TheInfoList

An electric field (sometimes E-field) is the physical field that surrounds electrically-
charged particle In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through Spa ...
s and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field for a system of charged particles. Electric fields originate from electric charges, or from time-varying
magnetic fields A magnetic field is a vector field that describes the magnetic influence on moving electric charge Electric charge is the physical property of matter that causes it to experience a force when placed in an electromagnetic field. There are two ...
. Electric fields and magnetic fields are both manifestations of the
electromagnetic force Electromagnetism is a branch of physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related ...

, one of the four
fundamental force#REDIRECT Fundamental interaction In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics ...
s (or interactions) of nature. Electric fields are important in many areas of
physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scie ...

, and are exploited practically in electrical technology. In atomic physics and
chemistry Chemistry is the scientific Science () is a systematic enterprise that builds and organizes knowledge Knowledge is a familiarity or awareness, of someone or something, such as facts A fact is an occurrence in the real world. T ...

, for instance, the electric field is the attractive force holding the
atomic nucleus The atomic nucleus is the small, dense region consisting of s and s at the center of an , discovered in 1911 by based on the 1909 . After the discovery of the neutron in 1932, models for a nucleus composed of protons and neutrons were quickl ...
and
electron The electron is a subatomic particle (denoted by the symbol or ) whose electric charge is negative one elementary charge. Electrons belong to the first generation (particle physics), generation of the lepton particle family, and are general ...

s together in atoms. It is also the force responsible for
chemical bonding A chemical bond is a lasting attraction between atom An atom is the smallest unit of ordinary matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday o ...
between atoms that result in
molecule A molecule is an electrically Electricity is the set of physical phenomena associated with the presence and motion Image:Leaving Yongsan Station.jpg, 300px, Motion involves a change in position In physics, motion is the phenomenon ...

s. The electric field is defined mathematically as a
vector field In vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Product ...

that associates to each point in space the (electrostatic or
Coulomb The coulomb (symbol: C) is the International System of Units International is an adjective (also used as a noun) meaning "between nations". International may also refer to: Music Albums * International (Kevin Michael album), ''International'' ( ...
) force per unit of
charge Charge or charged may refer to: Arts, entertainment, and media Films * ''Charge, Zero Emissions/Maximum Speed'', a 2011 documentary Music * Charge (David Ford album), ''Charge'' (David Ford album) * Charge (Machel Montano album), ''Charge'' (Mac ...
exerted on an infinitesimal positive
test charge In physical theories, a test particle, or test charge, is an idealized model of an object whose physical properties (usually mass Mass is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate ...
at rest at that point. The derived SI units for the electric field are
volts The volt is the derived unit for electric potential The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is the amount of work energy needed to move a unit of electric charge ...

per
meter The metre ( Commonwealth spelling) or meter (American spelling Despite the various English dialects spoken from country to country and within different regions of the same country, there are only slight regional variations in English ...

(V/m), exactly equivalent to
newtons The newton (symbol: N) is the International System of Units International is an adjective (also used as a noun) meaning "between nations". International may also refer to: Music Albums * International (Kevin Michael album), ''International'' ( ...
per
coulomb The coulomb (symbol: C) is the International System of Units International is an adjective (also used as a noun) meaning "between nations". International may also refer to: Music Albums * International (Kevin Michael album), ''International'' ( ...

(N/C).

# Description

The electric field is defined at each point in space as the force (per unit charge) that would be experienced by a vanishingly small positive
test charge In physical theories, a test particle, or test charge, is an idealized model of an object whose physical properties (usually mass Mass is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate ...
if held at that point. As the electric field is defined in terms of
force In physics, a force is an influence that can change the motion (physics), motion of an Physical object, object. A force can cause an object with mass to change its velocity (e.g. moving from a Newton's first law, state of rest), i.e., to acce ...

, and force is a
vector Vector may refer to: Biology *Vector (epidemiology) In epidemiology Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and risk factor, determinants of health and disease conditions in defined pop ...
(i.e. having both
magnitude Magnitude may refer to: Mathematics *Euclidean vector, a quantity defined by both its magnitude and its direction *Magnitude (mathematics), the relative size of an object *Norm (mathematics), a term for the size or length of a vector *Order of ...
and
direction Direction may refer to: *Relative direction, for instance left, right, forward, backwards, up, and down ** Anatomical terms of location for those used in anatomy *Cardinal direction Mathematics and science *Direction vector, a unit vector that ...
), it follows that an electric field is a
vector field In vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Product ...

