic force between two point charges is directly proportional to the product of the magnitudes of their charges and inversely proportional to the squared distance between them.[4] Coulomb discovered that bodies with like electrical charges repel:


Coulomb also showed that oppositely charged bodies attract according to an inverse-square law:


Here, ke is a constant, q1 and q2 are the quantities of each charge, and the scalar r is the distance between the charges.


The force is along the straight line joining the two charges. If the charges have the same sign, the electrostatic force between them makes them repel; if they have different signs, the force between them makes them attract.


Being an inverse-square law, the law is similar to Isaac Newton's inverse-square law of universal gravitation, but gravitational forces always make things attract, while electrostatic forces make charges attract or repel. Also, gravitational forces are much weaker than electrostatic forces.[2] Coulomb's law can be used to derive Gauss's law, and vice versa. In the case of a single point charge at rest, the two laws are equivalent, expressing the same physical law in different ways.[6] The law has been tested extensively, and observations have upheld the law on the scale from 10−16 m to 108 m.[6]

Scalar form[edit]

Coulomb's law can be stated as a simple mathematical expression. The scalar form gives the magnitude of the vector of the electrostatic force F between two point charges q1 and q2, but not its direction. If r is the distance between the charges, the magnitude of the force is where ε0 is the electric constant. If the product q1q2 is positive, the force between the two charges is repulsive; if the product is negative, the force between them is attractive.[19]

Coulomb constant[edit]

The Coulomb constant is a proportionality factor that appears in Coulomb's law and related formulas. Denoted , it is also called the electric force constant or electrostatic constant[23] hence the subscript 'e'. The Coulomb constant is given by . The constant is the vacuum electric permittivity (also known as the electric constant).[24] It should not be confused with , which is the dimensionless relative permittivity of the material in which the charges are immersed, or with their product , which is called "absolute permittivity of the material" and is still used in electrical engineering.


Since the 2019 redefinition of the SI base units,[25][26] the Coulomb constant, as calculated from CODATA 2018 recommended values, is[27]

In relativity[edit]

Coulomb's law can be used to gain insight into the form of the magnetic field generated by moving charges since by special relativity, in certain cases the magnetic field can be shown to be a transformation of forces caused by the electric field. When no acceleration is involved in a particle's history, Coulomb's law can be assumed on any test particle in its own inertial frame, supported by symmetry arguments in solving Maxwell's equation, shown above. Coulomb's law can be expanded to moving test particles to be of the same form. This assumption is supported by Lorentz force law which, unlike Coulomb's law is not limited to stationary test charges. Considering the charge to be invariant of observer, the electric and magnetic fields of a uniformly moving point charge can hence be derived by the Lorentz transformation of the four force on the test charge in the charge's frame of reference given by Coulomb's law and attributing magnetic and electric fields by their definitions given by the form of Lorentz force.[30] The fields hence found for uniformly moving point charges are given by:[31]where is the charge of the point source, is the position vector from the point source to the point in space, is the velocity vector of the charged particle, is the ratio of speed of the charged particle divided by the speed of light and is the angle between and .


This form of solutions need not obey Newton's third law as is the case in the framework of special relativity (yet without violating relativistic-energy momentum conservation).[32] Note that the expression for electric field reduces to Coulomb's law for non-relativistic speeds of the point charge and that the magnetic field in non-relativistic limit (approximating ) can be applied to electric currents to get the Biot–Savart law. These solutions, when expressed in retarded time also correspond to the general solution of Maxwell's equations given by solutions of Liénard–Wiechert potential, due to the validity of Coulomb's law within its specific range of application. Also note that the spherical symmetry for gauss law on stationary charges is not valid for moving charges owing to the breaking of symmetry by the specification of direction of velocity in the problem. Agreement with Maxwell's equations can also be manually verified for the above two equations.[33]

Coulomb, Charles Augustin (1788) [1785]. . Histoire de l'Académie Royale des Sciences. Imprimerie Royale. pp. 569–577.

"Premier mémoire sur l'électricité et le magnétisme"

Coulomb, Charles Augustin (1788) [1785]. . Histoire de l'Académie Royale des Sciences. Imprimerie Royale. pp. 578–611.

"Second mémoire sur l'électricité et le magnétisme"

Coulomb, Charles Augustin (1788) [1785]. . Histoire de l'Académie Royale des Sciences. Imprimerie Royale. pp. 612–638.

"Troisième mémoire sur l'électricité et le magnétisme"

Griffiths, David J. (1999). (3rd ed.). Prentice Hall. ISBN 978-0-13-805326-0.

Introduction to Electrodynamics

(1979) [1976]. Fundamentals of the Theory of Electricity (9th ed.). Moscow: Mir. pp. 23–27.

Tamm, Igor E.

Tipler, Paul A.; Mosca, Gene (2008). Physics for Scientists and Engineers (6th ed.). New York: W. H. Freeman and Company.  978-0-7167-8964-2. LCCN 2007010418.

ISBN

Young, Hugh D.; Freedman, Roger A. (2010). Sears and Zemansky's University Physics: With Modern Physics (13th ed.). Addison-Wesley (Pearson).  978-0-321-69686-1.

ISBN

on Project PHYSNET

Coulomb's Law

Archived 2009-02-21 at the Wayback Machine—a chapter from an online textbook

Electricity and the Atom

—a game created by the Molecular Workbench software

A maze game for teaching Coulomb's law

Walter Lewin, 8.02 Electricity and Magnetism, Spring 2002: Lecture 1 (video). MIT OpenCourseWare. License: Creative Commons Attribution-Noncommercial-Share Alike.

Electric Charges, Polarization, Electric Force, Coulomb's Law