Coulomb's law#Scalar form

File:Coulomb.jpg]]

Ancient cultures around the Mediterranean knew that certain objects, such as rods of amber, could be rubbed with cat's fur to attract light objects like feathers and pieces of paper. Thales of Miletus made the first recorded description of static electricity around 600 BC,{{cite journal |last1=Cork |first1=C.R. |title=Conductive fibres for electronic textiles |journal=Electronic Textiles |date=2015 |pages=3–20 |doi=10.1016/B978-0-08-100201-8.00002-3| isbn=9780081002018 }} when he noticed that friction could make a piece of amber attract small objects.{{Cite book |first=Joseph |last=Stewart |title=Intermediate Electromagnetic Theory |publisher=World Scientific |year=2001 | page=50 | isbn=978-981-02-4471-2}}{{Cite book |first=Brian |last=Simpson |title=Electrical Stimulation and the Relief of Pain |publisher=Elsevier Health Sciences |year=2003 |pages=6–7 |isbn=978-0-444-51258-1}}

In 1600, English scientist William Gilbert made a careful study of electricity and magnetism, distinguishing the lodestone effect from static electricity produced by rubbing amber. He coined the Neo-Latin word electricus ("of amber" or "like amber", from {{lang|grc|ἤλεκτρον}} [elektron], the Greek word for "amber") to refer to the property of attracting small objects after being rubbed.{{Cite book |first=Brian |last=Baigrie |title=Electricity and Magnetism: A Historical Perspective |publisher=Greenwood Press |year=2007 |pages=7–8 |isbn=978-0-313-33358-3}} This association gave rise to the English words "electric" and "electricity", which made their first appearance in print in Thomas Browne's Pseudodoxia Epidemica of 1646.{{Cite journal |first=Gordon |last=Chalmers |title=The Lodestone and the Understanding of Matter in Seventeenth Century England |journal=Philosophy of Science |year=1937 |volume=4 |issue=1 |pages=75–95 |doi=10.1086/286445|s2cid=121067746 }}

Early investigators of the 18th century who suspected that the electrical force diminished with distance as the force of gravity did (i.e., as the inverse square of the distance) included Daniel Bernoulli{{Cite book |last=Socin |first=Abel |title=Acta Helvetica Physico-Mathematico-Anatomico-Botanico-Medica |url=https://archive.org/stream/actahelveticaphy04helv#page/224/mode/2up |volume=4 |year=1760 |publisher=Basileae |language=la |pages=224–25}} and Alessandro Volta, both of whom measured the force between plates of a capacitor, and Franz Aepinus who supposed the inverse-square law in 1758.{{Cite book |last=Heilbron |first=J.L. |author-link=John L. Heilbron |title=Electricity in the 17th and 18th Centuries: A Study of Early Modern Physics |url=https://books.google.com/books?id=UlTLRUn1sy8C |isbn=978-0486406886 |location=Los Angeles, California |year=1979 |publisher=University of California Press |pages=[https://books.google.com/books?id=UlTLRUn1sy8C&pg=PA460 460–462] and [https://books.google.com/books?id=UlTLRUn1sy8C&pg=PA464 464] (including footnote 44)}}

Based on experiments with electrically charged spheres, Joseph Priestley of England was among the first to propose that electrical force followed an inverse-square law, similar to Newton's law of universal gravitation. However, he did not generalize or elaborate on this.{{Cite book |last=Schofield |first=Robert E. |title=The Enlightenment of Joseph Priestley: A Study of his Life and Work from 1733 to 1773 |year=1997 |publisher=University Park: Pennsylvania State University Press |pages=144–56 |isbn=978-0-271-01662-7}} In 1767, he conjectured that the force between charges varied as the inverse square of the distance.{{Cite book |last=Priestley |first=Joseph |title=The History and Present State of Electricity, with Original Experiments |url=https://archive.org/stream/historyandprese00priegoog#page/n6/mode/2up |year=1767 |location=London, England |page=732}}{{Cite book |first=Robert S. |last=Elliott |title=Electromagnetics: History, Theory, and Applications |url=http://eu.wiley.com/WileyCDA/WileyTitle/productCd-0780353846.html |year=1999 |publisher=Wiley |isbn=978-0-7803-5384-8 |access-date=2009-10-17 |archive-date=2014-03-10 |archive-url=https://web.archive.org/web/20140310020822/http://eu.wiley.com/WileyCDA/WileyTitle/productCd-0780353846.html |url-status=live }}

Image:Bcoulomb.png]]

In 1769, Scottish physicist John Robison announced that, according to his measurements, the force of repulsion between two spheres with charges of the same sign varied as {{math|x^−2.06}}.{{Cite book |last=Robison |first=John |editor-first=John |editor-last=Murray |title=A System of Mechanical Philosophy |url=https://archive.org/stream/asystemmechanic07wattgoog#page/n5/mode/2up |volume=4 |year=1822 |publisher=Printed for J. Murray |location=London, England}}

