View source for Coulomb's law

{{Use American English|date=August 2019}}
{{Short description|Fundamental physical law of electromagnetism}}
{{Electromagnetism|cTopic=Electrostatics}}

'''Coulomb's law''', or '''Coulomb's inverse-square law''', is an experimental [[physical law|law]]<ref>{{harvnb|Huray|2010|p=57}}</ref> of [[physics]] that quantifies the amount of force between two stationary, [[electric charge|electrically charged]] particles.  The electric force between charged bodies at rest is conventionally called ''electrostatic force''<ref>{{harvnb|Walker|Halliday|Resnick|2014|p=611}}</ref> or '''Coulomb force'''.<ref>{{harvnb|Walker|Halliday|Resnick|2014|p=609}}</ref>  The quantity of electrostatic force between stationary charges is always described by Coulomb's law.<ref>{{harvnb|Jackson|1999|p=24}}</ref>  The law was first published in 1785 by French physicist [[Charles-Augustin de Coulomb]], and was essential to the development of the [[classical electromagnetism|theory of electromagnetism]], maybe even its starting point,<ref>{{harvnb|Huray|2010|p=2}}</ref> because it was now possible to discuss quantity of electric charge in a meaningful way.<ref>{{cite book |last1=Roller |first1=Duane |author-link1= |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, MA |publisher=[[Harvard University Press]] |page=[https://archive.org/details/developmentofcon0000roll/page/79 79] |isbn=}}</ref>

In its [[scalar (physics)|scalar]] form, the law is:

:<math>F=k_e\frac{q_1 q_2}{r^2},</math>

where ''k''{{sub|''e''}} is [[Coulomb's constant]] (''k''{{sub|''e''}} ≈ {{val|9e9|u=N⋅m{{sup|2}}⋅C{{sup|−2}}}}),<ref>{{harvnb|Huray|2010|p=7}}</ref> ''q''{{sub|1}} and ''q''{{sub|2}} are the signed magnitudes of the charges, and the scalar ''r'' is the distance between the charges. The force of the interaction between the charges is attractive if the charges have opposite signs (i.e., ''F'' is negative) and repulsive if like-signed (i.e., ''F'' is positive).

Being an [[inverse-square law]], the law is analogous to [[Isaac Newton]]'s inverse-square [[Newton's law of universal gravitation|law of universal gravitation]], but gravitational forces are always attractive, while electrostatic forces can be attractive or repulsive.<ref name="Walker 2014 614">{{harvnb|Walker|Halliday|Resnick|2014|p=614}}</ref> Coulomb's law can be used to derive [[Gauss's law]], and vice versa. In the case of a single stationary point charge, the two laws are equivalent, expressing the same physical law in different ways.<ref>{{harvnb|Purcell|Morin|2013|p=[https://books.google.com/books?id=A2rS5vlSFq0C&pg=PA25 25]}}</ref> The law has been [[tests of electromagnetism|tested extensively]], and observations have upheld the law on a scale from 10<sup>−16</sup> m to 10<sup>8</sup> m.<ref>{{harvnb|Purcell|Morin|2013|p=[https://books.google.com/books?id=A2rS5vlSFq0C&pg=PA11 11]}}</ref>

