View source for Coulomb's law

{{Short description|Fundamental physical law of electromagnetism}}
[[File:CoulombsLaw scal.svg|thumb|The magnitude of the electrostatic [[force]] {{math|'''''F'''''}} between two [[point charge]]s {{math|''q''<sub>1</sub>}} and {{math|''q''<sub>2</sub>}} is directly proportional to the product of the magnitudes of charges and inversely proportional to the square of the distance between them. Like charges repel each other, and opposite charges attract each other.]]
{{Use British English|date=August 2019}}
{{Electromagnetism|cTopic=Electrostatics}}
'''Coulomb's inverse-square law''', or simply '''Coulomb's law''', is an experimental [[scientific law|law]]<ref name=":0">{{Cite book | last=Huray |first= Paul G.|title=Maxwell's equations|date=2010|publisher=Wiley| isbn=978-0-470-54991-9 |location=Hoboken, New Jersey|pages=8, 57|oclc=739118459}}</ref> of [[physics]] that calculates the amount of force between two [[electric charge|electrically charged]] particles at rest. This electric force is conventionally called the ''electrostatic force'' or '''Coulomb force'''.<ref name="halliday">{{cite book |last1=Halliday |first1=David |last2=Resnick |first2=Robert |last3=Walker |first3=Jearl |title=Fundamentals of Physics |date=2013 |publisher=John Wiley & Sons |isbn=9781118230718 |pages=609, 611}}</ref> 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 [[classical electromagnetism|theory of electromagnetism]] and maybe even its starting point,'''<ref name=":0" />''' as it allowed meaningful discussions of the amount of electric charge in a particle.<ref>{{cite book |last1=Roller |first1=Duane | last2=Roller |first2=D. H. D.|date=1954 |title=The development of the concept of electric charge: Electricity from the Greeks to Coulomb |url=https://archive.org/details/developmentofcon0000roll|url-access=registration |location=Cambridge, Massachusetts |publisher=[[Harvard University Press]] |page=[https://archive.org/details/developmentofcon0000roll/page/79 79] }}</ref>

The law states that the magnitude, or absolute value, of the attractive or repulsive electrostatic [[force]] between two point [[Electric charge|charges]] is directly proportional to the product of the magnitudes of their charges and inversely proportional to the squared distance between them.<ref name="1785a">{{cite book|last=Coulomb|year=1785 | chapter-url=https://books.google.com/books?id=by5EAAAAcAAJ&pg=PA569|chapter=Premier mémoire sur l'électricité et le magnétisme|trans-chapter=First dissertation on electricity and magnetism|title=Histoire de l'Académie Royale des Sciences|trans-title=History of the Royal Academy of Sciences|language=fr|pages=569–577}}</ref> Coulomb discovered that bodies with like electrical charges repel:

{{quote|It follows therefore from these three tests, that the repulsive force that the two balls – [that were] electrified with the same kind of electricity – exert on each other, follows the inverse proportion of the square of the distance.<ref name="1785b">{{cite book|last=Coulomb|year=1785|chapter-url=https://books.google.com/books?id=by5EAAAAcAAJ&pg=PA578|chapter=Second<!--sic--> mémoire sur l'électricité et le magnétisme|trans-chapter=Second dissertation on electricity and magnetism|title=Histoire de l'Académie Royale des Sciences|trans-title=History of the Royal Academy of Sciences|language=fr|pages=578–611|quote=Il résulte donc de ces trois essais, que l'action répulsive que les deux balles électrifées de la même nature d'électricité exercent l'une sur l'autre, suit la raison inverse du carré des distances.}}</ref>}}

Coulomb also showed that oppositely charged bodies attract according to an inverse-square law:
<math display="block">|F|=k_\text{e} \frac{|q_1||q_2|}{r^2}</math>

