Editing Electric field

{{Short description|Physical field surrounding an electric charge}}
{{Redirect|Electric fields|the Australian band|Electric Fields}}
{{Infobox physical quantity
|bgcolour={default}
|name = Electric field
|symbols = {{math|'''E'''}}
|unit = [[volt]] per [[meter]] (V/m) 
|baseunits = kg⋅m⋅s<sup>−3</sup>⋅A<sup>−1</sup>
|dimension='''M''' '''L''' '''T'''<sup>−3</sup> '''I'''<sup>−1</sup>}}
{{Electromagnetism|cTopic=Electrostatics}}

An '''electric field''' (sometimes called '''E-field'''<ref>{{cite journal |last1=Roche |first1=John |title=Introducing electric fields |journal=Physics Education |date=2016 |volume=51 |issue=5 |page=055005|doi=10.1088/0031-9120/51/5/055005 |bibcode=2016PhyEd..51e5005R |s2cid=125014664 }}</ref>) is a [[field (physics)|physical field]] that surrounds electrically [[charged particle]]s such as [[electron]]s. In classical [[electromagnetism]], the electric field of a single charge (or group of charges) describes their capacity to exert attractive or repulsive forces on another charged object. Charged particles exert attractive forces on each other when the sign of their charges are opposite, one being positive while the other is negative, and repel each other when the signs of the charges are the same. Because these forces are exerted mutually, two charges must be present for the forces to take place. These forces are described by [[Coulomb's law]], which says that the greater the magnitude of the charges, the greater the force, and the greater the distance between them, the weaker the force. Informally, the greater the charge of an object, the stronger its electric field. Similarly, an electric field is stronger nearer charged objects and weaker further away. Electric fields originate from electric charges and time-varying [[electric current]]s. Electric fields and [[magnetic fields]] are both manifestations of the electromagnetic field. Electromagnetism is one of the four [[fundamental interaction]]s of nature.

Electric fields are important in many areas of [[physics]], and are exploited in electrical technology. For example, in [[atomic physics]] and [[chemistry]], the interaction in the electric field between the [[atomic nucleus]] and electrons is the force that holds these particles together in atoms. Similarly, the interaction in the electric field between atoms is the force responsible for [[chemical bonding]] that result in [[molecule]]s.

The electric field is defined as a [[vector field]] that associates to each point in space the force per unit of [[Electric charge|charge]] exerted on an infinitesimal [[test charge]] at rest at that point.<ref>{{cite book |first=Richard|last=Feynman |title=The Feynman Lectures on Physics Vol II |publisher=Addison Wesley Longman |year=1970 |pages=1-3,1-4 |isbn=978-0-201-02115-8 |url=https://feynmanlectures.caltech.edu/II_01.html}}</ref><ref>{{cite book | last1 = Purcell | first1=Edward M. |last2= Morin | first2=David J.| title=''Electricity and Magnetism'' | edition= 3rd | publisher= Cambridge University Press | location = New York| year = 2013 | pages=15–16| isbn= 978-1-107-01402-2}}</ref><ref name="Serway" /> The [[SI]] unit for the electric field is the [[volt]] per [[meter]] (V/m), which is equal to the [[newton (unit)|newton]] per [[coulomb]] (N/C).<ref>{{SIbrochure9th}}, p. 23</ref>

== Description ==
[[File:VFPt image charge plane horizontal.svg|thumb|250px|Electric field of a positive point [[electric charge]] suspended over an infinite sheet of conducting material. The field is depicted by [[field line|electric field lines]], lines which follow the direction of the electric field in space.  The induced charge distribution in the sheet is not shown.]]
The electric field is defined at each point in space as the force that would be experienced by an [[infinitesimally small]] stationary [[test charge]] at that point divided by the charge.<ref name=uniphysics>{{citation| first = Francis | last = Sears| title = University Physics |edition=6th| publisher = Addison Wesley| year = 1982| isbn = 0-201-07199-1|display-authors=etal}}</ref>{{rp|469–70}} The electric field is defined in terms of [[force]], and force is a [[Euclidean vector|vector]] (i.e. having both [[Magnitude (mathematics)|magnitude]] and [[Direction (geometry)|direction]]), so it follows that an electric field may be described by a [[vector field]].<ref name=uniphysics/>{{rp|469–70}} The electric field acts between two charges similarly to the way that the [[gravitational field]] acts between two [[mass]]es, as they both obey an [[inverse-square law]] with distance.<ref name=Umashankar>{{citation| first = Korada | last = Umashankar| title = Introduction to Engineering Electromagnetic Fields| pages = 77–79| year = 1989| publisher = World Scientific| isbn = 9971-5-0921-0}}</ref> This is the basis for [[Coulomb's law]], which states that, for stationary charges, the electric field varies with the source charge and varies inversely with the square of the distance from the source. This means that if the source charge were doubled, the electric field would double, and if you move twice as far away from the source, the field at that point would be only one-quarter its original strength.