. Vector fields of this form are sometimes referred to as
force fields In speculative fiction, a force field, sometimes known as an energy shield, force shield, force bubble, defence shield or deflector shield, is a barrier made of things like energy, negative energy, dark energy, electromagnetic fields, gravitationa ...
. The electric field acts between two charges similarly to the way the
gravitational field In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "P ...

acts between two
mass Mass is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value ...
es, as they both obey an
inverse-square law 420px, S represents the light source, while r represents the measured points. The lines represent the flux emanating from the sources and fluxes. The total number of flux lines depends on the strength of the light source and is constant with in ...

with distance. This is the basis for
Coulomb's law between two point charge A point particle (ideal particle or point-like particle, often spelled pointlike particle) is an idealization of particle In the Outline of physical science, physical sciences, a particle (or corpuscule in older te ...
, 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
lines Long interspersed nuclear elements (LINEs) (also known as long interspersed nucleotide elements or long interspersed elements) are a group of non-LTR (long terminal repeat A long terminal repeat (LTR) is a pair of identical sequences of DNA ...
whose direction at each point is the same as the field's, a concept introduced by
Michael Faraday Michael Faraday (; 22 September 1791 – 25 August 1867) was an English scientist A scientist is a person who conducts scientific research The scientific method is an Empirical evidence, empirical method of acquiring knowledge ...

, whose term '
lines of force A line of force in Faraday Michael Faraday (; 22 September 1791 – 25 August 1867) was an English scientist A scientist is a person who conducts Scientific method, scientific research to advance knowledge in an Branches of science, ...
' is still sometimes used. This illustration has the useful property that the field's strength is proportional to the density of the lines. The field lines are the paths that a point positive charge would follow as it is forced to move within the field, similar to
trajectories A trajectory or flight path is the path that an object with mass Mass is both a property Property (''latin: Res Privata'') in the Abstract and concrete, abstract is what belongs to or with something, whether as an attribute or as a co ...

that masses follow within a gravitational field. Field lines due to stationary charges have several important properties, including always originating from positive charges and terminating at negative charges, they enter all good conductors at right angles, and they never cross or close in on themselves. 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. The study of electric fields created by stationary charges is called
electrostatics Electrostatics is a branch of physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related enti ...
. 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 Curl or CURL may refer to: Science and technology * Curl (mathematics) In vector calculus Vector calculus, or vector analysis, is concerned with derivative, differentiation and integral, integration of vector fields, primarily in 3-dimension ...
of the electric field is equal to the negative
time derivative A time derivative is a derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its Argument of a function, argument (input ...
of the magnetic field. In the absence of time-varying magnetic field, the electric field is therefore called
conservative Conservatism is an aesthetic Aesthetics, or esthetics (), is a branch of philosophy that deals with the nature of beauty and taste (sociology), taste, as well as the philosophy of art (its own area of philosophy that comes out of aest ...
(i.e. curl-free). This implies there are two kinds of electric fields: electrostatic fields and fields arising from time-varying magnetic fields. 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 An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the in ...
. The study of time varying magnetic and electric fields is called
electrodynamics Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electric charge, electrically charged particles. The electromagnetic force is carried by electromagneti ...
.

# Mathematical formulation

Electric fields are caused by
electric charges Electric charge is the physical property of matter that causes it to experience a force when placed in an electromagnetic field. There are two types of electric charge: ''positive'' and ''negative'' (commonly carried by protons and electrons resp ...
, described by
Gauss's law In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. ...
, and time varying
magnetic fields A magnetic field is a vector field that describes the magnetic influence on moving electric charge Electric charge is the physical property of matter that causes it to experience a force when placed in an electromagnetic field. There are two ...
, described by
Faraday's law of induction Faraday's law of induction (briefly, Faraday's law) is a basic law of electromagnetism Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electric ch ...
. 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 Maxwell's equations are a set of coupled partial differential equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), ...
that describe both fields as a function of charges and
currents Currents or The Current may refer to: Science and technology * Current (fluid), the flow of a liquid or a gas ** Air current, a flow of air ** Ocean current, a current in the ocean *** Rip current, a kind of water current ** Current (stream), c ...
.

