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An electric field (sometimes E-field) is the physical field that surrounds electrically
charged particle In physics, a charged particle is a particle with an electric charge. It may be an ion, such as a molecule or atom with a surplus or deficit of electrons relative to protons. It can also be an electron or a proton, or another elementary particle, ...
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 and time-varying
electric current An electric current is a stream of charged particles, such as electrons or ions, moving through an electrical conductor or space. It is measured as the net rate of flow of electric charge through a surface or into a control volume. The moving par ...
s. Electric fields and magnetic fields are both manifestations of the
electromagnetic field An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field (physics), field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is ...
, one of the four
fundamental interaction In physics, the fundamental interactions, also known as fundamental forces, are the interactions that do not appear to be reducible to more basic interactions. There are four fundamental interactions known to exist: the gravitational and elect ...
s (also called forces) 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 and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical science is that depar ...
, and are exploited in electrical technology. In
atomic physics Atomic physics is the field of physics that studies atoms as an isolated system of electrons and an atomic nucleus. Atomic physics typically refers to the study of atomic structure and the interaction between atoms. It is primarily concerned wit ...
and
chemistry Chemistry is the scientific study of the properties and behavior of matter. It is a natural science that covers the elements that make up matter to the compounds made of atoms, molecules and ions: their composition, structure, properties ...
, for instance, the electric field is the attractive force holding the
atomic nucleus The atomic nucleus is the small, dense region consisting of protons and neutrons at the center of an atom, discovered in 1911 by Ernest Rutherford based on the 1909 Geiger–Marsden experiments, Geiger–Marsden gold foil experiment. After th ...
and
electron The electron ( or ) is a subatomic particle with a negative one elementary charge, elementary electric charge. Electrons belong to the first generation (particle physics), generation of the lepton particle family, and are generally thought t ...
s together in atoms. It is also the force responsible for
chemical bonding A chemical bond is a lasting attraction between atoms or ions that enables the formation of Molecule, molecules and crystals. The bond may result from the Coulomb's law, electrostatic force between oppositely charged ions as in Ionic bonding, ...
between atoms that result in
molecule A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. In quantum physics, organic chemistry, and bioche ...
s. The electric field is defined as a vector field that associates to each point in space the electrostatic (
Coulomb The coulomb (symbol: C) is the unit of electric charge in the International System of Units (SI). In 2019 redefinition of the SI base units, the present version of the SI it is equal to the electric charge delivered by a 1 ampere constant curre ...
) force per unit of charge exerted on an infinitesimal positive test charge at rest at that point. The derived SI unit for the electric field is the
volt The volt (symbol: V) is the unit of electric potential, electric potential difference (voltage), and electromotive force in the International System of Units, International System of Units (SI). It is named after the Italian physicist Alessandro ...
per
meter The metre (British English, British spelling) or meter (American English, American spelling; American and British English spelling differences#-re, -er, see spelling differences) (from the French unit , from the Greek language, Greek noun , "m ...
(V/m), which is equal to the newton per
coulomb The coulomb (symbol: C) is the unit of electric charge in the International System of Units (SI). In 2019 redefinition of the SI base units, the present version of the SI it is equal to the electric charge delivered by a 1 ampere constant curre ...
(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 if held stationary 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 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 accelerate. Force can ...
, and force is a vector (i.e. having both magnitude and direction), it follows that an electric field is a vector field. Fields that may be defined in this manner are sometimes referred to as force fields. The electric field acts between two charges similarly to the way the
gravitational field In physics, a gravitational field is a scientific model, model used to explain the influences that a massive body extends into the space around itself, producing a force on another massive body. Thus, a gravitational Field (physics), field is us ...
acts between two
mass Mass is an Intrinsic and extrinsic properties, intrinsic property of a body. It was traditionally believed to be related to the physical quantity, quantity of matter in a Physical object, physical body, until the discovery of the atom and par ...
es, as they both obey an
inverse-square law In science, an inverse-square law is any scientific law stating that a specified physical quantity is Proportionality (mathematics)#Inverse proportionality, inversely proportional to the square (algebra), square of the distance from the source o ...
with distance. This is the basis for
Coulomb's law Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, ...
, 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 Line most often refers to: * Line (geometry), object with zero thickness and curvature that stretches to infinity * Telephone line, a single-user circuit on a telephone communication system Line, lines, The Line, or LINE may also refer to: Arts ...
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 who contributed to the study of electromagnetism and electrochemistry. His main discoveries include the principles underlying electromagnetic induction, ...
, whose term '
lines of force A line of force in Michael Faraday, Faraday's extended sense is synonymous with James Clerk Maxwell, Maxwell's line of induction. According to J.J. Thomson, Faraday usually discusses ''lines of force'' as chains of polarized particles in a dielectr ...
' is still sometimes used. This illustration has the useful property that the field's strength is proportional to the density of the lines. 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 that studies electric charges at Rest (physics), rest (static electricity). Since classical antiquity, classical times, it has been known that some materials, such as amber, attract lightweight particles af ...
. 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 (pronounced like "curl", UK: , US: ) is a computer software project providing a library (computing), library (libcurl) and command-line tool (curl) for transferring data using various network Protocol (computing), protocols. The name sta ...
of the electric field is equal to the negative
time derivative A time derivative is a derivative of a function with respect to time, usually interpreted as the rate of change of the value of the function. The variable denoting time is usually written as t. Notation A variety of notations are used to denote th ...
of the magnetic field. In the absence of time-varying magnetic field, the electric field is therefore called
conservative Conservatism is a Philosophy of culture, cultural, Social philosophy, social, and political philosophy that seeks to promote and to preserve traditional institutions, practices, and values. The central tenets of conservatism may vary in r ...
(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 (physics), field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is ...
. The study of time varying magnetic and electric fields is called electrodynamics.