== History ==
<!--Please grow the size in px for these two side images as the history section grows-->
[[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 papers. [[Thales of Miletus]] made a series of observations on [[static electricity]] around 600 BC, from which he believed that [[friction]] rendered amber [[magnetic]], in contrast to minerals such as [[magnetite]], which needed no rubbing.<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
| postscript = <!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}}}
</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
| postscript = <!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}}}
</ref> Thales was incorrect in believing the attraction was due to a magnetic effect, but later science would prove a link between [[magnetism]] and [[electricity]]. Electricity would remain little more than an intellectual curiosity for millennia until 1600, when the 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 [[New 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
| postscript = <!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}}}
</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
| postscript = <!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}}}</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=Latin 
|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. 
|authorlink=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#v=onepage&q&f=false 460–462] and [https://books.google.com/books?id=UlTLRUn1sy8C&pg=PA464#v=onepage&q&f=false 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}}<br /><blockquote>May we not infer from this experiment, that the attraction of electricity is subject to the same laws with that of gravitation, and is therefore according to the squares of the distances; since it is easily demonstrated, that were the earth in the form of a shell, a body in the inside of it would not be attracted to one side more than another?</blockquote></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 
| isbn = 978-0-7803-5384-8 
| postscript = <!--None-->}}</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 
|location = London, England }}<br />On [https://books.google.com/books?id=8pRDAAAAcAAJ&pg=PA68&redir_esc=y#v=onepage&q&f=false page 68], the author states that in 1769 he announced his findings regarding the force between spheres of like charge. On [https://books.google.com/books?id=8pRDAAAAcAAJ&pg=PA73#v=onepage&q&f=false page 73], the author states the force between spheres of like charge varies as x<sup>−2.06</sup>:<br /><blockquote>The result of the whole was, that the mutual repulsion of two spheres, electrified positively or negatively, was very nearly in the inverse proportion of the squares of the distances of their centres, or rather in a proportion somewhat greater, approaching to x<sup>−2.06</sup>.</blockquote>When making experiments with charged spheres of opposite charge the results were similar, as stated on [https://books.google.com/books?id=8pRDAAAAcAAJ&pg=PA73#v=onepage&q&f=false page 73]:<br /><blockquote>When the experiments were repeated with balls having opposite electricities, and which therefore attracted each other, the results were not altogether so regular and a few irregularities amounted to {{frac|6}} of the whole; but these anomalies were as often on one side of the medium as on the other. This series of experiments gave a result which deviated as little as the former (or rather less) from the inverse duplicate ratio of the distances; but the deviation was in defect as the other was in excess.</blockquote>Nonetheless, on [https://books.google.com/books?id=8pRDAAAAcAAJ&pg=PA74#v=onepage&q&f=false page 74] the author infers that the actual action is related exactly to the inverse duplicate of the distance:<br /><blockquote>We therefore think that it may be concluded, that the action between two spheres is exactly in the inverse duplicate ratio of the distance of their centres, and that this difference between the observed attractions and repulsions is owing to some unperceived cause in the form of the experiment.</blockquote>On [https://books.google.com/books?id=8pRDAAAAcAAJ&pg=PA75#v=onepage&q&f=false page 75], the authour compares the electric and gravitational forces:<br /><blockquote>Therefore we may conclude, that the law of electric attraction and repulsion is similar to that of gravitation, and that each of those forces diminishes in the same proportion that the square of the distance between the particles increases.</blockquote></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 |origyear=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. |chapterurl=https://archive.org/stream/electricalresear00caveuoft#page/104/mode/2up}}<br />On [https://archive.org/stream/electricalresear00caveuoft#page/111/mode/2up pages 111 and 112] the author states:<blockquote>We may therefore conclude that the electric attraction and repulsion must be inversely as some power of the distance between that of the 2 + {{frac|50}} th and that of the 2 − {{frac|50}} th, and there is no reason to think that it differs at all from the inverse duplicate ratio.</blockquote></ref>

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.

== The law ==

Coulomb's law states that:

{{quote|The magnitude of the electrostatic [[force]] of attraction or repulsion  between two point [[electric charge|charges]] is directly proportional to the product  of the magnitudes of charges and inversely proportional to the square of the distance between them.<ref name="1785a">Coulomb (1785a) [https://books.google.com/books?id=by5EAAAAcAAJ&pg=PA569#v=onepage&q&f=false "Premier mémoire sur l’électricité et le magnétisme,"] ''Histoire de l’Académie Royale des Sciences'', pp. 569–577 — Coulomb studied the repulsive force between bodies having electrical charges of the same sign:
{{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.''