## Electrostatics

In the special case of a
steady state In systems theory Systems theory is the interdisciplinary study of system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influen ...

(stationary charges and currents), the Maxwell-Faraday inductive effect disappears. The resulting two equations (Gauss's law $\nabla \cdot \mathbf = \frac$ and Faraday's law with no induction term $\nabla \times \mathbf = 0$), taken together, are equivalent to
Coulomb's law between two point charge A point particle (ideal particle or point-like particle, often spelled pointlike particle) is an idealization of particle In the Outline of physical science, physical sciences, a particle (or corpuscule in older te ...

, which states that a particle with electric charge $q_1$ at position $\mathbf_1$ exerts a force on a particle with charge $q_0$ at position $\mathbf_0$ of: :$\mathbf = \hat \mathbf_ \,,$ where $\hat \mathbf_$ is the
unit vector In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
in the direction from point $\mathbf_1$ to point $\mathbf_0$, and is the
electric constant Vacuum permittivity, commonly denoted (pronounced as "epsilon nought" or "epsilon zero") is the value of the absolute dielectric permittivity of classical vacuum. Alternatively it may be referred to as the permittivity of free space, the elec ...
(also known as "the absolute permittivity of free space") with units C2⋅m−2⋅N−1. Note that $\varepsilon_0$, the vacuum electric permittivity, must be substituted with $\varepsilon$,
permittivity In electromagnetism Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electric charge, electrically charged particles. The electromagnetic force is car ...
, when charges are in non-empty media. When the charges $q_0$ and $q_1$ 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 Coulomb's law, or Coulomb's inverse-square law, is an experimental law Law is a system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. A system ...
on any charge at position $\mathbf_0$ this expression can be divided by $q_0$ leaving an expression that only depends on the other charge (the ''source'' charge) :$\mathbf\left(\mathbf_0\right) = = \hat \mathbf_$ This is the ''electric field'' at point $\mathbf_0$ due to the point charge $q_1$; it is a
vector-valued function A vector-valued function, also referred to as a vector function, is a function (mathematics), mathematical function of one or more variables whose range of a function, range is a set of multidimensional Euclidean vector, vectors or infinite-dimensi ...

equal to the Coulomb force per unit charge that a positive point charge would experience at the position $\mathbf_0$. Since this formula gives the electric field magnitude and direction at any point $\mathbf_0$ in space (except at the location of the charge itself, $\mathbf_1$, where it becomes infinite) it defines a
vector field In vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Product ...

. 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 Image:Inverse square law.svg, 420px, S represents the light source, while r represents the measured points. The lines represent the flux emanating from the sources and fluxes. The total number of flux lines depends on the strength of the light so ...

of the distance from the charge. The Coulomb force on a charge of magnitude $q$ at any point in space is equal to the product of the charge and the electric field at that point :$\mathbf = q\mathbf$ The units of the electric field in the SI system are
newtons The newton (symbol: N) is the International System of Units International is an adjective (also used as a noun) meaning "between nations". International may also refer to: Music Albums * International (Kevin Michael album), ''International'' ( ...
per
coulomb The coulomb (symbol: C) is the International System of Units International is an adjective (also used as a noun) meaning "between nations". International may also refer to: Music Albums * International (Kevin Michael album), ''International'' ( ...

(N/C), or
volt The volt is the derived unit for electric potential The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of work (physics), work energy needed to move a ...

s per
meter The metre ( Commonwealth spelling) or meter (American spelling Despite the various English dialects spoken from country to country and within different regions of the same country, there are only slight regional variations in English ...

(V/m); in terms of the
SI base unit The SI base units are the standard units of measurement A unit of measurement is a definite magnitude Magnitude may refer to: Mathematics *Euclidean vector, a quantity defined by both its magnitude and its direction *Magnitude (mat ...

s they are kg⋅m⋅s−3⋅A−1.