In the early 1770s, the dependence of the force between charged bodies upon both distance and charge had already been discovered, but not published, by Henry Cavendish of England.{{cite book |editor-last=Maxwell |editor-first=James Clerk |title=The Electrical Researches of the Honourable Henry Cavendish... |url=https://archive.org/stream/electricalresear00caveuoft |edition=1st |orig-date=1879 |year=1967 |publisher=Cambridge University Press |location=Cambridge, England |pages=104–113 |chapter=Experiments on Electricity: Experimental determination of the law of electric force. |chapter-url=https://archive.org/stream/electricalresear00caveuoft#page/104/mode/2up}} In his notes, Cavendish wrote, "We may therefore conclude that the electric attraction and repulsion must be inversely as some power of the distance between that of the {{nowrap|2 + {{sfrac|50}}th}} and that of the {{nowrap|2 − {{sfrac|50}}th}}, and there is no reason to think that it differs at all from the inverse duplicate ratio".

Finally, in 1785, the French physicist Charles-Augustin de Coulomb published his first three reports of electricity and magnetism where he stated his law. This publication was essential to the development of the theory of electromagnetism. He used a torsion balance to study the repulsion and attraction forces of charged particles, and determined that the magnitude of the electric 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.

The torsion balance consists of a bar suspended from its middle by a thin fiber. The fiber acts as a very weak torsion spring. In Coulomb's experiment, the torsion balance was an insulating rod with a metal-coated ball attached to one end, suspended by a silk thread. The ball was charged with a known charge of static electricity, and a second charged ball of the same polarity was brought near it. The two charged balls repelled one another, twisting the fiber through a certain angle, which could be read from a scale on the instrument. By knowing how much force it took to twist the fiber through a given angle, Coulomb was able to calculate the force between the balls and derive his inverse-square proportionality law.

File:Coulombslawgraph.svg

Coulomb's law states that the electrostatic force $\backslash mathbf\{F\}\_1$ experienced by a charge, $q\_1$ at position $\backslash mathbf\{r\}\_1$, in the vicinity of another charge, $q\_2$ at position $\backslash mathbf\{r\}\_2$, in a vacuum is equal to

$$\backslash mathbf\{F\}\_1\; =\; \backslash frac\{q\_1q\_2\}\{4\backslash pi\backslash varepsilon\_0\}\; \{\backslash hat\backslash mathbf\; r\_\{12\}\backslash over$$

where $\backslash mathbf\{r\_\{12\}\; =\; r\_1\; -\; r\_2\}$ is the displacement vector between the charges, $\backslash hat\backslash mathbf\; r\_\{12\}$ a unit vector pointing from $q\_2$ to {{nowrap|$q\_1$,}} and $\backslash varepsilon\_0$ the electric constant. Here, $\backslash mathbf\{\backslash hat\{r\}\}\_\{12\}$ is used for the vector notation. The electrostatic force $\backslash mathbf\{F\}\_2$ experienced by $q\_2$, according to Newton's third law, is {{nowrap|$\backslash mathbf\{F\}\_2=-\backslash mathbf\{F\}\_1$.}}

If both charges have the same sign (like charges) then the product $q\_1q\_2$ is positive and the direction of the force on $q\_1$ is given by $\backslash widehat\{\backslash mathbf\{r\}\}\_\{12\}$; the charges repel each other. If the charges have opposite signs then the product $q\_1q\_2$ is negative and the direction of the force on $q\_1$ is {{nowrap|$-\backslash hat\{\backslash mathbf\{r\}\}\_\{12\}$;}} the charges attract each other.{{cite web|url=http://farside.ph.utexas.edu/teaching/em/lectures/node28.html|title=Coulomb's law|first=Richard|last=Fitzpatrick|date=2006-02-02|publisher=University of Texas|access-date=2007-11-30|archive-date=2015-07-09|archive-url=https://web.archive.org/web/20150709004404/http://farside.ph.utexas.edu/teaching/em/lectures/node28.html|url-status=live}}

The law of superposition allows Coulomb's law to be extended to include any number of point charges. The force acting on a point charge due to a system of point charges is simply the vector addition of the individual forces acting alone on that point charge due to each one of the charges. The resulting force vector is parallel to the electric field vector at that point, with that point charge removed.

Force $\backslash mathbf\{F\}$ on a small charge $q$ at position $\backslash mathbf\{r\}$, due to a system of $n$ discrete charges in vacuum is

$$\backslash mathbf\{F\}(\backslash mathbf\{r\})\; =\; \{q\backslash over4\backslash pi\backslash varepsilon\_0\}\; \backslash sum\_\{i=1\}^n\; q\_i\; \{\backslash hat\backslash mathbf\; r\_i\backslash over$$

where $q\_i$ is the magnitude of the {{math|i}}th charge, $\backslash mathbf\{r\}\_i$ is the vector from its position to $\backslash mathbf\{r\}$ and $\backslash hat\backslash mathbf\; r\_i$ is the unit vector in the direction of $\backslash mathbf\; r\_i$.