Translation: 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.|Coulomb (1785b) [https://books.google.com/books?id=by5EAAAAcAAJ&pg=PA578#v=onepage&q&f=false "Second mémoire sur l’électricité et le magnétisme,"] ''Histoire de l’Académie Royale des Sciences'', pages 578–611}} Coulomb also showed that oppositely charged bodies obey an inverse-square law of attraction.</ref>

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

[[Image:Coulombslaw.svg|center|A graphical representation of Coulomb's law]]

Coulomb's law can also be stated as a simple mathematical expression. The [[Scalar (mathematics)|scalar]] and [[Vector space|vector]] forms of the mathematical equation are
:<math>|\mathbf F|=k_e{|q_1q_2|\over r^2}\qquad</math> and <math>\qquad\mathbf F_1=k_e\frac{q_1q_2}{{|\mathbf r_{21}|}^2} \mathbf{\widehat{r}}_{21},\qquad</math> respectively,

where {{math|''k''{{sub|''e''}}}} is [[Coulomb's constant]] ({{math|''k''{{sub|''e''}}}}  = {{val|8.9875517873681764e9|u=N⋅m{{sup|2}}⋅C{{sup|−2}}}}), {{math|''q''{{sub|1}}}} and {{math|''q''{{sub|2}}}} are the signed magnitudes of the charges, the scalar {{math|''r''}} is the distance between the charges, the vector {{math|'''r'''{{sub|21}} {{=}} '''r'''{{sub|1}} − '''r'''{{sub|2}}}} is the vectorial distance between the charges, and {{math|'''r̂'''{{sub|21}} {{=}} {{sfrac|'''r'''{{sub|21}}|{{!}}'''r'''{{sub|21}}{{!}}}}}} (a unit vector pointing from {{math|''q''{{sub|2}}}} to {{math|''q''{{sub|1}}}}). The vector form of the equation calculates the force {{math|'''F'''{{sub|1}}}} applied on {{math|''q''{{sub|1}}}} by {{math|''q''{{sub|2}}}}. If {{math|'''r'''{{sub|12}}}} is used instead, then the effect on {{math|''q''{{sub|2}}}} can be found. It can be also calculated using [[Newton's third law]]: {{math|'''F'''{{sub|2}} {{=}} −'''F'''{{sub|1}}}}.

=== Units ===
When the [[electromagnetic theory]] is expressed in the [[International System of Units]], force is measured in [[Newton (unit)|newtons]], charge in [[coulomb]]s, and distance in [[metre|meter]]s. Coulomb's constant is given by {{math|''k''{{sub|''e''}} {{=}} {{sfrac|4π''ε''{{sub|0}}}}}}. The constant {{math|''ε''{{sub|0}}}} is the [[vacuum permittivity|vacuum electric permittivity]] (also known as "electric constant") <ref>{{SIbrochure9th}}, p. 15</ref> in C<sup>2</sup>⋅m<sup>−2</sup>⋅N<sup>−1</sup>. It should not be confused with {{math|''ε''{{sub|''r''}}}}, which is the dimensionless [[relative permittivity]] of the material in which the charges are immersed, or with their product {{math|''ε''{{sub|''a''}} {{=}} ''ε''{{sub|0}}''ε''{{sub|''r''}} }}, which is called "[[absolute permittivity]] of the material" and is still used in [[electrical engineering]].

The [[SI derived units]] for the electric field are [[volt]]s per meter, newtons per coulomb, or [[tesla (unit)|tesla]] meters per second.