## Superposition principle

Due to the
linearity Linearity is the property of a mathematical relationship (''function (mathematics), function'') that can be graph of a function, graphically represented as a straight Line (geometry), line. Linearity is closely related to ''Proportionality (math ...

of
Maxwell's equations Maxwell's equations are a set of coupled partial differential equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), ...
, electric fields satisfy the
superposition principle The superposition principle, also known as superposition property, states that, for all linear system In systems theory Systems theory is the interdisciplinary study of systems, i.e. cohesive groups of interrelated, interdependent parts that ...
, 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. This principle is useful in calculating the field created by multiple point charges. If charges $q_1, q_2, \dots, q_n$ are stationary in space at points $\mathbf_1,\mathbf_2,\dots,\mathbf_n$, 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: $\begin \mathbf(\mathbf) &= \mathbf_1(\mathbf) + \mathbf_2(\mathbf) + \mathbf_3(\mathbf) + \cdots \\$&= \hat \mathbf_1 + \hat \mathbf_2 + \hat \mathbf_3 + \cdots \\&= \sum_^N \hat \mathbf_k \end where $\mathbf$ is the
unit vector In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
in the direction from point $\mathbf_k$ to point $\mathbf$.

## Continuous charge distributions

The superposition principle allows for the calculation of the electric field due to a continuous distribution of charge $\rho\left(\mathbf\right)$ (where $\rho$ is the
charge density In electromagnetism Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electric charge, electrically charged particles. The electromagnetic force is c ...

in coulombs per cubic meter). By considering the charge $\rho\left(\mathbf\text{'}\right)dV$ in each small volume of space $dV$ at point $\mathbf\text{'}$ as a point charge, the resulting electric field, $d\mathbf\left(\mathbf\right)$, at point $\mathbf$ can be calculated as :$d\mathbf\left(\mathbf\right) = \hat \mathbf\text{'}$ where $\hat \mathbf\text{'}$ is the unit vector pointing from $\mathbf\text{'}$ to $\mathbf$. The total field is then found by "adding up" the contributions from all the increments of volume by over the volume of the charge distribution $V$: :$\mathbf\left(\mathbf\right) = \iiint_V \,\hat \mathbf\text{'}$ Similar equations follow for a surface charge with continuous charge distribution $\sigma\left(\mathbf\right)$ where $\sigma$ is the charge density in coulombs per square meter :$\mathbf\left(\mathbf\right) = \iint_S \, \hat \mathbf\text{'}$ and for line charges with continuous charge distribution $\lambda\left(\mathbf\right)$ where $\lambda$ is the charge density in coulombs per meter. :$\mathbf\left(\mathbf\right) = \int_P \, \hat \mathbf\text{'}$

## Electric potential

If a system is static, such that magnetic fields are not time-varying, then by Faraday's law, the electric field is curl-free. In this case, one can define an
electric potential The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of work (physics), work energy needed to move a unit of electric charge from a reference point to the spe ...

, that is, a function $\Phi$ such that This is analogous to the
gravitational potential In classical mechanics, the gravitational potential at a location is equal to the work Work may refer to: * Work (human activity), intentional activity people perform to support themselves, others, or the community ** Manual labour, physical wo ...

. The difference between the electric potential at two points in space is called the
potential difference Voltage, electric potential difference, electric pressure or electric tension is the difference in electric potential The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is def ...

(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 Magnetic vector potential, A, is the vector quantity in classical electromagnetism Classical electromagnetism or classical electrodynamics is a branch of theoretical physics that studies the interactions between electric charge Electric charg ...
, A, defined so that one can still define an electric potential $\Phi$ such that: :$\mathbf = - \nabla \Phi - \frac$ Where $\nabla \Phi$ is the
gradient In vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Prod ...

of the electric potential and $\frac$ is the
partial derivative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

of A with respect to time.
Faraday's law of induction Faraday's law of induction (briefly, Faraday's law) is a basic law of electromagnetism Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electric ch ...
can be recovered by taking the
curl Curl or CURL may refer to: Science and technology * Curl (mathematics) In vector calculus Vector calculus, or vector analysis, is concerned with derivative, differentiation and integral, integration of vector fields, primarily in 3-dimension ...
of that equation :$\nabla \times \mathbf = -\frac = -\frac$ which justifies, a posteriori, the previous form for E.