# Mathematical formulation

Electric fields are caused by electric charges, described by
Gauss's law In physics and electromagnetism, Gauss's law, also known as Gauss's flux theorem, (or sometimes simply called Gauss's theorem) is a law relating the distribution of electric charge to the resulting electric field. In its integral form, it stat ...
, and time varying magnetic fields, described by
Faraday's law of induction 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 inducti ...
. 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, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. Th ...
that describe both fields as a function of charges and currents.

## Electrostatics

In the special case of a
steady state In systems theory, a system or a process is in a steady state if the variables (called state variables) which define the behavior of the system or the process are unchanging in time. In continuous time, this means that for those properties ' ...
(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 Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, ...
, 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: $\mathbf = \frac \frac \hat \mathbf_ \,,$ where $\hat \mathbf_$ is the
unit vector In mathematics, a unit vector in a normed vector space is a Vector_(mathematics_and_physics), vector (often a vector (geometry), spatial vector) of Norm (mathematics), length 1. A unit vector is often denoted by a lowercase letter with a circumfle ...
in the direction from point $\mathbf_1$ to point $\mathbf_0$, and is the
electric constant Vacuum permittivity, commonly denoted (pronounced "epsilon nought" or "epsilon zero"), is the value of the Permittivity, absolute dielectric permittivity of classical vacuum. It may also be referred to as the permittivity of free space, the el ...
(also known as "the absolute permittivity of free space") with the unit C2⋅m−2⋅N−1. Note that $\varepsilon_0$, the vacuum electric permittivity, must be substituted with $\varepsilon$,
permittivity In electromagnetism, the absolute permittivity, often simply called permittivity and denoted by the Greek letter ''ε'' (Epsilon, epsilon), is a measure of the electric polarizability of a dielectric. A material with high permittivity polarizes ...
, 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 inverse-square law, or simply Coulomb's law, is an experimental scientific law, law of physics that quantifies the amount of force between two stationary, electric charge, electrically charged particles. The electric force between char ...
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) $\mathbf(\mathbf_0) = \frac = \frac \frac \hat \mathbf_$ 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 mathematical function In mathematics, a function from a set (mathematics), set to a set assigns to each element of exactly one element of .; the words map, mapping, tran ...
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. 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 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 SI unit of the electric field is the newton per
coulomb The coulomb (symbol: C) is the unit of electric charge in the International System of Units (SI). In 2019 redefinition of the SI base units, the present version of the SI it is equal to the electric charge delivered by a 1 ampere constant curre ...
(N/C), or
volt The volt (symbol: V) is the unit of electric potential, electric potential difference (voltage), and electromotive force in the International System of Units, International System of Units (SI). It is named after the Italian physicist Alessandro ...
per
meter The metre (British English, British spelling) or meter (American English, American spelling; American and British English spelling differences#-re, -er, see spelling differences) (from the French unit , from the Greek language, Greek noun , "m ...
(V/m); in terms of the
SI base unit The SI base units are the standard units of measurement defined by the International System of Units (SI) for the seven base quantities of what is now known as the International System of Quantities: they are notably a basic set from which a ...
s it is 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 (mat ...
of
Maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. Th ...