In this case, the principle of linear superposition is also used. For a continuous charge distribution, an integral over the region containing the charge is equivalent to an infinite summation, treating each infinitesimal element of space as a point charge $dq$. The distribution of charge is usually linear, surface or volumetric.

For a linear charge distribution (a good approximation for charge in a wire) where $\backslash lambda(\backslash mathbf\{r\}\text{'})$ gives the charge per unit length at position $\backslash mathbf\{r\}\text{'}$, and $d\backslash ell\text{'}$ is an infinitesimal element of length,{{cite web|url=http://dev.physicslab.org/Document.aspx?doctype=3&filename=Electrostatics_ContinuousChargedRod.xml|title=Charged rods|website=PhysicsLab.org|access-date=2007-11-06|archive-date=2014-10-10|archive-url=https://web.archive.org/web/20141010201853/http://dev.physicslab.org/Document.aspx?doctype=3&filename=Electrostatics_ContinuousChargedRod.xml|url-status=live}}

$$dq\text{'}\; =\; \backslash lambda(\backslash mathbf\{r\text{'}\})\; \backslash ,\; d\backslash ell\text{'}.$$

For a surface charge distribution (a good approximation for charge on a plate in a parallel plate capacitor) where $\backslash sigma(\backslash mathbf\{r\}\text{'})$ gives the charge per unit area at position $\backslash mathbf\{r\}\text{'}$, and $dA\text{'}$ is an infinitesimal element of area,

$$dq\text{'}\; =\; \backslash sigma(\backslash mathbf\{r\text{'}\})\backslash ,dA\text{'}.$$

For a volume charge distribution (such as charge within a bulk metal) where $\backslash rho(\backslash mathbf\{r\}\text{'})$ gives the charge per unit volume at position $\backslash mathbf\{r\}\text{'}$, and $dV\text{'}$ is an infinitesimal element of volume,

$$dq\text{'}\; =\; \backslash rho(\backslash boldsymbol\{r\text{'}\})\backslash ,dV\text{'}.$$

The force on a small test charge $q$ at position $\backslash boldsymbol\{r\}$ in vacuum is given by the integral over the distribution of charge

$$\backslash mathbf\{F\}(\backslash mathbf\{r\})\; =\; \backslash frac\{q\}\{4\backslash pi\backslash varepsilon\_0\}\backslash int\; dq\text{'}\; \backslash frac\{\backslash mathbf\{r\}\; -\; \backslash mathbf\{r\text{'}\}\}\{|\backslash mathbf\{r\}\; -\; \backslash mathbf\{r\text{'}\}|^3\}.$$

The "continuous charge" version of Coulomb's law is never supposed to be applied to locations for which $|\backslash mathbf\{r\}\; -\; \backslash mathbf\{r\text{'}\}|\; =\; 0$ because that location would directly overlap with the location of a charged particle (e.g. electron or proton) which is not a valid location to analyze the electric field or potential classically. Charge is always discrete in reality, and the "continuous charge" assumption is just an approximation that is not supposed to allow $|\backslash mathbf\{r\}\; -\; \backslash mathbf\{r\text{'}\}|\; =\; 0$ to be analyzed.

The constant of proportionality, $\backslash frac\{1\}\{4\backslash pi\backslash varepsilon\_0\}$, in Coulomb's law:

$$\backslash mathbf\{F\}\_1\; =\; \backslash frac\{q\_1q\_2\}\{4\backslash pi\backslash varepsilon\_0\}\; \{\backslash hat\backslash mathbf\; r\_\{12\}\backslash over$$

is a consequence of historical choices for units.{{rp|4-2}}

The constant $\backslash varepsilon\_0$ is the vacuum electric permittivity.{{SIbrochure9th|page=15}} Using the CODATA 2022 recommended value for $\backslash varepsilon\_0$,{{cite web |url=http://physics.nist.gov/cgi-bin/cuu/Value?ep0 |title=2022 CODATA Value: vacuum electric permittivity |work=The NIST Reference on Constants, Units, and Uncertainty |publisher=NIST |access-date=2024-10-27 |archive-date=2024-06-24 |archive-url=https://web.archive.org/web/20240624144904/https://physics.nist.gov/cgi-bin/cuu/Value?ep0 |url-status=live }} the Coulomb constant{{cite book |last=Serway |first=Raymond A. |last2=Jewett |first2=John W., Jr. |year=2014 |title=Physics for Scientists and Engineers |chapter=Some Physical Constants |at=Inside Cover |publisher=Cengage Learning |edition=Ninth |isbn=978-1-133-95405-7}} is

$$k\_\backslash text\{e\}=\backslash frac\{1\}\{4\backslash pi\backslash varepsilon\_0\}=\; 8.987\backslash \; 551\backslash \; 7862\; (14)\backslash times\; 10^9\backslash \; \backslash mathrm\{N\{\backslash cdot\}m^2\{\backslash cdot\}C^\{-2\}\}$$