Coulomb's law and [[Coulomb's constant]] can also be interpreted in various terms:
* [[Atomic units]]. In atomic units the force is expressed in [[hartree]]s per [[Bohr radius]], the charge in terms of the [[elementary charge]], and the distances in terms of the ''Bohr radius''.
* [[Electrostatic units]] or [[Gaussian units]]. In electrostatic units and Gaussian units, the unit charge (''esu'' or [[statcoulomb]]) is defined in such a way that the Coulomb constant {{math|''k''}} disappears because it has the value of one and becomes [[dimensionless]].
* [[Lorentz–Heaviside units]] (also called ''rationalized''). In Lorentz–Heaviside units the Coulomb constant is {{math|''k''{{sub|''e''}} {{=}} {{sfrac|4π}}}} and becomes [[dimensionless]].

[[Gaussian units]] and [[Lorentz–Heaviside units]] are both [[Centimetre–gram–second system of units|CGS unit systems]]. Gaussian units are more amenable for microscopic problems such as the electrodynamics of individual electrically charged particles.<ref name="Jackson 1999 784">{{harvnb|Jackson|1999|p=784}}</ref> SI units are more convenient for practical, large-scale phenomena, such as engineering applications.<ref name="Jackson 1999 784"/>

=== Electric field===
{{unreferenced|section|date=March 2019}}
{{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|test charge]]. In the simplest case, the field is considered to be generated solely by a single source [[point charge]]. The strength and direction of the Coulomb force {{math|'''F'''}} on a test charge {{math|''q''{{sub|t}}}} depends on the electric field {{math|'''E'''}} that it finds itself in, such that {{math|'''F''' {{=}} ''q''{{sub|t}}'''E'''}}. If the field is generated by a positive source point charge {{math|''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 {{math|''q''{{sub|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#The law|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]] {{math|''q''}} at a certain distance from it {{math|''r''}} in vacuum is given by:

:<math>|\boldsymbol{E}|={1\over4\pi\varepsilon_0}{|q|\over r^2}</math>

===Coulomb's constant===
{{main|Coulomb's constant}}

Coulomb's constant is a proportionality factor that appears in Coulomb's law as well as in other electric-related formulas. The value of this constant is dependent upon the medium that the charged objects are immersed in. Denoted {{math|''k''{{sub|''e''}}}}, it is also called the electric force constant or electrostatic constant,<ref name="Walker 2014 614"/>  hence the subscript {{math|''e''}}.

The exact value of Coulomb's constant in the case of air or vacuum is:
:<math>\begin{align}
k_e &= \frac{1}{4\pi\varepsilon_0}=\frac{c_0^2\mu_0}{4\pi}=c_0^2\times 10^{-7}\ \mathrm{H\cdot m}^{-1}\\
 &= 8.987\,551\,787\,368\,176\,4\times 10^9\ \mathrm{N\cdot m^2\cdot C}^{-2}
\end{align}</math>

===Limitations===
{{unreferenced|section|date=March 2019}}
There are three conditions to be fulfilled for the validity of Coulomb's law:
# 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 each other.

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|electrodynamic]] rules (incorporating the magnetic force) must be considered.

===Quantum field theory origin===
{{unreferenced|section|date=February 2018}}
[[File:Feynman diagram - Moller scattering 1.svg|thumb|The most basic Feynman diagram for QED interaction between two fermions]]
In simple terms, the [[Coulomb potential]] derives from the [[Quantum Electrodynamics|QED]] Lagrangian as follows. The [[Lagrangian (field theory)#Quantum electrodynamic Lagrangian|Lagrangian]] of quantum electrodynamics is normally written in [[natural units]], but in SI units, it is:

:<math>\mathcal{L}_\mathrm{QED} = \bar{\psi} ( i\hbar c \gamma^\mu D_\mu - m c^2) \psi - \frac{1}{4 \hbar c}F_{\mu \nu} F^{\mu \nu}</math>

where the covariant derivative (in SI units) is:

:<math>{D}_\mu = \partial_\mu + \frac{i g}{\hbar c} A_\mu</math>

where <math>g</math> is the [[coupling constant#gauge coupling|gauge coupling parameter]]. By putting the covariant derivative into the lagrangian explicitly, the interaction term (the term involving both <math>A</math> and <math>\psi</math>) is seen to be:

:<math>\mathcal{L}_\mathrm{int} = i g \bar{\psi} \gamma^\mu A_\mu \psi</math>

The most basic [[Feynman diagram]] for a QED interaction between two [[fermion]]s is the exchange of a single photon, with no loops. Following the [[Feynman rules]], this therefore contributes two QED vertex factors (<math>i g Q \gamma_\mu</math>) to the potential, where Q is the QED-[[Charge (physics)#abstract definition|charge operator]] (Q gives the charge in terms of the electron charge, and hence is exactly −1 for electrons, etc.). For the photon in the diagram, the Feynman rules demand the contribution of one bosonic massless [[propagator]] <math>\left(\frac{\hbar c}{k^2}\right)</math>. Ignoring the momentum on the external legs (the fermions), the potential is therefore:

:<math>V(\mathbf{r})=\frac{1}{(2\pi)^3} \int e^{\frac {i\mathbf{k \cdot r}}{\hbar}}
(i Q_1 g^2 \gamma_\mu)(i Q_2 g^2 \gamma_\nu) \left(\frac{\hbar c}{k^2}\right) \;d^3k</math>

which can be more usefully written as

:<math>V(\mathbf{r})= \frac{- g^2 \hbar c Q_1 Q_2}{4 \pi} \frac{1}{(2\pi)^3} \int e^{\frac {i\mathbf{k \cdot r}}{\hbar}} \frac {4\pi \eta_{\mu\nu} }{k^2} \;d^3k</math>

where <math>Q_i</math> is the QED-charge on the ''i''th particle. Recognising the integral as just being a [[Fourier transform]] enables the equation to be simplified:

:<math>V(\mathbf{r})= \frac{- g^2 \hbar c Q_1 Q_2 }{4 \pi} \frac{1}{r}.</math>

For various reasons, it is more convenient to define the [[fine-structure constant]] <math>\alpha = \frac{g^2}{4 \pi}</math>, and then define <math>e = \sqrt{4 \pi \alpha \varepsilon_0 \hbar c }</math>. Rearranging these definitions gives:

:<math>g^2 \hbar c = \frac{e^2}{\varepsilon_0}</math>

In natural units (<math>\hbar = 1</math>, <math>c = 1</math>, and <math>\varepsilon_0 = 1</math>), <math>g = e</math>. Continuing in SI units, the potential is therefore

:<math>V(\mathbf{r})= \frac{- e^2 }{4 \pi \varepsilon_0} \frac{Q_1 Q_2}{r}</math>

Defining <math>q_i = e Q_i</math>, as the macroscopic 'electric charge', makes e the macroscopic 'electric charge' for an electron, and enables the formula to be put into the familiar form of the Coulomb potential:

:<math>V(\mathbf{r})= \frac{-1}{4 \pi \varepsilon_0} \frac{q_1 q_2}{r}</math>

The force (<math>\frac{dV(\mathbf{r})}{dr}</math>) is therefore :

:<math>F(\mathbf{r})=\frac{1}{4\pi\varepsilon_0} \frac{q_1 q_2}{r^2}</math>

The derivation makes clear that the force law is only an approximation — it ignores the momentum of the input and output fermion lines, and ignores all quantum corrections (i.e. the myriad possible diagrams with internal loops).

The Coulomb potential, and its derivation, can be seen as a special case of the [[Yukawa potential]] (specifically, the case where the exchanged boson – the photon – has no rest mass).

==Scalar form==
[[Image:CoulombsLaw.svg|thumb|right|The absolute value of the force {{math|'''F'''}} between two [[point charge]]s {{math|''q''}} and {{math|''Q''}} relates to the distance between the point charges and to the simple product of their charges. The diagram shows that like charges repel each other, and opposite charges mutually attract.]]