, electric fields satisfy the
superposition principle The superposition principle, also known as superposition property, states that, for all linear systems, the net response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually. So tha ...
, 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: where $\mathbf$ is the
unit vector In mathematics, a unit vector in a normed vector space is a Vector_(mathematics_and_physics), vector (often a vector (geometry), spatial vector) of Norm (mathematics), length 1. A unit vector is often denoted by a lowercase letter with a circumfle ...
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, charge density is the amount of electric charge per unit length, surface area, or volume. Volume charge density (symbolized by the Greek letter ρ) is the quantity of charge per unit volume, measured in the Systeme Internat ...
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 $d\mathbf(\mathbf) = \frac\frac \hat \mathbf'$ 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 integrating over the volume of the charge distribution $V$: $\mathbf(\mathbf) = \frac \iiint_V \,\hat \mathbf'$ 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 $\mathbf(\mathbf) = \frac \iint_S \, \hat \mathbf'$ and for line charges with continuous charge distribution $\lambda\left(\mathbf\right)$ where $\lambda$ is the charge density in coulombs per meter. $\mathbf(\mathbf) = \frac \int_P \, \hat \mathbf'$

## 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 sp ...
, that is, a function $\Phi$ such that This is analogous to the
gravitational potential In classical mechanics Classical mechanics is a Theoretical physics, physical theory describing the motion of macroscopic objects, from projectiles to parts of Machine (mechanical), machinery, and astronomical objects, such as spacecraft ...
. The difference between the electric potential at two points in space is called the
potential difference Voltage, also known as electric pressure, electric tension, or (electric) potential difference, is the difference in electric potential The electric potential (also called the ''electric field potential'', potential drop, the electrostat ...
(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 In classical electromagnetism, magnetic vector potential (often called A) is the vector quantity defined so that its Curl (mathematics), curl is equal to the magnetic field: \nabla \times \mathbf = \mathbf. Together with the electric potential '' ...
, , 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, the gradient of a scalar-valued function, scalar-valued differentiable function of Function of several variables, several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "d ...
of the electric potential and $\frac$ is the
partial derivative In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...
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 predicting how a magnetic field will interact with an electric circuit to produce an electromotive force (emf)—a phenomenon known as electromagnetic inducti ...
can be recovered by taking the
curl cURL (pronounced like "curl", UK: , US: ) is a computer software project providing a library (computing), library (libcurl) and command-line tool (curl) for transferring data using various network Protocol (computing), protocols. The name sta ...
of that equation $\nabla \times \mathbf = -\frac = -\frac$ which justifies, a posteriori, the previous form for .

## 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 ( or ) is a subatomic particle with a negative one elementary charge, elementary electric charge. Electrons belong to the first generation (particle physics), generation of the lepton particle family, and are generally thought t ...
s as point sources where charge density is infinite on an infinitesimal section of space. A charge $q$ located at $\mathbf_0$ can be described mathematically as a charge density $\rho\left(\mathbf\right) = q\delta\left(\mathbf - \mathbf_0\right)$, where the
Dirac delta function In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented ...
(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 electric currents are unchanging. In that case,
Coulomb's law Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, ...
fully describes the field.