There are three conditions to be fulfilled for the validity of Coulomb's inverse square law:{{cite book |pages=465–468|publisher=CRC Press| doi=10.1201/b18636-105| isbn=978-0-429-22704-2 |chapter=Discussion on physics teaching innovation: Taking Coulomb's law as an example |title=Education Management and Management Science |date=2015 |author1=W. Shao |author2=B. Jiang |author3=J.K. Lv |editor=Dawei Zheng}}

1. The charges must have a spherically symmetric distribution (e.g. be point charges, or a charged metal sphere).
2. The charges must not overlap (e.g. they must be distinct point charges).
3. The charges must be stationary with respect to a nonaccelerating frame of reference.

The last of these is known as the electrostatic approximation. When movement takes place, an extra factor is introduced, which alters the force produced on the two objects. This extra part of the force is called the magnetic force. For slow movement, the magnetic force is minimal and Coulomb's law can still be considered approximately correct. A more accurate approximation in this case is, however, the Weber force. When the charges are moving more quickly in relation to each other or accelerations occur, Maxwell's equations and Einstein's theory of relativity must be taken into consideration.

{{main|Electric field}}

File:Electric field one charge changing.gif

An electric field is a vector field that associates to each point in space the Coulomb force experienced by a unit test charge.{{cite book| last1=Feynman | first1=Richard P. | url=https://feynmanlectures.caltech.edu/II_04.html|title=The Feynman Lectures on Physics Vol II | date=1970 | publisher=Addison-Wesley | isbn=9780201021158}} The strength and direction of the Coulomb force $\backslash mathbf\; F$ on a charge $q\_t$ depends on the electric field $\backslash mathbf\{E\}$ established by other charges that it finds itself in, such that $\backslash mathbf\{F\}\; =\; q\_t\; \backslash mathbf\{E\}$. In the simplest case, the field is considered to be generated solely by a single source point charge. More generally, the field can be generated by a distribution of charges who contribute to the overall by the principle of superposition.

If the field is generated by a positive source point charge $q$, the direction of the electric field points along lines directed radially outwards from it, i.e. in the direction that a positive point test charge $q\_t$ would move if placed in the field. For a negative point source charge, the direction is radially inwards.

The magnitude of the electric field {{math|E}} can be derived from Coulomb's law. By choosing one of the point charges to be the source, and the other to be the test charge, it follows from Coulomb's law that the magnitude of the electric field {{math|E}} created by a single source point charge Q at a certain distance from it r in vacuum is given by

$$|\backslash mathbf\{E\}|\; =\; k\_\backslash text\{e\}\; \backslash frac$$

A system of n discrete charges $q\_i$ stationed at $\backslash mathbf\; r\_i\; =\; \backslash mathbf\; r-\backslash mathbf\; r\_i$ produces an electric field whose magnitude and direction is, by superposition

$$\backslash mathbf\{E\}(\backslash mathbf\{r\})\; =\; \{1\backslash over4\backslash pi\backslash varepsilon\_0\}\; \backslash sum\_\{i=1\}^n\; q\_i\; \{\backslash hat\backslash mathbf\; r\_i\backslash over$$

{{See also|Coulomb explosion}}

Coulomb's law holds even within atoms, correctly describing the force between the positively charged atomic nucleus and each of the negatively charged electrons. This simple law also correctly accounts for the forces that bind atoms together to form molecules and for the forces that bind atoms and molecules together to form solids and liquids. Generally, as the distance between ions increases, the force of attraction, and binding energy, approach zero and ionic bonding is less favorable. As the magnitude of opposing charges increases, energy increases and ionic bonding is more favorable.

{{Duplication|dupe=Gauss's_law#Relation_to_Coulomb's_law}}

{{Excerpt|Gauss's law|Deriving Gauss's law from Coulomb's law}}

Strictly speaking, Coulomb's law cannot be derived from Gauss's law alone, since Gauss's law does not give any information regarding the curl of {{math|E}} (see Helmholtz decomposition and Faraday's law). However, Coulomb's law can be proven from Gauss's law if it is assumed, in addition, that the electric field from a point charge is spherically symmetric (this assumption, like Coulomb's law itself, is exactly true if the charge is stationary, and approximately true if the charge is in motion).