Here, {{math|''k''{{sub|e}}}} is a constant, {{math|''q''{{sub|1}}}} and {{math|''q''{{sub|2}}}} 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 [[Newton's law of universal gravitation|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.<ref name="halliday"/> 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.<ref name="Purcell">{{cite book |last1=Purcell |first1=Edward M. |title=Electricity and magnetism |date=21 January 2013 | location=Cambridge |isbn=9781107014022 |edition=3rd}}</ref> The law has been [[tests of electromagnetism|tested extensively]], and observations have upheld the law on the scale from 10<sup>−16</sup> m to 10<sup>8</sup> m.<ref name="Purcell"/>

== History ==
[[File:Coulomb.jpg|thumb|left|Charles-Augustin de Coulomb]]
Ancient cultures around the [[Mediterranean Sea|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,<ref>{{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 }}</ref> when he noticed that [[friction]] could make a piece of amber attract small objects.<ref name="stewart">{{Cite book |first=Joseph |last=Stewart |title=Intermediate Electromagnetic Theory |publisher=World Scientific |year=2001 | page=50 | isbn=978-981-02-4471-2}}</ref><ref>{{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}}</ref>

In 1600, English scientist [[William Gilbert (astronomer)|William Gilbert]] made a careful study of electricity and magnetism, distinguishing the [[lodestone]] effect from static electricity produced by rubbing amber.<ref name="stewart" /> 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.<ref>{{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}}</ref> 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.<ref>{{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 }}</ref>

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]]<ref>{{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}}</ref> 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.<ref>{{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)}}</ref>

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.<ref>{{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}}</ref> In 1767, he conjectured that the force between charges varied as the inverse square of the distance.<ref>{{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}}</ref><ref>{{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 }}</ref>

[[Image:Bcoulomb.png|thumb|Coulomb's [[torsion balance]]]]
In 1769, Scottish physicist [[John Robison (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''<sup>−2.06</sup>}}.<ref>{{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}}</ref>

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.<ref>{{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}}</ref> 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 [[classical electromagnetism|theory of electromagnetism]].<ref name="1785a"/> He used a [[torsion balance]] to study the repulsion and attraction forces of [[charged particle]]s, and determined that the magnitude of the electric force between two [[point charge]]s 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 [[Insulator (electricity)|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 [[scientific instrument|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.

== Scalar form ==
Coulomb's law can be stated as a simple mathematical expression. The [[Scalar (mathematics)|scalar]] form gives the magnitude of the vector of the electrostatic force {{math|'''F'''}} between two point charges {{math|''q''<sub>1</sub>}} and {{math|''q''<sub>2</sub>}}, but not its direction. If {{mvar|r}} is the distance between the charges, the magnitude of the force is
<math display="block">|\mathbf{F}|=\frac{|q_1q_2|}{4\pi\varepsilon_0 r^2},</math>
where {{math|''ε''<sub>0</sub>}} is the [[electric constant]]. If the product {{math|''q''<sub>1</sub>''q''<sub>2</sub>}} is positive, the force between the two charges is repulsive; if the product is negative, the force between them is attractive.<ref>{{cite web|url=http://hyperphysics.phy-astr.gsu.edu/hbase/electric/elefor.html#c1|title=Coulomb's law|website=Hyperphysics|access-date=2007-11-30|archive-date=2019-04-13|archive-url=https://web.archive.org/web/20190413184345/http://hyperphysics.phy-astr.gsu.edu/hbase/electric/elefor.html#c1|url-status=live}}</ref>

== Vector form ==
[[File:Coulombslawgraph.svg|thumb|right|350px|In the image, the vector {{math|'''F'''{{sub|1}}}} is the force experienced by {{math|''q''{{sub|1}}}}, and the vector {{math|'''F'''{{sub|2}}}} is the force experienced by {{math|''q''{{sub|2}}}}. When {{math|''q''{{sub|1}}''q''{{sub|2}} > 0}} the forces are repulsive (as in the image) and when {{math|''q''{{sub|1}}''q''{{sub|2}} < 0}} the forces are attractive (opposite to the image). The magnitude of the forces will always be equal.]]