The electric field can be visualized with a set of [[field line|lines]] whose direction at each point is the same as those of the field, a concept introduced by [[Michael Faraday]],<ref name="elec_princ_p73">{{citation |last = Morely & Hughes |title = Principles of Electricity |edition=5th |year = 1970 |page = 73 |publisher = Longman |isbn = 0-582-42629-4 }}</ref> whose term '[[Line of force|lines of force]]' is still sometimes used. This illustration has the useful property that, when drawn so that each line represents the same amount of [[flux]], the [[Field strength|strength of the field]] is proportional to the density of the lines.<ref name="Tou">{{cite book |last1 = Tou |first1 = Stephen |title = Visualization of Fields and Applications in Engineering |publisher = John Wiley and Sons |date = 2011 |pages = 64 |url = https://books.google.com/books?id=4kjP1ALPTfUC&q=%22field+line%22+vector+field+tangent&pg=PT64 |isbn = 9780470978467 }}</ref> Field lines due to stationary charges have several important properties, including that they always originate from positive charges and terminate at negative charges, they enter all good conductors at right angles, and they never cross or close in on themselves.<ref name=uniphysics/>{{rp|479}} The field lines are a representative concept; the field actually permeates all the intervening space between the lines. More or fewer lines may be drawn depending on the precision to which it is desired to represent the field.<ref name="elec_princ_p73"/> The study of electric fields created by stationary charges is called [[electrostatics]].

[[Faraday's law of induction|Faraday's law]] describes the relationship between a time-varying magnetic field and the electric field. One way of stating Faraday's law is that the [[Curl (mathematics)|curl]] of the electric field is equal to the negative [[time derivative]] of the magnetic field.<ref name=griffiths>{{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 |oclc=40251748 }}</ref>{{rp|327}} In the absence of time-varying magnetic field, the electric field is therefore called [[conservative vector field|conservative]] (i.e. curl-free).<ref name=griffiths />{{rp|24,90–91}} This implies there are two kinds of electric fields: electrostatic fields and fields arising from time-varying magnetic fields.<ref name=griffiths/>{{rp|305–307}} While the curl-free nature of the static electric field allows for a simpler treatment using electrostatics, time-varying magnetic fields are generally treated as a component of a unified [[electromagnetic field]]. The study of magnetic and electric fields that change over time is called [[electrodynamics]].

== Mathematical formulation ==
{{main|Mathematical descriptions of the electromagnetic field}}

Electric fields are caused by [[electric charges]], described by [[Gauss's law]],<ref>Purcell, p. 25: "Gauss's Law: the flux of the electric field E through any closed surface ... equals 1/''e'' times the total charge enclosed by the surface."</ref> and time varying [[magnetic fields]], described by [[Faraday's law of induction]].<ref>Purcell, p 356: "Faraday's Law of Induction."</ref> Together, these laws are enough to define the behavior of the electric field. However, since the magnetic field is described as a function of electric field, the equations of both fields are coupled and together form [[Maxwell's equations]] that describe both fields as a function of charges and [[Electric current|currents]].

[[File:Cat demonstrating static cling with styrofoam peanuts.jpg|thumb|upright=1.4|Evidence of an electric field: [[styrofoam peanut]]s clinging to a cat's fur due to [[static electricity]]. The [[triboelectric effect]] causes an [[electrostatic charge]] to build up on the fur due to the cat's motions. The electric field of the charge causes polarization of the molecules of the styrofoam due to [[electrostatic induction]], resulting in a slight attraction of the light plastic pieces to the charged fur. This effect is also the cause of [[static cling]] in clothes.]]

=== Electrostatics ===
{{main|Coulomb's law}}
In the special case of a [[steady state]] (stationary charges and currents), the Maxwell-Faraday [[inductive effect]] disappears. The resulting two equations (Gauss's law <math>\nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0}</math> and Faraday's law with no induction term <math>\nabla \times \mathbf{E} = 0</math>), taken together, are equivalent to [[Coulomb's law#Electric field|Coulomb's law]], which states that a particle with electric charge <math>q_1</math> at position <math>\mathbf r_1</math> exerts a force on a particle with charge <math>q_0</math> at position <math>\mathbf r_0</math> of:<ref>Purcell, p7: "... the interaction between electric charges ''at rest'' is described by Coulomb's Law: two stationary electric charges repel or attract each other with a force proportional to the product of the magnitude of the charges and inversely proportional to the square of the distance between them.</ref>
<math display="block">\mathbf{F}_{01} = \frac{q_1q_0}{4\pi\varepsilon_0} {\hat\mathbf r_{01}\over {|\mathbf r_{01}|}^2} = \frac{q_1q_0}{4\pi\varepsilon_0} {\mathbf r_{01}\over {|\mathbf r_{01}|}^3}</math>
where
* <math> \mathbf{F}_{01} </math> is the force on charged particle <math> q_0 </math> caused by charged particle <math> q_1 </math>.
* {{math|''ε''{{sub|0}}}} is the [[permittivity of free space]].
* <math> \hat \mathbf{r}_{01} </math> is a [[unit vector]] directed from <math> \mathbf r_1 </math> to <math> \mathbf r_0 </math>.
* <math> \mathbf{r}_{01} </math> is the [[displacement vector]] from <math> \mathbf r_1 </math> to <math> \mathbf r_0 </math>.