{{math proof|title=Outline of proof|proof=

Taking {{mvar|S}} in the integral form of Gauss's law to be a spherical surface of radius {{mvar|r}}, centered at the point charge {{mvar|Q}}, we have

$$\backslash oint\_S\backslash mathbf\{E\}\backslash cdot\; d\backslash mathbf\{A\}\; =\; \backslash frac\{Q\}\{\backslash varepsilon\_0\}$$

By the assumption of spherical symmetry, the integrand is a constant which can be taken out of the integral. The result is

$$4\backslash pi\; r^2\backslash hat\{\backslash mathbf\{r\}\}\backslash cdot\backslash mathbf\{E\}(\backslash mathbf\{r\})\; =\; \backslash frac\{Q\}\{\backslash varepsilon\_0\}$$

where {{math|r̂}} is a unit vector pointing radially away from the charge. Again by spherical symmetry, {{math|E}} points in the radial direction, and so we get

$$\backslash mathbf\{E\}(\backslash mathbf\{r\})\; =\; \backslash frac\{Q\}\{4\backslash pi\; \backslash varepsilon\_0\}\; \backslash frac\{\backslash hat\{\backslash mathbf\{r\}\}\}\{r^2\}$$

which is essentially equivalent to Coulomb's law. Thus the inverse-square law dependence of the electric field in Coulomb's law follows from Gauss's law.

}}

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{{Broken anchor|date=2025-06-09|bot=User:Cewbot/log/20201008/configuration|target_link=Lorentz force#Lorentz force law as the definition of E and B|reason= The anchor (Lorentz force law as the definition of E and B) has been deleted.|diff_id=1294583823}}.{{Cite book |last1=Rosser |first1=W. G. V. |url=https://link.springer.com/book/10.1007/978-1-4899-6559-2 |title=Classical Electromagnetism via Relativity |year=1968 |isbn=978-1-4899-6258-4 |pages=29–42 |language=en |doi=10.1007/978-1-4899-6559-2 |access-date=2022-10-09 |archive-date=2022-10-09 |archive-url=https://web.archive.org/web/20221009151104/https://link.springer.com/book/10.1007/978-1-4899-6559-2 |url-status=live }} The fields hence found for uniformly moving point charges are given by:{{Cite book |last=Heaviside |first=Oliver |url=https://en.wikisource.org/wiki/Electromagnetic_effects_of_a_moving_charge |title=Electromagnetic waves, the propagation of potential, and the electromagnetic effects of a moving charge |year=1894 |access-date=2022-10-09 |archive-date=2022-10-09 |archive-url=https://web.archive.org/web/20221009151113/https://en.wikisource.org/wiki/Electromagnetic_effects_of_a_moving_charge |url-status=live }}$$\backslash mathbf\{E\}\; =\; \backslash frac\; q\; \{4\; \backslash pi\; \backslash epsilon\_0\; r^3\}\; \backslash frac\; \{1-\; \backslash beta^2\}\; \{(1-\; \backslash beta^2\; \backslash sin^2\; \backslash theta)^\{3/2\}\}\; \backslash mathbf\{r\}$$$$\backslash mathbf\{B\}\; =\; \backslash frac\; q\; \{4\; \backslash pi\; \backslash epsilon\_0\; r^3\}\; \backslash frac\; \{1-\; \backslash beta^2\}\; \{(1-\; \backslash beta^2\; \backslash sin^2\; \backslash theta)^\{3/2\}\}\; \backslash frac\{\backslash mathbf\{v\}\; \backslash times\; \backslash mathbf\{r\}\}\{c^2\}\; =\; \backslash frac\{\backslash mathbf\{v\}\; \backslash times\; \backslash mathbf\{E\}\}\{c^2\}$$where $q$ is the charge of the point source, $\backslash mathbf\{r\}$ is the position vector from the point source to the point in space, $\backslash mathbf\{v\}$ is the velocity vector of the charged particle, $\backslash beta$ is the ratio of speed of the charged particle divided by the speed of light and $\backslash theta$ is the angle between $\backslash mathbf\{r\}$ and $\backslash mathbf\{v\}$.

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).{{Cite book |last=Griffiths |first=David J. |title=Introduction to electrodynamics |date=1999 |publisher=Prentice Hall |isbn=0-13-805326-X |edition=3rd |location=Upper Saddle River, NJ |page=517 |oclc=40251748}} 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 $\backslash beta\backslash ll\; 1$) 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.{{Cite book |last=Purcell |first=Edward |url=http://dx.doi.org/10.1017/cbo9781139005043 |title=Electricity and Magnetism |date=2011-09-22 |publisher=Cambridge University Press |doi=10.1017/cbo9781139005043 |isbn=978-1-107-01360-5 |access-date=2022-10-09 |archive-date=2023-12-30 |archive-url=https://web.archive.org/web/20231230135350/https://www.cambridge.org/core/books/electricity-and-magnetism/2340243C0ED54EEB07C3B6F6895FE8B9 |url-status=live }}