## Parallels between electrostatic and gravitational fields

Coulomb's law, which describes the interaction of electric charges: $\mathbf = q \left(\frac \frac\right) = q \mathbf$ is similar to
Newton's law of universal gravitation Newton's law of universal gravitation is usually stated as that every particle attracts every other particle in the universe with a force that is proportional to the product of their masses and inversely proportional to the square of the dist ...
: $\mathbf = m\left(-GM\frac\right) = m\mathbf$ (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,
conservative Conservatism is a Philosophy of culture, cultural, Social philosophy, social, and political philosophy that seeks to promote and to preserve traditional institutions, practices, and values. The central tenets of conservatism may vary in r ...
and obey an
inverse-square law In science, an inverse-square law is any scientific law stating that a specified physical quantity is Proportionality (mathematics)#Inverse proportionality, inversely proportional to the square (algebra), square of the distance from the source o ...
.

## 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, also known as electric pressure, electric tension, or (electric) potential difference, is the difference in electric potential between two points. In a Electrostatics, static electric field, it corresponds to the Work (electrical), w ...
(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, also known as electric pressure, electric tension, or (electric) potential difference, is the difference in electric potential The electric potential (also called the ''electric field potential'', potential drop, the electrostat ...
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: $\nabla \times \mathbf = \mu_0\left(\mathbf + \varepsilon_0 \frac \right) ,$ where $\mathbf$ is the
current density In electromagnetism, current density is the amount of charge per unit time that flows through a unit area of a chosen cross section. The current density vector is defined as a vector (geometric), vector whose magnitude is the electric current per ...
, $\mu_0$ is the
vacuum permeability The vacuum magnetic permeability (variously ''vacuum permeability'', ''permeability of free space'', ''permeability of vacuum''), also known as the magnetic constant, is the magnetic permeability in a classical vacuum. It is a physical constant, ...
, and $\varepsilon_0$ is the
vacuum permittivity Vacuum permittivity, commonly denoted (pronounced "epsilon nought" or "epsilon zero"), is the value of the Permittivity, absolute dielectric permittivity of classical vacuum. It may also be referred to as the permittivity of free space, the el ...
. That is, both electric currents (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 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 (physics), field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is ...
. 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 (disambiguation), 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 he ...
: $\mathbf = q\mathbf + q\mathbf \times \mathbf$

# 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 (physics), field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is ...
is $u_\text = \frac , \mathbf, ^2 + \frac , \mathbf, ^2$ where is the
permittivity In electromagnetism, the absolute permittivity, often simply called permittivity and denoted by the Greek letter ''ε'' (Epsilon, epsilon), is a measure of the electric polarizability of a dielectric. A material with high permittivity polarizes ...
of the medium in which the field exists, $\mu$ its
magnetic permeability In electromagnetism, permeability is the measure of magnetization that a material obtains in response to an applied magnetic field. Permeability is typically represented by the (italicized) Greek letter Mu (letter), ''μ''. The term was coined ...
, and and are the electric and magnetic field vectors. As and 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 $U_\text = \frac \int_ \left( \varepsilon , \mathbf, ^2 + \frac , \mathbf, ^2 \right) dV \, .$