Coulomb's law in vector form states that the electrostatic force <math display="inline">\mathbf{F}_1</math> experienced by a charge, <math>q_1</math> at position <math>\mathbf{r}_1</math>, in the vicinity of another charge, <math>q_2</math> at position <math>\mathbf{r}_2</math>, in a vacuum is equal to<ref name=":1" />
<math display="block">\mathbf{F}_1 = \frac{q_1q_2}{4\pi\varepsilon_0} \frac{\mathbf{r}_1-\mathbf{r}_2}{|\mathbf{r}_1 - \mathbf{r}_2|^3} = \frac{q_1q_2}{4\pi\varepsilon_0} \frac{\mathbf{\hat{r}}_{12}}{|\mathbf{r}_{12}|^2}</math>

where <math display="inline">\boldsymbol{r}_{12} = \boldsymbol{r}_1 - \boldsymbol{r}_2</math> is the vectorial distance between the charges, <math display="inline">\widehat{\mathbf{r}}_{12}=\frac{\mathbf{r}_{12}}{|\mathbf{r}_{12}|}</math> a unit vector pointing from <math display="inline">q_2</math> to {{nowrap|<math display="inline">q_1</math>,}} and <math>\varepsilon_0</math> the [[electric constant]]. Here, <math display="inline">\mathbf{\hat{r}}_{12}</math> is used for the vector notation.

The vector form of Coulomb's law is simply the scalar definition of the law with the direction given by the [[unit vector]], {{nowrap|<math display="inline">\widehat{\mathbf{r}}_{12}</math>,}} parallel with the line ''from'' charge <math>q_2</math> ''to'' charge <math>q_1</math>.<ref name="uTexas">{{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}}</ref> If both charges have the same [[Plus and minus signs|sign]] (like charges) then the [[Scalar multiplication|product]] <math>q_1q_2</math> is positive and the direction of the force on <math>q_1</math> is given by <math display="inline">\widehat{\mathbf{r}}_{12}</math>; the charges repel each other. If the charges have opposite signs then the product <math>q_1q_2</math> is negative and the direction of the force on <math>q_1</math> is {{nowrap|<math display="inline">-\hat{\mathbf{r}}_{12}</math>;}} the charges attract each other.

The electrostatic force <math display="inline">\mathbf{F}_2</math> experienced by <math>q_2</math>, according to [[Newton's laws of motion|Newton's third law]], is {{nowrap|<math display="inline">\mathbf{F}_2=-\mathbf{F}_1</math>.}}

=== System of discrete charges ===
The [[superposition principle|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 <math display="inline">\mathbf{F}</math> on a small charge <math>q</math> at position <math display="inline">\mathbf{r}</math>, due to a system of <math display="inline">N</math> discrete charges in vacuum is<ref name=":1" />

<math display="block">\mathbf{F}(\mathbf{r}) = {q\over4\pi\varepsilon_0} \sum_{i=1}^N q_i \frac{\mathbf{r} - \mathbf{r}_i}{|\mathbf{r}- \mathbf{r}_i|^3} = {q\over 4\pi\varepsilon_0} \sum_{i=1}^N q_i {\hat{\mathbf{R}}_i\over|\mathbf{R}_i|^2},</math>

where <math>q_i</math> and <math display="inline">\mathbf{r}_i</math> are the magnitude and position respectively of the {{math|''i''}}th charge, <math display="inline">\hat{\mathbf{R}}_i</math> is a unit vector in the direction of <math display="inline">\mathbf{R}_i = \mathbf{r}-\mathbf{r}_i</math>, a vector pointing from charges <math>q_i</math> to <math>q</math>.<ref name="uTexas" />

=== Continuous charge distribution ===
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 <math>dq</math>. The distribution of charge is usually linear, surface or volumetric.