When it is of interest to know the magnitude of the electrostatic force (and not its direction), it may be easiest to consider a scalar version of the law. The [[scalar (mathematics)|scalar form]] of Coulomb's Law relates the magnitude and sign of the electrostatic force {{math|'''F'''}} acting simultaneously on two point charges {{math|''q''{{sub|1}}}} and {{math|''q''{{sub|2}}}} as follows:

:<math>|\boldsymbol{F}|=k_e{|q_1q_2|\over r^2}</math>

where {{math|''r''}} is the separation distance and {{math|''k''{{sub|''e''}}}} is Coulomb's constant. If the product {{math|''q''{{sub|1}}''q''{{sub|2}}}} is positive, the force between the two charges is repulsive; if the product is negative, the force between them is attractive.<ref>[http://hyperphysics.phy-astr.gsu.edu/hbase/electric/elefor.html#c1 Coulomb's law], Hyperphysics</ref>

==Vector form==
[[File:Coulombslaw.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 states that the electrostatic force {{math|'''F'''{{sub|1}}}} experienced by a charge, {{math|''q''{{sub|1}}}} at position {{math|'''r'''{{sub|1}}}}, in the vicinity of another charge, {{math|''q''{{sub|2}}}} at position {{math|'''r'''{{sub|2}}}}, in a vacuum is equal to:

:<math>\boldsymbol{F_1}={q_1q_2\over4\pi\varepsilon_0}{(\boldsymbol{r_1-r_2})\over|\boldsymbol{r_1-r_2}|^3}={q_1q_2\over4\pi\varepsilon_0}{\boldsymbol{\widehat{r}_{21}}\over |\boldsymbol{r_{21}}|^2},</math>

where {{math|'''r'''{{sub|21}} {{=}} '''r'''{{sub|1}} − '''r'''{{sub|2}}}}, the unit vector {{math|'''r̂'''{{sub|21}} {{=}} {{sfrac|'''r'''{{sub|21}}|{{!}}'''r'''{{sub|21}}{{!}}}}}}, and {{math|''ε''{{sub|0}}}} is the [[electric constant]].

The vector form of Coulomb's law is simply the scalar definition of the law with the direction given by the [[unit vector]], {{math|'''r̂'''{{sub|21}}}}, parallel with the line ''from'' charge {{math|''q''{{sub|2}}}} ''to'' charge {{math|''q''{{sub|1}}}}.<ref name="uTexas">[http://farside.ph.utexas.edu/teaching/em/lectures/node28.html Coulomb's law], University of Texas</ref> If both charges have the same [[Plus and minus signs|sign]] (like charges) then the [[Scalar multiplication|product]] {{math|''q''{{sub|1}}''q''{{sub|2}}}} is positive and the direction of the force on {{math|''q''{{sub|1}}}} is given by {{math|'''r̂'''{{sub|21}}}}; the charges repel each other. If the charges have opposite signs then the product {{math|''q''{{sub|1}}''q''{{sub|2}}}} is negative and the direction of the force on {{math|''q''{{sub|1}}}} is given by {{math|−'''r̂'''{{sub|21}} {{=}} '''r̂'''{{sub|12}}}}; the charges attract each other.

The electrostatic force {{math|'''F'''{{sub|2}}}} experienced by {{math|''q''{{sub|2}}}}, according to [[Newton's laws of motion|Newton's third law]], is {{math|'''F'''{{sub|2}} {{=}} −'''F'''{{sub|1}}}}.

===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.