Note that <math>\varepsilon_0</math> must be replaced with <math>\varepsilon</math>, [[permittivity]], when charges are in non-empty media.
When the charges <math>q_0</math> and <math>q_1</math> have the same sign this force is positive, directed away from the other charge, indicating the particles repel each other. When the charges have unlike signs the force is negative, indicating the particles attract.
To make it easy to calculate the [[Coulomb force]] on any charge at position <math>\mathbf r_0</math> this expression can be divided by <math>q_0</math> leaving an expression that only depends on the other charge (the ''source'' charge)<ref name="Purcell">{{cite book | last1 = Purcell | first1 = Edward | title = Electricity and Magnetism | edition = 2nd | publisher = Cambridge University Press | date = 2011 | pages = 8–9 | url = https://books.google.com/books?id=Z3bkNh6h4WEC&pg=PA8 | isbn = 978-1139503556 }}</ref><ref name="Serway">{{cite book | last1 = Serway | first1 = Raymond A. | last2 = Vuille | first2 = Chris | title = College Physics | edition = 10th | publisher = Cengage Learning | date = 2014 | pages = 532–533 | url = https://books.google.com/books?id=xETAAgAAQBAJ&q=work+energy+capacitor&pg=PA522 | isbn = 978-1305142824 }}</ref>
<math display="block">\mathbf{E}_{1} (\mathbf r_0) = \frac{ \mathbf{F}_{01} } {q_0} = \frac{q_1}{4\pi\varepsilon_0} {\hat\mathbf r_{01}\over {|\mathbf r_{01}|}^2} = \frac{q_1}{4\pi\varepsilon_0} {\mathbf r_{01}\over {|\mathbf r_{01}|}^3}</math>
where: 
* <math>\mathbf{E}_{1} (\mathbf r_0) </math> is the component of the electric field at <math> q_0 </math> due to <math> q_1 </math>.

This is the ''electric field'' at point <math>\mathbf r_0</math> due to the point charge <math>q_1</math>; it is a [[vector-valued function]] equal to the Coulomb force per unit charge that a positive point charge would experience at the position <math>\mathbf r_0</math>.
Since this formula gives the electric field magnitude and direction at any point <math>\mathbf r_0</math> in space (except at the location of the charge itself, <math>\mathbf r_1</math>, where it becomes infinite) it defines a [[vector field]].
From the above formula it can be seen that the electric field due to a point charge is everywhere directed away from the charge if it is positive, and toward the charge if it is negative, and its magnitude decreases with the [[inverse square law|inverse square]] of the distance from the charge.

The Coulomb force on a charge of magnitude <math>q</math> at any point in space is equal to the product of the charge and the electric field at that point
<math display="block">\mathbf{F} = q\mathbf{E} .</math>
The [[Systeme International|SI]] unit of the electric field is the [[newton (unit)|newton]] per [[coulomb]] (N/C), or [[volt]] per [[meter]] (V/m); in terms of the [[SI base unit]]s it is kg⋅m⋅s<sup>−3</sup>⋅A<sup>−1</sup>.

=== Superposition principle ===
Due to the [[Linear differential equation|linearity]] of [[Maxwell's equations]], electric fields satisfy the [[superposition principle]], which states that the total electric field, at a point, due to a collection of charges is equal to the vector sum of the electric fields at that point due to the individual charges.<ref name="Serway" /> This principle is useful in calculating the field created by multiple point charges. If charges <math>q_1, q_2, \dots, q_n</math> are stationary in space at points <math>\mathbf r_1,\mathbf r_2,\dots,\mathbf r_n</math>, in the absence of currents, the superposition principle says that the resulting field is the sum of fields generated by each particle as described by Coulomb's law:
<math display="block">\begin{align}
\mathbf{E}(\mathbf{r}) = \mathbf{E}_1(\mathbf{r}) + \mathbf{E}_2(\mathbf{r}) + \dots + \mathbf{E}_n(\mathbf{r})  = {1\over4\pi\varepsilon_0} \sum_{i=1}^n q_i {\hat\mathbf r_i\over {|\mathbf r_i|}^2} = {1\over 4\pi\varepsilon_0} \sum_{i=1}^n q_i {\mathbf r_i\over {|\mathbf r_i|}^3}  
\end{align}</math>
where 
* <math>\hat\mathbf r_i</math> is the unit vector in the direction from point <math>\mathbf r_i</math> to point <math>\mathbf r</math>
* <math>\mathbf r_i</math> is the displacement vector from point <math>\mathbf r_i</math> to point <math>\mathbf r</math>.

=== Continuous charge distributions ===
The superposition principle allows for the calculation of the electric field due to a distribution of [[charge density]] <math>\rho(\mathbf r)</math>. By considering the charge <math>\rho(\mathbf r')dv</math> in each small volume of space <math>dv</math> at point <math>\mathbf r'</math> as a point charge, the resulting electric field, <math>d\mathbf{E}(\mathbf r)</math>, at point <math>\mathbf r</math> can be calculated as 
<math display="block">d\mathbf{E}(\mathbf r) = \frac{\rho(\mathbf r')}{4\pi\varepsilon_0}{\hat\mathbf r'\over {|\mathbf r'|}^2} dv = \frac{\rho(\mathbf r')}{4\pi\varepsilon_0} {\mathbf r'\over {|\mathbf r'|}^3} dv </math>
where 
* <math>\hat \mathbf{r}'</math> is the unit vector pointing from <math>\mathbf r'</math> to <math>\mathbf r</math>. 
* <math>\mathbf r'</math> is the displacement vector from <math>\mathbf r'</math> to <math>\mathbf r</math>. 
The total field is found by summing the contributions from all the increments of volume by [[Integral|integrating]] the charge density over the volume <math>V</math>:
<math display="block">\mathbf{E}(\mathbf r) = \frac{1}{4\pi\varepsilon_0} \iiint_V \, \rho(\mathbf r') {\mathbf r'\over {|\mathbf r'|}^3} dv</math>