{{See also|Electric potential}}

{{Jargon cleanup|date=October 2020}}

File:Feynman diagram - Moller scattering 1.svg

The Coulomb potential admits continuum states (with E > 0), describing electron-proton scattering, as well as discrete bound states, representing the hydrogen atom.{{cite book|last1=Griffiths|first1=David J.|title=Introduction to quantum mechanics|date=16 August 2018|isbn=978-1-107-18963-8|edition=Third|location=Cambridge, United Kingdom}} It can also be derived within the non-relativistic limit between two charged particles, as follows:

Under Born approximation, in non-relativistic quantum mechanics, the scattering amplitude $\backslash mathcal\{A\}(|\; \backslash mathbf\{p\}\; \backslash rangle\; \backslash to\; |\; \backslash mathbf\{p\}\text{'}\backslash rangle)$ is:

$$\backslash mathcal\{A\}(|\; \backslash mathbf\{p\}\; \backslash rangle\; \backslash to\; |\; \backslash mathbf\{p\}\text{'}\backslash rangle)\; -\; 1\; =\; 2\backslash pi\; \backslash delta(E\_p\; -\; E\_\{p\text{'}\})(-i)\backslash int\; d^3\backslash mathbf\; r\; \backslash ,\; V(\backslash mathbf\; r)\; e^\{-i(\backslash mathbf\; p\; -\; \backslash mathbf\; p\text{'})\backslash mathbf\; r\}$$

This is to be compared to the:

$$\backslash int\; \backslash frac\{d^3k\}\{(2\backslash pi)^3\}\; e^\{i\; k\; r\_0\}\; \backslash langle\; p\text{'},k\; |S|\; p,k\; \backslash rangle$$

where we look at the (connected) S-matrix entry for two electrons scattering off each other, treating one with "fixed" momentum as the source of the potential, and the other scattering off that potential.

Using the Feynman rules to compute the S-matrix element, we obtain in the non-relativistic limit with $m\_0\; \backslash gg\; |\; \backslash mathbf\; p|$

$$\backslash langle\; p\text{'},k\; |\; S\; |\; p,k\; \backslash rangle\; |\_\{conn\}\; =\; -i\backslash frac\{e^2\}\{|\; \backslash mathbf\; p\; -\; \backslash mathbf\; p\text{'}|^2\; -\; i\backslash varepsilon\}(2m)^2\backslash delta(E\_\{p,k\}\; -\; E\_\{p\text{'},k\})(2\backslash pi)^4\backslash delta(\backslash mathbf\; p\; -\; \backslash mathbf\; p\text{'})$$

Comparing with the QM scattering, we have to discard the $(2m)^2$ as they arise due to differing normalizations of momentum eigenstate in QFT compared to QM and obtain:

$$\backslash int\; V(\backslash mathbf\; r)e^\{-i(\backslash mathbf\; p\; -\; \backslash mathbf\; p\text{'})\; \backslash mathbf\; r\}d^3\backslash mathbf\; r\; =\; \backslash frac\{e^2\}\{|\; \backslash mathbf\; p\; -\; \backslash mathbf\; p\text{'}|^2\; -\; i\backslash varepsilon\}$$

where Fourier transforming both sides, solving the integral and taking $\backslash varepsilon\; \backslash to\; 0$ at the end will yield

$$V(r)\; =\; \backslash frac\{e^2\}\{4\backslash pi\; r\}$$

as the Coulomb potential.{{Cite web|title=Quantum Field Theory I + II|url=https://www.thphys.uni-heidelberg.de/~weigand/QFT2-14/SkriptQFT2.pdf|website=Institute for Theoretical Physics, Heidelberg University|access-date=2020-09-24|archive-date=2021-05-04|archive-url=https://web.archive.org/web/20210504114001/https://www.thphys.uni-heidelberg.de/~weigand/QFT2-14/SkriptQFT2.pdf|url-status=live}}

However, the equivalent results of the classical Born derivations for the Coulomb problem are thought to be strictly accidental.{{Cite book|last=Baym, Gordon|title=Lectures on quantum mechanics|year=2018| isbn=978-0-429-49926-5|location=Boca Raton|oclc=1028553174}}{{Cite book|last=Gould | first= Robert J. |title=Electromagnetic processes|date=21 July 2020| isbn=978-0-691-21584-6| location=Princeton, N.J. | oclc=1176566442}}

The Coulomb potential, and its derivation, can be seen as a special case of the Yukawa potential, which is the case where the exchanged boson – the photon – has no rest mass.