The force {{math|'''F'''}} on a small charge {{math|''q''}} at position {{math|'''r'''}}, due to a system of {{math|''N''}} discrete charges in vacuum is:

: <math>\boldsymbol{F(r)} = {q\over4\pi\varepsilon_0} \sum_{i=1}^N q_i{\boldsymbol{r-r_i}\over|\boldsymbol{r-r_i}|^3} = {q\over4\pi\varepsilon_0} \sum_{i=1}^Nq_i {\boldsymbol{\widehat{R_i}}\over|\boldsymbol{R_i}|^2},</math>

where {{math|''q''{{sub|''i''}}}} and {{math|'''r'''{{sub|''i''}}}} are the magnitude and position respectively of the {{math|''i''}}th charge, {{math|'''R̂'''{{sub|''i''}}}} is a unit vector in the direction of {{math|'''R'''{{sub|''i''}} {{=}} '''r''' − '''r'''{{sub|''i''}}}} (a vector pointing from charges {{math|''q''{{sub|''i''}}}} to {{math|''q''}}).<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''}}. 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|''λ''('''r′''')}} gives the charge per unit length at position {{math|'''r′'''}}, and {{math|''dℓ′''}} is an infinitesimal element of length,

:<math>dq = \lambda(\boldsymbol{r'})\,d\ell'.</math><ref>[http://dev.physicslab.org/Document.aspx?doctype=3&filename=Electrostatics_ContinuousChargedRod.xml Charged rods], PhysicsLab.org</ref>

For a surface charge distribution (a good approximation for charge on a plate in a parallel plate [[capacitor]]) where {{math|''σ''('''r′''')}} gives the charge per unit area at position {{math|'''r′'''}}, and {{math|''dA′''}} is an infinitesimal element of area,

:<math>dq = \sigma(\boldsymbol{r'})\,dA'.</math>

For a volume charge distribution (such as charge within a bulk metal) where {{math|''ρ''('''r′''')}} gives the charge per unit volume at position {{math|'''r′'''}}, and {{math|''dV′''}} is an infinitesimal element of volume,

:<math>dq = \rho(\boldsymbol{r'})\,dV'.</math><ref name="uTexas"/>

The force on a small test charge {{math|''q′''}} at position {{math|'''r'''}} in vacuum is given by the integral over the distribution of charge:

:<math>\boldsymbol{F} = {q'\over 4\pi\varepsilon_0}\int dq {\boldsymbol{r} - \boldsymbol{r'} \over |\boldsymbol{r} - \boldsymbol{r'}|^3}.</math>

== Simple experiment to verify Coulomb's law ==

[[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''}} and same-sign charge {{math|''q''}}, hanging from two ropes of negligible mass of length {{math|''l''}}. The forces acting on each sphere are three: the weight {{math|''mg''}}, the rope tension {{math|''T''}} and the electric force {{math|''F''}}.

In the equilibrium state:
{{NumBlk||
:<math>T \ \sin \theta_1 =F_1 \,\!</math>
|{{EquationRef|1}}}}

and:
{{NumBlk||
:<math>T \ \cos \theta_1 =mg \,\!</math>
|{{EquationRef|2}}}}

Dividing ({{EquationNote|1}}) by ({{EquationNote|2}}):
{{NumBlk||
:<math>\frac {\sin \theta_1}{\cos \theta_1 }=
\frac {F_1}{mg}\Rightarrow F_1= mg \tan \theta_1 </math>
|{{EquationRef|3}}}}

Let {{math|''L''{{sub|1}}}} be the distance between the charged spheres; the repulsion force between them {{math|''F''{{sub|1}}}}, assuming Coulomb's law is correct, is equal to

{{NumBlk||
: <math> F_1 = \frac{q^2}{4 \pi \varepsilon_0 L_1^2}</math>
|{{EquationRef|Coulomb's law}}}}

so:
{{NumBlk||
: <math>\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|{{sfrac|''q''|2}}}}. In the equilibrium state, the distance between the charges will be {{math|''L''{{sub|2}} < ''L''{{sub|1}}}} and the repulsion force between them will be:
{{NumBlk||
:<math>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|''F''{{sub|2}} {{=}} ''mg'' tan ''θ''{{sub|2}}}}. And:
:<math>\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>\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|''θ''{{sub|1}}}} and {{math|''θ''{{sub|2}}}} and the distance between the charges {{math|''L''{{sub|1}}}} and {{math|''L''{{sub|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||
:<math>\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>\frac{\frac{L_1}{2\ell}}{\frac{L_2}{2\ell}}\approx 4 {\left ( \frac {L_2}{L_1} \right ) }^2 \Rightarrow \,\!</math> <math>\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 check that the division approximates the theoretical value.