Similar equations follow for a surface charge with [[surface charge density]] <math>\sigma(\mathbf r')</math> on surface <math> S </math>
<math display="block">\mathbf{E}(\mathbf r) = \frac{1}{4\pi\varepsilon_0} \iint_S \, \sigma(\mathbf r') {\mathbf r'\over {|\mathbf r'|}^3} da,</math>
and for line charges with [[linear charge density]] <math>\lambda(\mathbf r')</math> on line <math> L </math>
<math display="block">\mathbf{E}(\mathbf r) = \frac{1}{4\pi\varepsilon_0} \int_L \,\lambda(\mathbf r') {\mathbf r'\over {|\mathbf r'|}^3} d \ell.</math>

=== Electric potential ===
{{main|Electric potential}}
{{See also|Helmholtz decomposition|Conservative vector field#Irrotational vector fields}}

If a system is static, such that magnetic fields are not time-varying, then by Faraday's law, the electric field is [[Conservative vector field|curl-free]]. In this case, one can define an [[electric potential]], that is, a function <math>\varphi</math> such that {{nowrap|<math> \mathbf{E} = -\nabla \varphi </math>.}}<ref>{{cite web|url=http://physicspages.com/pdf/Griffiths%20EM/Griffiths%20Problems%2002.20.pdf|title=Curl & Potential in Electrostatics|last=gwrowe|date=8 October 2011| work=physicspages.com|access-date=2 November 2020|archive-url=https://web.archive.org/web/20190322101416/http://physicspages.com/pdf/Griffiths%20EM/Griffiths%20Problems%2002.20.pdf|archive-date=22 March 2019}}</ref> This is analogous to the [[gravitational potential]]. The difference between the electric potential at two points in space is called the [[potential difference]] (or voltage) between the two points.

In general, however, the electric field cannot be described independently of the magnetic field. Given the [[magnetic vector potential]], {{math|'''A'''}}, defined so that {{tmath|1= \mathbf{B} = \nabla \times \mathbf{A} }}, one can still define an electric potential <math> \varphi</math> such that:
<math display="block"> \mathbf{E} = - \nabla \varphi - \frac { \partial \mathbf{A} } { \partial t } ,</math>
where <math>\nabla \varphi</math> is the [[gradient]] of the electric potential and <math>\frac { \partial \mathbf{A} } { \partial t }</math> is the [[partial derivative]] of {{math|'''A'''}} with respect to time.

[[Faraday's law of induction]] can be recovered by taking the [[Curl (mathematics)|curl]] of that equation <ref>{{cite book| title=Maxwell's Equations| first1=Paul G.| last1=Huray| publisher=Wiley-IEEE| year=2009| isbn=978-0-470-54276-7| url=https://books.google.com/books?id=0QsDgdd0MhMCp| page=205}}{{Dead link|date=March 2024 |bot=InternetArchiveBot |fix-attempted=yes }}</ref>
<math display="block">\nabla \times \mathbf{E} = -\frac{\partial (\nabla \times \mathbf{A})} {\partial t} = -\frac{\partial \mathbf{B}} {\partial t} ,</math>
which justifies, a posteriori, the previous form for {{math|'''E'''}}.

=== Continuous vs. discrete charge representation ===
{{Main|Charge density}}
The equations of electromagnetism are best described in a continuous description. However, charges are sometimes best described as discrete points; for example, some models may describe [[electron]]s as point sources where charge density is infinite on an infinitesimal section of space.

A charge <math>q</math> located at <math>\mathbf{r}_0</math> can be described mathematically as a charge density {{tmath|1= \rho(\mathbf{r}) = q\delta(\mathbf{r} - \mathbf{r}_0)}}, where the [[Dirac delta function]] (in three dimensions) is used. Conversely, a charge distribution can be approximated by many small point charges.

== Electrostatic fields ==
{{Main|Electrostatics}}
[[Image:VFPt charges plus minus thumb.svg|220px|thumb|right|Illustration of the electric field surrounding a positive (red) and a negative (blue) charge]]

Electrostatic fields are electric fields that do not change with time. Such fields are present when systems of charged matter are stationary, or when [[constant current| electric currents]] are unchanging. In that case, [[Coulomb's law]] fully describes the field.<ref>Purcell, pp. 5–7.</ref>

=== Parallels between electrostatic and gravitational fields ===
{{See also|Gravitoelectromagnetism}}
Coulomb's law, which describes the interaction of electric charges:
<math display="block">\mathbf{F} = q \left(\frac{Q}{4\pi\varepsilon_0} \frac{\mathbf{\hat{r}}}{|\mathbf{r}|^2}\right) = q \mathbf{E}</math>
is similar to [[Newton's law of universal gravitation]]:
<math display="block">\mathbf{F} = m\left(-GM\frac{\mathbf{\hat{r}}}{|\mathbf{r}|^2}\right) = m\mathbf{g}</math>
(where <math display="inline">\mathbf{\hat{r}} = \mathbf{\frac{r}{|r|}}</math>).

This suggests similarities between the electric field {{math|'''E'''}} and the gravitational field {{math|'''g'''}}, or their associated potentials. Mass is sometimes called "gravitational charge".<ref>{{cite journal |last1=Salam |first1=Abdus |title=Quarks and leptons come out to play |journal=New Scientist |date=16 December 1976 |volume=72 |page=652 |url=https://books.google.com/books?id=WIbyn2jxGhoC&pg=PA652 }}{{Dead link|date=March 2024 |bot=InternetArchiveBot |fix-attempted=yes }}</ref>

Electrostatic and gravitational forces both are [[central force|central]], [[conservative force|conservative]] and obey an [[inverse-square law]].