For a linear charge distribution (a good approximation for charge in a wire) where <math>\lambda(\mathbf{r}')</math> gives the charge per unit length at position <math>\mathbf{r}'</math>, and <math>d\ell'</math> is an infinitesimal element of length,<ref>{{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}}</ref>
<math display="block">dq' = \lambda(\mathbf{r'}) \, d\ell'.</math>

For a surface charge distribution (a good approximation for charge on a plate in a parallel plate [[capacitor]]) where <math>\sigma(\mathbf{r}')</math> gives the charge per unit area at position <math>\mathbf{r}'</math>, and <math>dA'</math> is an infinitesimal element of area,
<math display="block">dq' = \sigma(\mathbf{r'})\,dA'.</math>

For a volume charge distribution (such as charge within a bulk metal) where <math>\rho(\mathbf{r}')</math> gives the charge per unit volume at position <math>\mathbf{r}'</math>, and <math>dV'</math> is an infinitesimal element of volume,<ref name="uTexas" />
<math display="block">dq' = \rho(\boldsymbol{r'})\,dV'.</math>

The force on a small test charge <math>q</math> at position <math>\boldsymbol{r}</math> in vacuum is given by the integral over the distribution of charge
<math display="block">\mathbf{F}(\mathbf{r}) = \frac{q}{4\pi\varepsilon_0}\int dq' \frac{\mathbf{r} - \mathbf{r'}}{|\mathbf{r} - \mathbf{r'}|^3}.</math>

The "continuous charge" version of Coulomb's law is never supposed to be applied to locations for which <math>|\mathbf{r} - \mathbf{r'}| = 0</math> 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 <math>|\mathbf{r} - \mathbf{r'}| = 0</math> to be analyzed.
<!-- Coulomb's law in media other than vacuum. (the only mention was in "The value of this constant is dependent upon the medium that the charged objects are immersed in.") -->

== Coulomb constant ==
The Coulomb constant is a proportionality factor that appears in Coulomb's law and related formulas. Denoted <math>k_\text{e}</math>, it is also called the electric force constant or electrostatic constant<ref>{{Cite book | title=Fundamentals of physics| last1=Walker| first1=Jearl | date=2014 | publisher=Wiley |last2=Halliday |first2=David |last3=Resnick |first3=Robert |isbn=9781118230732 |edition=10th |location=Hoboken, NJ| oclc=950235056| page=614}}</ref> hence the subscript 'e'. The Coulomb constant is given by <math display="inline">k_\text{e} = \frac{1}{4\pi\varepsilon_0}</math>. The constant <math>\varepsilon_0</math> is the [[vacuum permittivity|vacuum electric permittivity]] (also known as the ''electric constant'').<ref>{{SIbrochure9th|page=15}}</ref> It should not be confused with <math>\varepsilon_r</math>, which is the dimensionless [[relative permittivity]] of the material in which the charges are immersed, or with their product <math>\varepsilon_a = \varepsilon_0\varepsilon_r</math>, which is called "[[absolute permittivity]] of the material" and is still used in [[electrical engineering]].

Since the [[2019 redefinition of the SI base units]],<ref name="SI-statement">{{citation | url = https://www.bipm.org/utils/common/pdf/SI-statement.pdf | title = BIPM statement: Information for users about the proposed revision of the SI | access-date = 2020-04-15 | archive-date = 2018-01-21 | archive-url = https://web.archive.org/web/20180121160000/https://www.bipm.org/utils/common/pdf/SI-statement.pdf | url-status = dead }}</ref><ref>{{cite web | url = http://www.bipm.org/en/committees/cipm/meeting/105.html | title = Decision CIPM/105-13 (October 2016) | access-date = 2020-04-15 | archive-date = 2017-08-24 | archive-url = https://web.archive.org/web/20170824095943/http://www.bipm.org/en/committees/cipm/meeting/105.html | url-status = live }} The day is the 144th anniversary of the [[Metre Convention]].</ref> the Coulomb constant, as calculated from [[CODATA]] 2018 recommended values, is<ref name="ke">Derived from {{math|1=''k''<sub>\text{e}</sub> = 1 / (4''πε''<sub>0</sub>)}} – {{cite web |url=http://physics.nist.gov/cgi-bin/cuu/Value?ep0 |title=2018 CODATA Value: vacuum electric permittivity |date=20 May 2019 |work=The NIST Reference on Constants, Units, and Uncertainty |publisher=[[National Institute of Standards and Technology|NIST]] |access-date=2019-05-20 |archive-date=2016-06-03 |archive-url=https://web.archive.org/web/20160603013303/http://physics.nist.gov/cgi-bin/cuu/Value?ep0 |url-status=live }}</ref>
<math display="block">k_\text{e} = 8.987\,551\,792\,3\,(14)\times 10^9\ \mathrm{N{\cdot}m^2{\cdot}C^{-2}} .</math>