{{Overly detailed|date=October 2020|section}}File:Verificacion ley coulomb.png

It is possible to verify Coulomb's law with a simple experiment. Consider two small spheres of mass $m$ and same-sign charge $q$, hanging from two ropes of negligible mass of length $l$. The forces acting on each sphere are three: the weight $mg$, the rope tension $\backslash mathbf\; T$ and the electric force $\backslash mathbf\; F$. In the equilibrium state:

{{NumBlk||$$\backslash mathbf\; T\; \backslash sin\; \backslash theta\_1\; =\backslash mathbf\; F\_1$$|{{EquationRef|1}}}}

and

{{NumBlk||$$\backslash mathbf\; T\; \backslash cos\; \backslash theta\_1\; =\; mg$$|{{EquationRef|2}}}}

Dividing ({{EquationNote|1}}) by ({{EquationNote|2}}):

{{NumBlk||$$\backslash frac\; \{\backslash sin\; \backslash theta\_1\}\{\backslash cos\; \backslash theta\_1\; \}=\backslash frac\; \{F\_1\}\{mg\}\backslash Rightarrow\; F\_1=\; mg\; \backslash tan\; \backslash theta\_1$$|{{EquationRef|3}}}}

Let $\backslash mathbf\; L\_1$ be the distance between the charged spheres; the repulsion force between them $\backslash mathbf\; F\_1$, assuming Coulomb's law is correct, is equal to

{{NumBlk||$$F\_1\; =\; \backslash frac\{q^2\}\{4\; \backslash pi\; \backslash varepsilon\_0\; L\_1^2\}$$

|{{EquationRef|Coulomb's law}}}}

so:

{{NumBlk||$$\backslash frac\{q^2\}\{4\; \backslash pi\; \backslash varepsilon\_0\; L\_1^2\}=mg\; \backslash tan\; \backslash theta\_1$$|{{EquationRef|4}}}}

If we now discharge one of the spheres, and we put it in contact with the charged sphere, each one of them acquires a charge $\backslash frac\{q\}\{2\}$. In the equilibrium state, the distance between the charges will be and the repulsion force between them will be:

{{NumBlk||$$F\_2\; =\; \backslash frac\{\{(\backslash frac\{q\}\{2\})\}^2\}\{4\; \backslash pi\; \backslash varepsilon\_0\; L\_2^2\}=\backslash frac\{\backslash frac\{q^2\}\{4\}\}\{4\; \backslash pi\; \backslash varepsilon\_0\; L\_2^2\}$$|{{EquationRef|5}}}}

We know that $\backslash mathbf\; F\_2\; =\; mg\; \backslash tan\; \backslash theta\_2$ and:

$$\backslash frac\{\backslash frac\{q^2\}\{4\}\}\{4\; \backslash pi\; \backslash varepsilon\_0\; L\_2^2\}=mg\; \backslash tan\; \backslash theta\_2$$

Dividing ({{EquationNote|4}}) by ({{EquationNote|5}}), we get:

{{NumBlk||$$\backslash frac\{\backslash left(\backslash cfrac\{q^2\}\{4\; \backslash pi\; \backslash varepsilon\_0\; L\_1^2\}\; \backslash right)\}\{\backslash left(\backslash cfrac\{\backslash frac\{q^2\}\{4\}\}\{4\; \backslash pi\; \backslash varepsilon\_0\; L\_2^2\}\backslash right)\}\; =\; \backslash frac\{mg\; \backslash tan\; \backslash theta\_1\}\{mg\; \backslash tan\; \backslash theta\_2\}\; \backslash Rightarrow\; 4\; \{\backslash left(\; \backslash frac\; \{L\_2\}\{L\_1\}\; \backslash right)\; \}^2\; =\; \backslash frac\{\; \backslash tan\; \backslash theta\_1\}\{\; \backslash tan\; \backslash theta\_2\}$$|{{EquationRef|6}}}}

Measuring the angles $\backslash theta\_1$ and $\backslash theta\_2$ and the distance between the charges $\backslash mathbf\; L\_1$ and $\backslash mathbf\; L\_2$ is sufficient to verify that the equality is true taking into account the experimental error. In practice, angles can be difficult to measure, so if the length of the ropes is sufficiently great, the angles will be small enough to make the following approximation:

{{NumBlk||$$\backslash tan\; \backslash theta\; \backslash approx\; \backslash sin\; \backslash theta\; =\; \backslash frac\{\backslash frac\{L\}\{2\}\}\{\backslash ell\}\; =\; \backslash frac\{L\}\{2\backslash ell\}\; \backslash Rightarrow\; \backslash frac\{\; \backslash tan\; \backslash theta\_1\}\{\; \backslash tan\; \backslash theta\_2\}\; \backslash approx\; \backslash frac\{\backslash frac\{L\_1\}\{2\backslash ell\}\}\{\backslash frac\{L\_2\}\{2\backslash ell\}\}$$|{{EquationRef|7}}}}

Using this approximation, the relationship ({{EquationNote|6}}) becomes the much simpler expression:

{{NumBlk||$$\backslash frac\{\backslash frac\{L\_1\}\{2\backslash ell\}\}\{\backslash frac\{L\_2\}\{2\backslash ell\}\}\backslash approx\; 4\; \{\backslash left\; (\; \backslash frac\; \{L\_2\}\{L\_1\}\; \backslash right\; )\; \}^2\; \backslash Rightarrow\; \backslash frac\{L\_1\}\{L\_2\}\backslash approx\; 4\; \{\backslash left\; (\; \backslash frac\; \{L\_2\}\{L\_1\}\; \backslash right\; )\; \}^2\; \backslash Rightarrow\; \backslash frac\{L\_1\}\{L\_2\}\backslash approx\backslash sqrt[3]\{4\}$$|{{EquationRef|8}}}}

In this way, the verification is limited to measuring the distance between the charges and checking that the division approximates the theoretical value.