=== Uniform fields ===
[[Image:VFPt capacitor-square-plate.svg|220px|thumb|right|Illustration of the electric field between two parallel [[conductive]] plates of finite size (known as a [[parallel plate capacitor]]). In the middle of the plates, far from any edges, the electric field is very nearly uniform.]]
A uniform field is one in which the electric field is constant at every point. It can be approximated by placing two conducting [[Capacitor#Parallel-plate capacitor|plates]] parallel to each other and maintaining a [[voltage]] (potential difference) between them; it is only an approximation because of boundary effects (near the edge of the planes, the electric field is distorted because the plane does not continue). Assuming infinite planes, the magnitude of the electric field {{math|''E''}} is:
<math display="block"> E = - \frac{\Delta V}{d} ,</math>
where {{math|Δ''V''}} is the [[potential difference]] between the plates and {{math|''d''}} is the distance separating the plates. The negative sign arises as positive charges repel, so a positive charge will experience a force away from the positively charged plate, in the opposite direction to that in which the voltage increases. In micro- and nano-applications, for instance in relation to semiconductors, a typical magnitude of an electric field is in the order of {{val|e=6|u=V⋅m<sup>−1</sup>}}, achieved by applying a voltage of the order of 1 volt between conductors spaced 1&nbsp;μm apart.

== Electromagnetic fields ==
[[File:Electrostatic induction.svg|thumb|upright=1.7|The electric field ''(lines with arrows)'' of a charge <span style="color:red;">(+)</span> induces surface charges ''(<span style="color:red;">red</span> and <span style="color:blue;">blue</span> areas)'' on metal objects due to [[electrostatic induction]].]]
{{main|Electromagnetic field}}

Electromagnetic fields are electric and magnetic fields, which may change with time, for instance when charges are in motion. Moving charges produce a magnetic field in accordance with [[Ampère's circuital law]] ([[Maxwell's equations|with Maxwell's addition]]), which, along with Maxwell's other equations, defines the magnetic field, <math>\mathbf{B}</math>, in terms of its curl:
<math display="block">\nabla \times \mathbf{B} = \mu_0\left(\mathbf{J} + \varepsilon_0 \frac{\partial \mathbf{E}} {\partial t} \right) ,</math>
where <math>\mathbf{J}</math> is the [[current density]], <math>\mu_0</math> is the [[vacuum permeability]], and <math>\varepsilon_0</math> is the [[vacuum permittivity]].

Both the [[electric current density]] and the [[partial derivative]] of the electric field with respect to time, contribute to the curl of the magnetic field. In addition, the [[Maxwell–Faraday equation]] states 
<math display="block">\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}.</math>
These represent two of [[Maxwell's equations|Maxwell's four equations]] and they intricately link the electric and magnetic fields together, resulting in the [[electromagnetic field]]. The equations represent a set of four coupled multi-dimensional partial differential equations which, when solved for a system, describe the combined behavior of the electromagnetic fields. In general, the force experienced by a test charge in an electromagnetic field is given by the [[Lorentz force law]]:
<math display="block">\mathbf{F} = q\mathbf{E} + q\mathbf{v} \times \mathbf{B} .</math>

== Energy in the electric field ==

The total energy per unit volume stored by the [[electromagnetic field]] is<ref name="Griffiths">{{cite book|title=Introduction to Electrodynamics|edition=3|first=D.J.|last=Griffiths|publisher=Cambridge University Press|date= 2017 |url=https://archive.org/details/introductiontoel0000grif/page/356/mode/2up  |page=357, eq. 8.5  |isbn=9781108420419}}</ref>
<math display="block"> u_\text{EM} = \frac{\varepsilon}{2} |\mathbf{E}|^2 + \frac{1}{2\mu} |\mathbf{B}|^2 </math>
where {{mvar|ε}} is the [[permittivity]] of the medium in which the field exists, <math>\mu</math> its [[magnetic permeability]], and {{math|'''E'''}} and {{math|'''B'''}} are the electric and magnetic field vectors.

As {{math|'''E'''}} and {{math|'''B'''}} fields are coupled, it would be misleading to split this expression into "electric" and "magnetic" contributions. In particular, an electrostatic field in any given frame of reference in general transforms into a field with a magnetic component in a relatively moving frame. Accordingly, decomposing the electromagnetic field into an electric and magnetic component is frame-specific, and similarly for the associated energy.

The total energy {{math|''U''{{sub|EM}}}} stored in the electromagnetic field in a given volume {{math|''V''}} is
<math display="block">U_\text{EM} = \frac{1}{2} \int_{V} \left( \varepsilon |\mathbf{E}|^2 + \frac{1}{\mu} |\mathbf{B}|^2 \right) dV \, .</math>

== Electric displacement field ==
{{main|electric displacement field}}

=== Definitive equation of vector fields ===
{{See also|List of electromagnetism equations}}

In the presence of matter, it is helpful to extend the notion of the electric field into three vector fields:<ref name="auto">{{cite book|title=Electromagnetism|edition=2|first1=I.S.|last1=Grant|first2=W.R.|last2=Phillips|publisher=John Wiley & Sons|year=2008|isbn=978-0-471-92712-9}}</ref>
<math display="block">\mathbf{D} = \varepsilon_0 \mathbf{E} + \mathbf{P}</math>
where {{math|'''P'''}} is the [[electric polarization]] – the volume density of [[electric dipole moment]]s, and {{math|'''D'''}} is the [[electric displacement field]]. Since {{math|'''E'''}} and {{math|'''P'''}} are defined separately, this equation can be used to define {{math|'''D'''}}. The physical interpretation of {{math|'''D'''}} is not as clear as {{math|'''E'''}} (effectively the field applied to the material) or {{math|'''P'''}} (induced field due to the dipoles in the material), but still serves as a convenient mathematical simplification, since Maxwell's equations can be simplified in terms of [[Maxwell's equations#Macroscopic formulation|free charges and currents]].