== Limitations ==
There are three conditions to be fulfilled for the validity of Coulomb's inverse square law:<ref>{{cite book|title=Discussion on physics teaching innovation: Taking Coulomb's law as an example|date=2015-07-28| work=Education Management and Management Science|pages=465–468|publisher=CRC Press| doi=10.1201/b18636-105| isbn=978-0-429-22704-2}}</ref>
# The charges must have a spherically symmetric distribution (e.g. be point charges, or a charged metal sphere).
# The charges must not overlap (e.g. they must be distinct point charges).
# The charges must be stationary with respect to a nonaccelerating frame of reference.

The last of these is known as the [[electrostatics#Electrostatic approximation|electrostatic approximation]]. When movement takes place, [[Albert Einstein|Einstein]]'s [[theory of relativity]] must be taken into consideration, and a result, 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, and is described by [[magnetic field]]s. For slow movement, the magnetic force is minimal and Coulomb's law can still be considered approximately correct, but when the charges are moving more quickly in relation to each other, the full [[electrodynamics]] rules (incorporating the magnetic force) must be considered.

== Electric field ==
{{main|Electric field}}
[[File:Electric field one charge changing.gif|thumb|If two charges have the same sign, the electrostatic force between them is repulsive; if they have different sign, the force between them is attractive.]]

An electric field is a [[vector field]] that associates to each point in space the Coulomb force experienced by a [[Test particle#Test particles in plasma physics or electrodynamics|unit test charge]].<ref name=":1">{{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}}</ref> The strength and direction of the Coulomb force <math display="inline">\mathbf F</math> on a charge <math display="inline">q_t</math> depends on the electric field <math display="inline">\mathbf{E}</math> established by other charges that it finds itself in, such that <math display="inline">\mathbf{F} = q_t \mathbf{E}</math>. 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 [[Superposition principle|principle of superposition]].

If the field is generated by a positive source point charge <math display="inline">q</math>, 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 <math display="inline">q_t</math> 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
<math display="block">|\mathbf{E}| = k_\text{e} \frac{|q|}{r^2}</math>

A system ''N'' of charges <math>q_i</math> stationed at <math display="inline">\mathbf{r}_i</math> produces an electric field whose magnitude and direction is, by superposition
<math display="block">\mathbf{E}(\mathbf{r}) = {1\over4\pi\varepsilon_0} \sum_{i=1}^N q_i
\frac{\mathbf{r}-\mathbf{r}_i}{|\mathbf{r}-\mathbf{r}_i|^3}</math>

== Atomic forces ==
{{See also|Coulomb explosion}}
Coulomb's law holds even within [[atom]]s, correctly describing the [[force]] between the positively charged [[atomic nucleus]] and each of the negatively charged [[electron]]s. This simple law also correctly accounts for the forces that bind atoms together to form [[molecule]]s and for the forces that bind atoms and molecules together to form solids and liquids. Generally, as the distance between [[ion]]s 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.

== Relation to Gauss's law ==
{{Duplication|dupe=Gauss's_law#Relation_to_Coulomb's_law}}

=== Deriving Gauss's law from Coulomb's law ===
[[Gauss's law]] can be derived from Coulomb's law and the assumption that electric field obeys the [[superposition principle]], which says that the resulting field is the vector sum of fields generated by each particle (or the integral, if the charges are distributed in a region of space).