## Continuous vs. discrete charge representation

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 The electron is a subatomic particle (denoted by the symbol or ) whose electric charge is negative one elementary charge. Electrons belong to the first generation (particle physics), generation of the lepton particle family, and are general ...

s as point sources where charge density is infinite on an infinitesimal section of space. A charge $q$ located at $\mathbf$ can be described mathematically as a charge density $\rho\left(\mathbf\right)=q\delta\left(\mathbf\right)$, where the
Dirac delta function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is n ...
(in three dimensions) is used. Conversely, a charge distribution can be approximated by many small point charges.

# Electrostatic fields

Electrostatic fields are electric fields that do not change with time. Such fields are present when systems of charged matter are stationary, or when are unchanging. In that case,
Coulomb's law between two point charge A point particle (ideal particle or point-like particle, often spelled pointlike particle) is an idealization of particle In the Outline of physical science, physical sciences, a particle (or corpuscule in older te ...
fully describes the field.

## Parallels between electrostatic and gravitational fields

Coulomb's law, which describes the interaction of electric charges: :$\mathbf=q\left\left(\frac\frac\right\right)=q\mathbf$ is similar to
Newton's law of universal gravitation Newton's law of universal gravitation is usually stated as that every particle In the Outline of physical science, physical sciences, a particle (or corpuscule in older texts) is a small wikt:local, localized physical body, object to which can ...
: :$\mathbf=m\left\left(-GM\frac\right\right)=m\mathbf$ (where $\mathbf = \mathbf$). This suggests similarities between the electric field E and the gravitational field g, or their associated potentials. Mass is sometimes called "gravitational charge". Electrostatic and gravitational forces both are
central Central is an adjective usually referring to being in the center (disambiguation), center of some place or (mathematical) object. Central may also refer to: Directions and generalised locations * Central Africa, a region in the centre of Africa ...
,
conservative Conservatism is an aesthetic Aesthetics, or esthetics (), is a branch of philosophy that deals with the nature of beauty and taste (sociology), taste, as well as the philosophy of art (its own area of philosophy that comes out of aest ...
and obey an
inverse-square law 420px, S represents the light source, while r represents the measured points. The lines represent the flux emanating from the sources and fluxes. The total number of flux lines depends on the strength of the light source and is constant with in ...

.

## Uniform fields

A uniform field is one in which the electric field is constant at every point. It can be approximated by placing two conducting plates parallel to each other and maintaining a
voltage Voltage, electric potential difference, electric pressure or electric tension is the difference in electric potential The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is the ...

(potential difference) between them; it is only an approximation because of boundary effects (near the edge of the planes, electric field is distorted because the plane does not continue). Assuming infinite planes, the magnitude of the electric field ''E'' is: :$E = - \frac$ where Δ''V'' is the
potential difference Voltage, electric potential difference, electric pressure or electric tension is the difference in electric potential The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is def ...

between the plates and ''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 , achieved by applying a voltage of the order of 1 volt between conductors spaced 1 µm apart.

# Electrodynamic fields

Electrodynamic fields are electric fields which do change with time, for instance when charges are in motion. In this case, a magnetic field is produced in accordance with Ampère's circuital law ( with Maxwell's addition), which, along with Maxwell's other equations, defines the magnetic field, $\mathbf$, in terms of its curl: :$\nabla \times \mathbf = \mu_0\left\left(\mathbf + \varepsilon_0 \frac \right\right) ,$ where $\mathbf$ is the
current density In electromagnetism Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electric charge, electrically charged particles. The electromagnetic force is c ...

, $\mu_0$ is the
vacuum permeability Vacuum permeability is the magnetic permeability in a classical vacuum. ''Vacuum permeability'' is derived from production of a magnetic field by an electric current or by a moving electric charge and in all other formulas for magnetic-field prod ...
, and $\varepsilon_0$ is the
vacuum permittivity Vacuum permittivity, commonly denoted (pronounced as "epsilon nought" or "epsilon zero") is the value of the absolute dielectric permittivity of classical vacuum. Alternatively may be referred to as the permittivity of free space, the electr ...
. That is, both
electric currents An electric current is a stream of charged particle In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, ...
(i.e. charges in uniform motion) and the (partial) time derivative of the electric field directly contributes to the magnetic field. In addition, the
Maxwell–Faraday equation Faraday's law of induction (briefly, Faraday's law) is a basic law of electromagnetism predicting how a magnetic field will interact with an electric circuit to produce an electromotive force (EMF)—a phenomenon known as electromagnetic inductio ...
states :$\nabla \times \mathbf = -\frac .$ These represent two of Maxwell's four equations and they intricately link the electric and magnetic fields together, resulting in the
electromagnetic field An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the in ...
. 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 Lorentz is a name derived from the Roman surname, Laurentius, which means "from Laurentum". It is the German form of Laurence. Notable people with the name include: Given name * Lorentz Aspen (born 1978), Norwegian heavy metal pianist and keyb ...
: : $\mathbf = q\mathbf + q\mathbf \times \mathbf$