=== Constitutive relation ===
{{Main|Constitutive equation}}

The {{math|'''E'''}} and {{math|'''D'''}} fields are related by the [[permittivity]] of the material, {{math|''ε''}}.<ref>{{cite book|title=Electricity and Modern Physics|edition=2|first1=G.A.G.|last1=Bennet|first2=Edward|last2=Arnold|year=1974|publisher=Edward Arnold|isbn=0-7131-2459-8}}</ref><ref name="auto"/>

For linear, [[Homogeneity and heterogeneity|homogeneous]], [[Isotropy|isotropic]] materials {{math|'''E'''}} and {{math|'''D'''}} are proportional and constant throughout the region, there is no position dependence: 
<math display="block">\mathbf{D}(\mathbf{r}) = \varepsilon\mathbf{E}(\mathbf{r}) .</math>

For inhomogeneous materials, there is a position dependence throughout the material:<ref>{{Cite book|last1=Landau|first1=Lev Davidovich|author-link1=Lev Landau|last2=Lifshitz|first2=Evgeny M.| author-link2= Evgeny Lifshitz |url=https://books.google.com/books?id=XX0xxgEACAAJ|title=Electrodynamics of Continuous Media|date=1963|series=[[Course of Theoretical Physics]] |volume=8 |publisher=Pergamon|isbn=978-0-7581-6499-5|page=285|language=en|section=68 the propagation of waves in an inhomogeneous medium |quote=In Maxwell's equations… ''ε'' is a function of the co-ordinates.}}</ref>
<math display="block">\mathbf{D}(\mathbf{r}) = \varepsilon (\mathbf{r})\mathbf{E}(\mathbf{r})</math>

For anisotropic materials the {{math|'''E'''}} and {{math|'''D'''}} fields are not parallel, and so {{math|'''E'''}} and {{math|'''D'''}} are related by the [[Permittivity|permittivity tensor]] (a 2nd order [[tensor field]]), in component form:
<math display="block">D_i = \varepsilon_{ij} E_j</math>

For non-linear media, {{math|'''E'''}} and {{math|'''D'''}} are not proportional. Materials can have varying extents of linearity, homogeneity and isotropy.

== Relativistic effects on electric field ==

=== Point charge in uniform motion ===
The invariance of the form of [[Maxwell's equations]] under [[Lorentz transformation]] can be used to derive the electric field of a uniformly moving point charge. The charge of a particle is considered frame invariant, as supported by experimental evidence.<ref name=":0">{{Cite book |last1=Purcell |first1=Edward M. |url=https://www.cambridge.org/highereducation/books/electricity-and-magnetism/0F97BB6C5D3A56F19B9835EDBEAB087C |title=Electricity and Magnetism |last2=Morin |first2=David J. |date=2013-01-21 |website=Higher Education from Cambridge University Press |isbn=9781139012973 |pages=241–251 |language=en |doi=10.1017/cbo9781139012973 |access-date=2022-07-04}}</ref> Alternatively the electric field of uniformly moving point charges can be derived from the [[Lorentz transformation]] of [[four-force]] experienced by test charges in the source's [[rest frame]] given by [[Coulomb's law]] and assigning electric field and magnetic field by their definition 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}}</ref> However the following equation is only applicable when no acceleration is involved in the particle's history where [[Coulomb's law]] can be considered or symmetry arguments can be used for solving [[Maxwell's equations]] in a simple manner. The electric field of such a uniformly moving point charge is hence 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}}</ref>
<math display="block">\mathbf{E} = \frac q {4 \pi \varepsilon_0 r^3}   \frac {1- \beta^2} {(1- \beta^2 \sin^2 \theta)^{3/2}} \mathbf{r} ,</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>\beta</math> is the ratio of observed speed of the charge particle to the speed of light and <math>\theta</math> is the angle between <math>\mathbf{r}</math> and the observed velocity of the charged particle. 

The above equation reduces to that given by Coulomb's law for non-relativistic speeds of the point charge. Spherical symmetry is not satisfied due to breaking of symmetry in the problem by specification of direction of velocity for calculation of field. To illustrate this, field lines of moving charges are sometimes represented as unequally spaced radial lines which would appear equally spaced in a co-moving reference frame.<ref name=":0" />