{{math proof|title=Outline of proof| proof=
Coulomb's law states that the electric field due to a stationary [[point charge]] is:
<math display="block">\mathbf{E}(\mathbf{r}) = \frac{q}{4\pi \varepsilon_0} \frac{\mathbf{e}_r}{r^2},</math>
where
* {{math|'''e'''<sub>''r''</sub>}} is the radial [[unit vector]],
* {{mvar|r}} is the radius, {{math|{{abs|'''r'''}}}},
* {{math|''ε''<sub>0</sub>}} is the [[electric constant]],
* {{mvar|q}} is the charge of the particle, which is assumed to be located at the [[origin (mathematics)|origin]].

Using the expression from Coulomb's law, we get the total field at {{math|'''r'''}} by using an integral to sum the field at {{math|'''r'''}} due to the infinitesimal charge at each other point {{math|'''s'''}} in space, to give
<math display="block">\mathbf{E}(\mathbf{r}) = \frac{1}{4\pi\varepsilon_0} \int \frac{\rho(\mathbf{s})(\mathbf{r}-\mathbf{s})}{|\mathbf{r}-\mathbf{s}|^3} \, d^3 \mathbf{s}</math>
where {{mvar|ρ}} is the charge density. If we take the divergence of both sides of this equation with respect to '''r''', and use the known theorem<ref>{{cite book | last=Griffiths | first=David J. | title=Introduction to Electrodynamics | edition=4th | publisher=Prentice Hall | year=2013 | page=50 }}</ref>

<math display="block">\nabla \cdot \frac{\mathbf{r}}{|\mathbf{r}|^3}=-\nabla^2\frac1{|\mathbf{r}|} = 4\pi \delta(\mathbf{r})</math>

where {{math|''δ''('''r''')}} is the [[Dirac delta function]], the result is

<math display="block">\nabla\cdot\mathbf{E}(\mathbf{r}) = \frac{1}{\varepsilon_0} \int \rho(\mathbf{s})\, \delta(\mathbf{r}-\mathbf{s})\, d^3 \mathbf{s}</math>

Using the "[[Dirac delta function#Translation|sifting property]]" of the Dirac delta function, we arrive at
<math display="block">\nabla\cdot\mathbf{E}(\mathbf{r}) = \frac{\rho(\mathbf{r})}{\varepsilon_0},</math>
which is the differential form of Gauss' law, as desired.
}}

Note that since Coulomb's law only applies to stationary charges, there is no reason to expect Gauss's law to hold for moving charges based on this derivation alone. In fact, Gauss's law does hold for moving charges, and in this respect Gauss's law is more general than Coulomb's law.

=== Deriving Coulomb's law from Gauss'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 (mathematics)|curl]] of {{math|'''E'''}} (see [[Helmholtz decomposition]] and [[Faraday's law of induction|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' law to be a spherical surface of radius {{mvar|r}}, centered at the point charge {{mvar|Q}}, we have
<math display="block">\oint_S\mathbf{E}\cdot d\mathbf{A} = \frac{Q}{\varepsilon_0} </math>

By the assumption of spherical symmetry, the integrand is a constant which can be taken out of the integral. The result is
<math display="block">4\pi r^2\hat{\mathbf{r}}\cdot\mathbf{E}(\mathbf{r}) = \frac{Q}{\varepsilon_0}</math>
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
<math display="block">\mathbf{E}(\mathbf{r}) = \frac{Q}{4\pi \varepsilon_0} \frac{\hat{\mathbf{r}}}{r^2}</math>
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' law.
}}