==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==

===Deriving Gauss's law from Coulomb's law===

Strictly speaking, [[Gauss's law]] cannot be derived from Coulomb's law alone, since Coulomb's law gives the electric field due to an individual [[point charge]] only. However, Gauss's law ''can'' be proven from Coulomb's law if it is assumed, in addition, that the electric field obeys the [[superposition principle]]. The superposition principle says that the resulting field is the vector sum of fields generated by each particle (or the integral, if the charges are distributed smoothly in space).

:{| class="toccolours collapsible collapsed" width="80%" style="text-align:left"
!Outline of proof
|-
|[[Coulomb's law]] states that the electric field due to a stationary [[point charge]] is:

:<math>\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>\mathbf{E}(\mathbf{r}) = \frac{1}{4\pi\varepsilon_0} \int \frac{\rho(\mathbf{s})(\mathbf{r}-\mathbf{s})}{|\mathbf{r}-\mathbf{s}|^3} \, \mathrm{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>See, for example, {{cite book | last=Griffiths | first=David J. | title=Introduction to Electrodynamics |edition=4th | publisher=Prentice Hall | year=2013 | page=50 }}</ref>

:<math>\nabla \cdot \left(\frac{\mathbf{r}}{|\mathbf{r}|^3}\right) = 4\pi \delta(\mathbf{r})</math>

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

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

Using the "[[Dirac delta function#Translation|sifting property]]" of the Dirac delta function, we arrive at

:<math>\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).

:{| class="toccolours collapsible collapsed" width="80%" style="text-align:left"
!Outline of 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>\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>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>\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.
|}

==See also==
{{commons category|Coulomb's law}}
{{Portal|Electronics}}
* [[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]]

==Notes==
{{Reflist}}

==References==
{{Refbegin}}
* {{cite book |last1=Huray |first1=Paul G. |title=Maxwell's equations |date=2010 |publisher=Wiley |location=Hoboken, NJ| isbn=0470542764 | ref = harv}}
* {{cite book  |last=Jackson | first=J. D.|year=1999  |edition=3rd | title=Classical Electrodynamics  |location=New York  |publisher=[[John Wiley & Sons|Wiley]] |oclc=318176085 |isbn=978-0-471-30932-1 |origyear=1962 |ref=harv}}
* {{cite book | last1= Purcell | first1= Edward M. |first2=David J. |last2=Morin | title= Electricity and Magnetism | publisher= Cambridge University Press | date= 2013 | edition= 3rd | url= https://books.google.com/books?id=A2rS5vlSFq0C | isbn= 9781107014022 | ref=harv}}
* {{Cite book|url=|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| ref=harv}}
{{Refend}}

==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 |origyear=1785 |chapterurl=https://books.google.com/books?id=by5EAAAAcAAJ&pg=PA569#v=onepage&q&f=false}}
* {{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 |origyear=1785 |chapterurl=https://books.google.com/books?id=by5EAAAAcAAJ&pg=PA578#v=onepage&q&f=false}}
* {{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 |origyear=1785 |chapterurl=https://books.google.com/books?id=by5EAAAAcAAJ&pg=PA612#v=onepage&q&f=false}}
* {{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=[[Igor Tamm|Tamm, Igor E.]]|title= Fundamentals of the Theory of Electricity |edition=9th | publisher=Mir |year=1979 |url=https://archive.org/details/TammElectricity/page/n25|pages=23–27|origyear=1976| location=Moscow| isbn= }}
* {{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==
* [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]—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.

[[Category:Electrostatics]]
[[Category:Force]]
[[Category:Scientific laws]]