# 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: $\mathbf = \varepsilon_0 \mathbf + \mathbf$ where P is the
electric polarization In classical electromagnetism, polarization density (or electric polarization, or simply polarization) is the vector field that expresses the density of permanent or induced electric dipole moments in a dielectric material. When a dielectric is p ...
– 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 is the
electric displacement field In physics, the electric displacement field (denoted by D) or electric induction is a vector field that appears in Maxwell's equations. It accounts for the effects of Charge density#Free, bound and total charge, free and bound charge within mater ...
. Since E and P are defined separately, this equation can be used to define . The physical interpretation of D is not as clear as E (effectively the field applied to the material) or (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, the absolute permittivity, often simply called permittivity and denoted by the Greek letter ''ε'' (Epsilon, epsilon), is a measure of the electric polarizability of a dielectric. A material with high permittivity polarizes ...
of the material, ''ε''. For linear,
homogeneous Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the Uniformity (chemistry), uniformity of a Chemical substance, substance or organism. A material or image that is homogeneous is uniform in compos ...
,
isotropic Isotropy is uniformity in all orientations; it is derived . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence '' anisotropy''. ''Anisotropy'' is also used to describ ...
materials E and D are proportional and constant throughout the region, there is no position dependence: $\mathbf(\mathbf) = \varepsilon\mathbf(\mathbf)$ For inhomogeneous materials, there is a position dependence throughout the material: $\mathbf(\mathbf) = \varepsilon (\mathbf)\mathbf(\mathbf)$ For anisotropic materials the and fields are not parallel, and so and are related by the permittivity tensor (a 2nd order
tensor field In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold). Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis ...
), in component form: $D_i = \varepsilon_ E_j$ For non-linear media, and 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 Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. Th ...
under
Lorentz transformation In physics, the Lorentz transformations are a six-parameter family of Linear transformation, linear coordinate transformation, transformations from a Frame of Reference, coordinate frame in spacetime to another frame that moves at a constant velo ...
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. Alternatively the electric field of uniformly moving point charges can be derived from the
Lorentz transformation In physics, the Lorentz transformations are a six-parameter family of Linear transformation, linear coordinate transformation, transformations from a Frame of Reference, coordinate frame in spacetime to another frame that moves at a constant velo ...
of
four-force In the special theory of relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between Spacetime, space and time. In Albert Einstein's original treatment, the ...
experienced by test charges in the source's rest frame given by
Coulomb's law Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, ...
and assigning electric field and magnetic field by their definition given by the form of
Lorentz force In physics (specifically in electromagnetism) the Lorentz force (or electromagnetic force) is the combination of electric and magnetic force on a point charge due to electromagnetic fields. A particle of charge moving with a velocity in an elect ...
. However the following equation is only applicable when no acceleration is involved in the particle's history where
Coulomb's law Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, ...
can be considered or symmetry arguments can be used for solving
Maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. Th ...
in a simple manner. The electric field of such a uniformly moving point charge is hence given by: $\mathbf = \frac q \frac \mathbf$ where $q$ is the charge of the point source, $\mathbf$ is the position vector from the point source to the point in space, $\beta$ is the ratio of observed speed of the charge particle to the speed of light and $\theta$ is the angle between $\mathbf$ 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. Spherically 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.

## Propagation of disturbances in electric fields

Special theory of relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between Spacetime, space and time. In Albert Einstein's original treatment, the theory is based on two Postulates of ...
imposes the
principle of locality In physics, the principle of locality states that an object is influenced directly only by its immediate surroundings. A theory that includes the principle of locality is said to be a "local theory". This is an alternative to the concept of ins ...
, that requires cause and effect to be time-like separated events where the causal efficacy does not travel faster than the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special relativity, special theory of relativity, is ...
. 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 electromagnetic disturbances travel at the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special relativity, special theory of relativity, is ...
. Advanced time, which also provides a solution for maxwell's law are ignored as an unphysical solution.For the motion of a
charged particle In physics, a charged particle is a particle with an electric charge. It may be an ion, such as a molecule or atom with a surplus or deficit of electrons relative to protons. It can also be an electron or a proton, or another elementary 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 The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special relativity, special theory of relativity, is ...
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 An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field (physics), field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is ...
, 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' law In physics and electromagnetism, Gauss's law, also known as Gauss's flux theorem, (or sometimes simply called Gauss's theorem) is a law relating the distribution of electric charge to the resulting electric field. In its integral form, it stat ...
. 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 An electromagnetic pulse (EMP), also a transient electromagnetic disturbance (TED), is a brief burst of electromagnetic energy. Depending upon the source, the origin of an EMP can be natural or artificial, and can occur as an electromagnetic fie ...
of radiation is generated that connects at the boundary of this disturbance travelling outwards at the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special relativity, special theory of relativity, is ...
. In general, any accelerating point charge radiates
electromagnetic waves In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic field, electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, inf ...
however, non radiating acceleration is possible in a systems of charges.