=== Propagation of disturbances in electric fields ===
{{See also|Paradox of radiation of charged particles in a gravitational field}}
[[Special relativity|Special theory of relativity]] imposes the [[principle of locality]], that requires cause and effect to be time-like separated events where the [[Causality|causal efficacy]] does not travel faster than the [[speed of light]].<ref>{{Cite book |author=Naber, Gregory L. |title=The Geometry of Minkowski spacetime: an introduction to the mathematics of the special theory of relativity |date=2012 |publisher=Springer |isbn=978-1-4419-7837-0 |pages=4–5 |oclc=804823303}}</ref> [[Maxwell's equations|Maxwell's laws]] are found to confirm to this view since the general solutions of fields are given in terms of retarded time which indicate that [[Electromagnetism|electromagnetic]] disturbances travel at the [[speed of light]]. Advanced time, which also provides a solution for [[Maxwell's equations|Maxwell's law]] are ignored as an unphysical solution.[[File:Bremsstrahlung.gif|thumb|An illustrative example showing bremsstrahlung radiation: Field lines and modulus of the electric field generated by a (negative) charge first moving at a constant speed and then stopping quickly to show the electromagnetic wave generated and propagation of disturbances in electromagnetic field.]]For the motion of a [[charged particle]], considering for example the case of a moving particle with the above described electric field coming to an abrupt stop, the electric fields at points far from it do not immediately revert to that classically given for a stationary charge. On stopping, the field around the stationary points begin to revert to the expected state and this effect propagates outwards at the [[speed of light]] while the electric field lines far away from this will continue to point radially towards an assumed moving charge. This [[virtual particle]] will never be outside the range of propagation of the disturbance in [[electromagnetic field]], since charged particles are restricted to have speeds slower than that of light, which makes it impossible to construct a [[Gaussian surface]] in this region that violates [[Gauss's law]]. Another technical difficulty that supports this is that charged particles travelling faster than or equal to speed of light no longer have a unique retarded time. Since electric field lines are continuous, an [[electromagnetic pulse]] of radiation is generated that connects at the boundary of this disturbance travelling outwards at the [[speed of light]].<ref>{{Cite book |last=Purcell |first=Edward M. |title=Electricity and Magnetism |date=2013 |author2=David J. Morin |isbn=978-1-139-01297-3 |edition=Third |location=Cambridge |pages=251–255 |oclc=1105718330}}</ref> In general, any accelerating point charge radiates [[Electromagnetic radiation|electromagnetic waves]] however, [[Nonradiation condition|non-radiating acceleration]] is possible in a systems of charges.

=== Arbitrarily moving point charge ===
{{See also|Jefimenko's equations#Heaviside–Feynman formula}}
For arbitrarily moving point charges, propagation of potential fields such as Lorenz gauge fields at the speed of light needs to be accounted for by using [[Liénard–Wiechert potential]].<ref>{{Cite book |last=Griffiths |first=David J. |title=Introduction to electrodynamics |publisher=[[Cambridge University Press]] |year=2017 |isbn=978-1-108-42041-9 |edition=4th |location=United Kingdom |pages=454 |language=en |oclc=1021068059}}</ref> Since the potentials satisfy [[Maxwell's equations]], the fields derived for point charge also satisfy [[Maxwell's equations]]. The electric field is expressed as:<ref name=":1">{{Cite book |last=Jackson |first=John David |title=Classical electrodynamics |date=1999 |publisher=Wiley |isbn=0-471-30932-X |edition=3rd |location=New York |pages=664–665 |oclc=38073290}}</ref>
<math display="block">\mathbf{E}(\mathbf{r}, \mathbf{t}) = \frac{1}{4 \pi \varepsilon_0} \left(\frac{q(\mathbf{n}_s - \boldsymbol{\beta}_s)}{\gamma^2 (1 - \mathbf{n}_s \cdot \boldsymbol{\beta}_s)^3 |\mathbf{r} - \mathbf{r}_s|^2} + \frac{q \mathbf{n}_s \times \big((\mathbf{n}_s - \boldsymbol{\beta}_s) \times \dot{\boldsymbol{\beta}_s}\big)}{c(1 - \mathbf{n}_s \cdot \boldsymbol{\beta}_s)^3 |\mathbf{r} - \mathbf{r}_s|} \right)_{t = t_r}</math>
where <math>q</math> is the charge of the point source, <math display="inline">{t_r}</math> is [[retarded time]] or the time at which the source's contribution of the electric field originated, <math display="inline">{r}_s(t)</math> is the position vector of the particle, <math display="inline">{n}_s(\mathbf{r},t)</math> is a unit vector pointing from charged particle to the point in space, <math display="inline"> \boldsymbol{\beta}_s(t)</math> is the velocity of the particle divided by the speed of light, and <math display="inline">\gamma(t)</math> is the corresponding [[Lorentz factor]]. The retarded time is given as solution of:
<math>t_r=\mathbf{t}-\frac{|\mathbf{r} - \mathbf{r}_s(t_r)|}{c}</math>

The uniqueness of solution for <math display="inline">{t_r}</math> for given <math>\mathbf{t}</math>, <math>\mathbf{r}</math> and <math>r_s(t)</math> is valid for charged particles moving slower than speed of light. [[Electromagnetic radiation]] of accelerating charges is known to be caused by the acceleration dependent term in the electric field from which relativistic correction for [[Larmor formula#Relativistic generalization|Larmor formula]] is obtained.<ref name=":1" /> 

There exist yet another set of solutions for Maxwell's equation of the same form but for advanced time <math display="inline">{t_a}</math> instead of retarded time given as a solution of:

<math>t_a=\mathbf{t}+\frac{|\mathbf{r} - \mathbf{r}_s(t_a)|}{c}</math>

Since the physical interpretation of this indicates that the electric field at a point is governed by the particle's state at a point of time in the future, it is considered as an unphysical solution and hence neglected. However, there have been theories exploring the advanced time solutions of [[Maxwell's equations]], such as [[Wheeler–Feynman absorber theory|Feynman Wheeler absorber theory]].

The above equation, although consistent with that of uniformly moving point charges as well as its non-relativistic limit, are not corrected for quantum-mechanical effects.