== In relativity ==
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 equations|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|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|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#Lorentz force law as the definition of E and B|Lorentz force]].<ref>{{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 }}</ref> The fields hence found for uniformly moving point charges are given by:<ref>{{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 }}</ref><math display="block">\mathbf{E} = \frac q {4 \pi \epsilon_0 r^3}   \frac {1- \beta^2} {(1- \beta^2 \sin^2 \theta)^{3/2}} \mathbf{r}</math><math display="block">\mathbf{B} = \frac q {4 \pi \epsilon_0 r^3}   \frac {1- \beta^2} {(1- \beta^2 \sin^2 \theta)^{3/2}} \frac{\mathbf{v} \times \mathbf{r}}{c^2} = \frac{\mathbf{v} \times \mathbf{E}}{c^2}</math>where <math>q</math> is the charge of the point source, <math>\mathbf{r}</math> is the position vector from the point source to the point in space, <math>\mathbf{v}</math> is the velocity vector of the charged particle, <math>\beta</math> is the ratio of speed of the charged particle divided by the speed of light and <math>\theta</math> is the angle between <math>\mathbf{r}</math> and <math>\mathbf{v}</math>.

This form of solutions need not obey [[Newton's laws of motion|Newton's third law]] as is the case in the framework of [[special relativity]] (yet without violating relativistic-energy momentum conservation).<ref>{{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}}</ref> 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 <math>\beta\ll 1</math>) 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.<ref>{{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 }}</ref>

== Coulomb potential ==
{{See also|Electric potential}}
<!--define Coulomb potential-->

=== Quantum field theory ===
{{Jargon cleanup|date=October 2020}}
[[File:Feynman diagram - Moller scattering 1.svg|thumb|The most basic Feynman diagram for QED interaction between two fermions]]
The '''Coulomb potential''' admits continuum states (with ''E'' > 0), describing electron-proton [[Scattering theory|scattering]], as well as discrete bound states, representing the hydrogen atom.<ref name="griffiths">{{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}}</ref> It can also be derived within the [[Non-relativistic quantum electrodynamics|non-relativistic limit]] between two charged particles, as follows:

Under [[Born approximation]], in non-relativistic quantum mechanics, the scattering amplitude <math display="inline"> \mathcal{A}(| \mathbf{p} \rangle \to | \mathbf{p}'\rangle)</math> is:
<math display="block">\mathcal{A}(| \mathbf{p} \rangle \to | \mathbf{p}'\rangle) - 1 = 2\pi \delta(E_p - E_{p'})(-i)\int d^3\mathbf r \, V(\mathbf r) e^{-i(\mathbf p - \mathbf p')\mathbf r}</math>
This is to be compared to the:
<math display="block">\int \frac{d^3k}{(2\pi)^3} e^{i k r_0} \langle p',k |S| p,k \rangle</math>
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 <math>m_0 \gg | \mathbf p|</math>
<math display="block">\langle p',k | S | p,k \rangle |_{conn} = -i\frac{e^2}{| \mathbf p - \mathbf p'|^2 - i\varepsilon}(2m)^2\delta(E_{p,k} - E_{p',k})(2\pi)^4\delta(\mathbf p - \mathbf p')</math>

Comparing with the QM scattering, we have to discard the <math>(2m)^2</math> as they arise due to differing normalizations of momentum eigenstate in QFT compared to QM and obtain:
<math display="block">\int V(\mathbf r)e^{-i(\mathbf p - \mathbf p') \mathbf r}d^3\mathbf r = \frac{e^2}{| \mathbf p - \mathbf p'|^2 - i\varepsilon}</math>
where Fourier transforming both sides, solving the integral and taking <math>\varepsilon \to 0</math> at the end will yield
<math display="block">V(r) = \frac{e^2}{4\pi r}</math>
as the Coulomb potential.<ref>{{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}}</ref>

However, the equivalent results of the classical Born derivations for the Coulomb problem are thought to be strictly accidental.<ref>{{Cite book|last=Baym, Gordon|title=Lectures on quantum mechanics|year=2018| isbn=978-0-429-49926-5|location=Boca Raton|oclc=1028553174}}</ref><ref>{{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}}</ref>

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.<ref name="griffiths" />

== Verification ==
{{Overly detailed|date=October 2020|section}}[[File:Verificacion ley coulomb.png|250px|thumb|Experiment to verify Coulomb's law.]]

