Editing Electric field (section)

== 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.