{{Portal|Electronics}}

{{cols}}

• Biot–Savart law
• Darwin Lagrangian
• Electromagnetic force
• Gauss's law
• Method of image charges
• Molecular modelling
• Newton's law of universal gravitation, which uses a similar structure, but for mass instead of charge
• Static forces and virtual-particle exchange
• Casimir effect{{colend}}

{{Reflist|30em}}Spavieri, G., Gillies, G. T., & Rodriguez, M. (2004). Physical implications of Coulomb’s Law. Metrologia, 41(5), S159–S170. doi:10.1088/0026-1394/41/5/s06 

• {{cite book | last=Coulomb |first=Charles Augustin | title=Histoire de l'Académie Royale des Sciences |year=1788 |publisher = Imprimerie Royale |chapter=Premier mémoire sur l'électricité et le magnétisme |pages=569–577 |orig-date=1785 |chapter-url=https://books.google.com/books?id=by5EAAAAcAAJ&pg=PA569}}
• {{cite book | last=Coulomb |first=Charles Augustin | title=Histoire de l'Académie Royale des Sciences |year=1788 |publisher = Imprimerie Royale |chapter=Second mémoire sur l'électricité et le magnétisme |pages=578–611 |orig-date=1785 |chapter-url=https://books.google.com/books?id=by5EAAAAcAAJ&pg=PA578}}
• {{cite book | last=Coulomb |first=Charles Augustin | title=Histoire de l'Académie Royale des Sciences |year=1788 |publisher = Imprimerie Royale |chapter=Troisième mémoire sur l'électricité et le magnétisme |pages=612–638 |orig-date=1785 |chapter-url=https://books.google.com/books?id=by5EAAAAcAAJ&pg=PA612}}
• {{cite book |last=Griffiths |first=David J. |title=Introduction to Electrodynamics |edition=3rd |publisher=Prentice Hall |year=1999 |isbn=978-0-13-805326-0 |url-access=registration |url=https://archive.org/details/introductiontoel00grif_0 }}
• {{cite book | author=Tamm, Igor E. | author-link=Igor Tamm |title= Fundamentals of the Theory of Electricity |edition=9th | publisher=Mir |year=1979 |url=https://archive.org/details/TammElectricity|pages=[https://archive.org/details/TammElectricity/page/n26 23]–27|orig-date=1976| location=Moscow}}
• {{cite book | last1=Tipler | first1=Paul A. | last2=Mosca | first2=Gene | title=Physics for Scientists and Engineers |edition=6th | publisher=W. H. Freeman and Company |location=New York | year=2008 | isbn=978-0-7167-8964-2 | lccn=2007010418}}
• {{cite book | last1= Young |first1 = Hugh D. | last2 = Freedman| first2 = Roger A. | title = Sears and Zemansky's University Physics: With Modern Physics| edition = 13th| year = 2010| publisher = Addison-Wesley (Pearson)| isbn = 978-0-321-69686-1}}

{{Commonscat}}

• [http://www.physnet.org/modules/pdf_modules/m114.pdf Coulomb's Law] on [http://www.physnet.org Project PHYSNET]
• [http://www.lightandmatter.com/html_books/4em/ch01/ch01.html Electricity and the Atom] {{Webarchive|url=https://web.archive.org/web/20090221030155/http://www.lightandmatter.com/html_books/4em/ch01/ch01.html |date=2009-02-21 }}—a chapter from an online textbook
• [http://mw2.concord.org/public/student/game/electrostatic_maze5.html A maze game for teaching Coulomb's law]—a game created by the Molecular Workbench software
• [https://web.archive.org/web/20090314044312/http://ocw.mit.edu/OcwWeb/Physics/8-02Electricity-and-MagnetismSpring2002/VideoAndCaptions/detail/embed01.htm Electric Charges, Polarization, Electric Force, Coulomb's Law] Walter Lewin, 8.02 Electricity and Magnetism, Spring 2002: Lecture 1 (video). MIT OpenCourseWare. License: Creative Commons Attribution-Noncommercial-Share Alike.

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Category:Electromagnetism

Category:Electrostatics

Category:Eponymous laws of physics

Category:Force

Category:Scientific laws