== Common formulæ==
{| class="wikitable"
|+
!Charge configuration
!Figure
! colspan="2" |Electric field
|-
|Infinite wire
| [[File:Charged infinite wire problem.svg|frameless|235x235px]]
| colspan="2" |<math>\mathbf E = \frac{\lambda}{2\pi\varepsilon_0x} \hat\mathbf x,</math> 
 
where <math>\lambda</math> is uniform linear charge density.
|-
|Infinitely large surface
|[[File:Charged infinite plane problem.svg|frameless|235x235px]]
| colspan="2" |<math>\mathbf E = \frac{\sigma}{2\varepsilon_0}\hat\mathbf x,</math>
where <math>\sigma</math> is uniform surface charge density.
|-
|Infinitely long cylindrical volume
|[[File:Charged infinite cylinder problem.svg|frameless|235x235px]]
| colspan="2" |<math>\mathbf E = \frac{\lambda}{2\pi\varepsilon_0x}\hat\mathbf x , </math>
where <math>\lambda</math> is uniform linear charge density.
|-
|Spherical volume
|[[File:Charged solid sphere problem.svg|frameless|235x235px]]
|<math>\mathbf E = \frac{Q}{4\pi\varepsilon_0x^2}\hat\mathbf x, </math> 
outside the sphere, where <math>Q</math> is the total charge uniformly distributed in the volume.
|<math>\mathbf E = \frac{Qr}{4\pi\varepsilon_0R^3}\hat\mathbf r,</math>
inside the sphere, where <math>Q</math> is the total charge uniformly distributed in the volume.
|-
|Spherical surface
|[[File:Charged spherical surface problem.svg|frameless|235x235px]]
|<math>\mathbf E = \frac{Q}{4\pi\varepsilon_0x^2} \hat\mathbf x,</math> 
outside the sphere, where <math>Q</math> is the total charge uniformly distributed on the surface.
|<math>\mathbf E = 0,</math> 
inside the sphere for uniform charge distribution.
|-
|Charged Ring
|[[File:Charged ring problem.svg|frameless|235x235px]]
| colspan="2" | <math>\mathbf E = \frac{Qx}{4\pi\varepsilon_0(R^2+x^2)^{3/2}}\hat\mathbf x,</math> 
on the axis, where <math>Q</math> is the total charge uniformly distributed on the ring.
|-
|Charged Disc
|[[File:Charged disc problem.svg|frameless|235x235px]]
| colspan="2" | <math>\mathbf E = \frac{\sigma}{2\varepsilon_0} \left[1-\frac x \sqrt{x^2+R^2} \right] \hat\mathbf x,</math>
on the axis, where <math>\sigma</math> is the uniform surface charge density.
|-
|Electric Dipole
|[[File:Electric dipole - axial and equatorial problem.svg|frameless|235x235px]]
| <math>\mathbf E = -\frac{\mathbf p}{4\pi\varepsilon_0 r^3},</math>

on the equatorial plane, where <math>\mathbf p</math> is the electric dipole moment.
| <math>\mathbf E = \frac{\mathbf p}{2\pi\varepsilon_0x^3} ,</math>

on the axis (given that <math>x \gg d</math>), where <math>x</math> can also be negative to indicate position at the opposite direction on the axis, and <math>\mathbf p</math> is the electric dipole moment.
|}
Electric field infinitely close to a conducting surface in electrostatic equilibrium having charge density <math>\sigma</math> at that point is <math display="inline">\frac{\sigma}{\varepsilon_0} \hat\mathbf x</math> since charges are only formed on the surface and the surface at the infinitesimal scale resembles an infinite 2D plane. In the absence of external fields, spherical conductors exhibit a uniform charge distribution on the surface and hence have the same electric field as that of uniform spherical surface distribution.

== See also ==
* [[Classical electromagnetism]]
* [[Relativistic electromagnetism]]
* [[Electricity]]
* [[History of electromagnetic theory]]
* [[Electromagnetic field]]
* [[Magnetism]]
* [[Teltron tube]]
* [[Teledeltos]], a conductive paper that may be used as a simple analog computer for modelling fields

== References ==
{{reflist}}
* {{cite book |author1=Purcell, Edward |author2=Morin, David |title= Electricity and Magnetism| edition= 3rd | publisher=Cambridge University Press, New York |year= 2013 |isbn=978-1-107-01402-2}}
* {{cite book |author=Browne, Michael |title= Physics for Engineering and Science|edition= 2nd |publisher= McGraw-Hill, Schaum, New York | year= 2011|isbn=978-0-07-161399-6}}

== External links ==
{{Commons category|Electric field}}
* [http://hyperphysics.phy-astr.gsu.edu/hbase/electric/elefie.html Electric field in "Electricity and Magnetism", R Nave] – [[Hyperphysics]], [[Georgia State University]]
* [http://teacher.pas.rochester.edu/phy122/Lecture_Notes/ Frank Wolfs's lectures] at [[University of Rochester]], chapters 23 and 24
* [http://www.lightandmatter.com/html_books/0sn/ch10/ch10.html Fields] {{Webarchive|url=https://web.archive.org/web/20100527194640/http://www.lightandmatter.com/html_books/0sn/ch10/ch10.html |date=2010-05-27 }} – a chapter from an online textbook

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{{DEFAULTSORT:Electric Field}}
[[Category:Electrostatics]]
[[Category:Electromagnetic quantities]]
[[Category:Electromagnetism]]