# Energy in the electric field

The total energy per unit volume stored by the
electromagnetic field An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the in ...
is :$u_\text = \frac , \mathbf, ^2 + \frac , \mathbf, ^2$ where ''ε'' is the
permittivity In electromagnetism Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electric charge, electrically charged particles. The electromagnetic force is car ...
of the medium in which the field exists, $\mu$ its
magnetic permeability In electromagnetism Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electric charge, electrically charged particles. The electromagnetic force is ...
, and E and B are the electric and magnetic field vectors. As E and 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 ''U'' stored in the electromagnetic field in a given volume ''V'' is :$U_\text = \frac \int_ \left\left( \varepsilon , \mathbf, ^2 + \frac , \mathbf, ^2 \right\right) dV \, .$

# The electric displacement field

## Definitive equation of vector fields

In the presence of matter, it is helpful to extend the notion of the electric field into three vector fields:Electromagnetism (2nd Edition), I.S. Grant, W.R. Phillips, Manchester Physics, John Wiley & Sons, 2008, :$\mathbf=\varepsilon_0\mathbf+\mathbf\!$ where P is the
electric polarization In classical electromagnetism Classical electromagnetism or classical electrodynamics is a branch of theoretical physics that studies the interactions between electric charge Electric charge is the physical property of matter that causes it t ...
– the volume density of
electric dipole moment The electric dipole moment is a measure of the separation of positive and negative electrical charges within a system, that is, a measure of the system's overall Chemical polarity, polarity. The International System of Units, SI unit for electric ...
s, and D is the
electric displacement field In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succ ...
. Since E and P are defined separately, this equation can be used to define D. The physical interpretation of D is not as clear as E (effectively the field applied to the material) or 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 free charges and currents.

## Constitutive relation

The E and D fields are related by the
permittivity In electromagnetism Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electric charge, electrically charged particles. The electromagnetic force is car ...
of the material, ''ε''. For linear,
homogeneous Homogeneity and heterogeneity are concepts often used in the sciences Science (from the Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originall ...
,
isotropic Isotropy is uniformity in all orientations; it is derived from the Greek ''isos'' (ἴσος, "equal") and ''tropos'' (τρόπος, "way"). Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by t ...
materials E and D are proportional and constant throughout the region, there is no position dependence: :$\mathbf\left(\mathbf\right) = \varepsilon\mathbf\left(\mathbf\right)$ For inhomogeneous materials, there is a position dependence throughout the material: :$\mathbf\left(\mathbf\right) = \varepsilon \left(\mathbf\right)\mathbf\left(\mathbf\right)$ For anisotropic materials the E and D fields are not parallel, and so E and D are related by the permittivity tensor (a 2nd order
tensor field In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
), in component form: :$D_i=\varepsilon_E_j$ For non-linear media, E and D are not proportional. Materials can have varying extents of linearity, homogeneity and isotropy.

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Classical electromagnetism Classical electromagnetism or classical electrodynamics is a branch of theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and ...
*
Electricity Electricity is the set of physical Physical may refer to: *Physical examination, a regular overall check-up with a doctor *Physical (album), ''Physical'' (album), a 1981 album by Olivia Newton-John **Physical (Olivia Newton-John song), "Physi ...

*
History of electromagnetic theory The history of electromagnetic theory begins with ancient measures to understand atmospheric electricity Atmospheric electricity is the study of electrical charges in the Earth's atmosphere (or that of another planet). The movement of charge b ...
* Optical field * Magnetism * Teltron tube * Teledeltos, a conductive paper that may be used as a simple analog computer for modelling fields

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