## Arbitrarily moving point charge

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. Since the potentials satisfy
maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. Th ...
, the fields derived for point charge also satisfy
Maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. Th ...
. The electric field is expressed as: $\mathbf(\mathbf, \mathbf) = \frac \left(\frac + \frac \right)_$ where $q$ is the charge of the point source,  is retarded time or the time at which the source's contribution of the electric field originated, $_s(t)$ is the position vector of the particle, $_s(\mathbf,t)$ is a unit vector pointing from charged particle to the point in space, $\boldsymbol_s(t)$ is the velocity of the particle divided by the speed of light, and $\gamma(t)$ is the corresponding
Lorentz factor The Lorentz factor or Lorentz term is a quantity (physics), quantity expressing how much the measurements of time, length, and other physical properties change for an object while that object is moving. The expression appears in several equations ...
. The retarded time is given as solution of: $t_r=\mathbf-\frac$ The uniqueness of solution for  for given $\mathbf$, $\mathbf$ and $r_s\left(t\right)$ is valid for charged particles moving slower than speed of light.
Electromagnetic radiation In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic field, electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, inf ...
of accelerating charges is known to be caused by the acceleration dependent term in the electric field from which relativistic correction for Larmor formula is obtained. There exist yet another set of solutions for maxwell's equation of the same form but for advanced time  instead of retarded time given as a solution of: $t_a=\mathbf+\frac$ 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 Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. Th ...
, such as 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.

# Some Common Electric Field Values

*Infinite Wire having Uniform charge density $\lambda$ has Electric Field at a distance $x$ from it as $\frac \hat$ *Infinitely large surface having charge density $\sigma$ has Electric Field at a distance $x$ from it as $\frac \hat$ *Infinitely long cylinder having Uniform charge density $\lambda$ that is charge contained along unit length of the cylinder has Electric Field at a distance $x$ from it as $\frac \hat$ while it is $0$ everywhere inside the cylinder *Uniformly Charged non-conducting sphere of radius $R$, volume charge density $\rho$ and total charge $Q$ has Electric Field at a distance $x$ from it as $\frac \hat$ while the electric field at a point $\vec$ inside sphere from its center is given by $\frac\vec$ *Uniformly Charged conducting sphere of radius $R$, surface charge density $\sigma$ and total charge $Q$ has Electric Field at a distance $x$ from it as $\frac \hat$ while the electric field inside is $0$ *Electric field infinitely close to a conducting surface in electrostatic equilibrium having charge density $\sigma$ at that point is $\frac \hat$ *Uniformly Charged Ring having total charge $Q$ has Electric Field at a distance $x$ along its axis as $\frac \hat$' *Uniformly charged disc of radius $R$ and charge density $\sigma$ has Electric Field at a distance $x$ along its axis from it as *Electric field due to dipole of dipole moment $\vec$ at a distance $x$ from their center along equatorial plane is given as $-\frac$ and the same along the axial line is approximated to $\frac$ for $x$ much bigger than the distance between dipoles. Further generalization is given by
multipole expansion A multipole expansion is a Series (mathematics), mathematical series representing a Function (mathematics), function that depends on angles—usually the two angles used in the spherical coordinate system (the polar and Azimuth, azimuthal angles) f ...
.

*
Classical electromagnetism Classical electromagnetism or classical electrodynamics is a branch of theoretical physics that studies the interactions between electric charges and electrical current, currents using an extension of the classical Newtonian model; It is, therefo ...
* Relativistic Electromagnetism *
Electricity Electricity is the set of physics, physical Phenomenon, phenomena associated with the presence and motion of matter that has a property of electric charge. Electricity is related to magnetism, both being part of the phenomenon of electromagne ...
* History of electromagnetic theory * Optical field *
Magnetism Magnetism is the class of physical attributes that are mediated by a magnetic field, which refers to the capacity to induce attractive and repulsive phenomena in other entities. Electric currents and the magnetic moments of elementary particles ...
* Teltron tube * Teledeltos, a conductive paper that may be used as a simple analog computer for modelling fields

* *