It is possible to verify Coulomb's law with a simple experiment. Consider two small spheres of mass <math>m</math> and same-sign charge <math>q</math>, hanging from two ropes of negligible mass of length <math>l</math>. The forces acting on each sphere are three: the weight <math>mg</math>, the rope tension <math>\mathbf T</math> and the electric force <math>\mathbf F</math>. In the equilibrium state: 
{{NumBlk||<math display="block">\mathbf T \sin \theta_1 =\mathbf F_1</math>|{{EquationRef|1}}}}
and
{{NumBlk||<math display="block">\mathbf T \cos \theta_1 = mg</math>|{{EquationRef|2}}}}
Dividing ({{EquationNote|1}}) by ({{EquationNote|2}}):
{{NumBlk||<math display="block">\frac {\sin \theta_1}{\cos \theta_1 }=\frac {F_1}{mg}\Rightarrow F_1= mg \tan \theta_1 </math>|{{EquationRef|3}}}}
Let <math>\mathbf L_1</math> be the distance between the charged spheres; the repulsion force between them <math>\mathbf F_1</math>, assuming Coulomb's law is correct, is equal to
{{NumBlk||<math display="block"> F_1 = \frac{q^2}{4 \pi \varepsilon_0 L_1^2}</math>
|{{EquationRef|Coulomb's law}}}}
so:
{{NumBlk||<math display="block">\frac{q^2}{4 \pi \varepsilon_0 L_1^2}=mg \tan \theta_1 </math>|{{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 <math display="inline">\frac{q}{2}</math>. In the equilibrium state, the distance between the charges will be <math display="inline">\mathbf L_2 < \mathbf L_1</math> and the repulsion force between them will be:
{{NumBlk||<math display="block">F_2 = \frac{{(\frac{q}{2})}^2}{4 \pi \varepsilon_0 L_2^2}=\frac{\frac{q^2}{4}}{4 \pi \varepsilon_0 L_2^2} </math>|{{EquationRef|5}}}}
We know that <math>\mathbf F_2 = mg \tan \theta_2</math> and:
<math display="block">\frac{\frac{q^2}{4}}{4 \pi \varepsilon_0 L_2^2}=mg \tan \theta_2</math>
Dividing ({{EquationNote|4}}) by ({{EquationNote|5}}), we get: 
{{NumBlk||<math display="block">\frac{\left(\cfrac{q^2}{4 \pi \varepsilon_0 L_1^2} \right)}{\left(\cfrac{\frac{q^2}{4}}{4 \pi \varepsilon_0 L_2^2}\right)} = \frac{mg \tan \theta_1}{mg \tan \theta_2} \Rightarrow 4 {\left( \frac {L_2}{L_1} \right) }^2 = \frac{ \tan \theta_1}{ \tan \theta_2}</math>|{{EquationRef|6}}}}
Measuring the angles <math>\theta_1</math> and <math>\theta_2</math> and the distance between the charges <math>\mathbf L_1</math> and <math>\mathbf L_2</math> 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||<math display="block">\tan \theta \approx \sin \theta = \frac{\frac{L}{2}}{\ell} = \frac{L}{2\ell} \Rightarrow \frac{ \tan \theta_1}{ \tan \theta_2} \approx \frac{\frac{L_1}{2\ell}}{\frac{L_2}{2\ell}}</math>|{{EquationRef|7}}}}
Using this approximation, the relationship ({{EquationNote|6}}) becomes the much simpler expression:
{{NumBlk||<math display="block">\frac{\frac{L_1}{2\ell}}{\frac{L_2}{2\ell}}\approx 4 {\left ( \frac {L_2}{L_1} \right ) }^2 \Rightarrow \frac{L_1}{L_2}\approx 4 {\left ( \frac {L_2}{L_1} \right ) }^2 \Rightarrow \frac{L_1}{L_2}\approx\sqrt[3]{4}</math>|{{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.

== See also ==
{{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}}

== References ==
{{Reflist|30em}}

== Related reading ==
* {{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}}

== External links ==
{{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.

{{Authority control}}

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