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An electric field (sometimes E-field) is the
physical field In physics, a field is a physical quantity, represented by a scalar, vector, or tensor, that has a value for each point in space and time. For example, on a weather map, the surface temperature is described by assigning a number to each point o ...
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 pa ...
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 currents. Electric fields and magnetic fields are both manifestations of the electromagnetic field, 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 electro ...
s (also called forces) of nature. Electric fields are important in many areas of
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, and are exploited in electrical technology. In atomic physics and chemistry, 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 gold foil experiment. After the discovery of the neutron ...
and
electron The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no ...
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 molecules and crystals. The bond may result from the electrostatic force between oppositely charged ions as in ionic bonds, or through the sharing o ...
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) 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'' (Machel Montano album) * ''Charge!!'', an album by The Aqu ...
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, charge, or size) are assumed to be negligible except for the property being studied, which is considered to be insuf ...
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 (SI). It is named after the Italian physicist Alessandro Volta (1745–1827). Defin ...
per
meter The metre (British spelling) or meter (American spelling; see spelling differences) (from the French unit , from the Greek noun , "measure"), symbol m, is the primary unit of length in the International System of Units (SI), though its pref ...
(V/m), which is equal to the newton per coulomb (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, charge, or size) are assumed to be negligible except for the property being studied, which is considered to be insuf ...
if held stationary at that point. As the electric field is defined in terms of force, and force is a
vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
(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), 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 acts between two
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different eleme ...
es, as they both obey an inverse-square law 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 that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is convention ...
, 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 inducti ...
, whose term '
lines of force A line of force in Faraday's extended sense is synonymous with Maxwell's line of induction. According to J.J. Thomson, Faraday usually discusses ''lines of force'' as chains of polarized particles in a dielectric, yet sometimes Faraday discusses ...
' 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. 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 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 cultural, social, and political philosophy that seeks to promote and to preserve traditional institutions, practices, and values. The central tenets of conservatism may vary in relation to the culture and civilization in ...
(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. The study of time varying magnetic and electric fields is called
electrodynamics In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions o ...
.


Mathematical formulation

Electric fields are caused by
electric charges Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons respectiv ...
, 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 sta ...
, and time varying
magnetic fields A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...
, described by Faraday's law of induction. 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. ...
that describe both fields as a function of charges and
currents Currents, Current 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 (stre ...
.


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 ''p' ...
(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 that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is convention ...
, 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 (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat"). The term ''direction v ...
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 absolute dielectric permittivity of classical vacuum. It may also be referred to as the permittivity of free space, the electric const ...
(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 Vacuum permittivity, commonly denoted (pronounced "epsilon nought" or "epsilon zero"), is the value of the absolute dielectric permittivity of classical vacuum. It may also be referred to as the permittivity of free space, the electric const ...
, must be substituted with \varepsilon,
permittivity In electromagnetism, the absolute permittivity, often simply called permittivity and denoted by the Greek letter ''ε'' (epsilon), is a measure of the electric polarizability of a dielectric. A material with high permittivity polarizes more in ...
, 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 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 of one or more variables whose range is a set of multidimensional vectors or infinite-dimensional vectors. The input of a vector-valued function could ...
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 In science, an inverse-square law is any scientific law stating that a specified physical quantity is inversely proportional to the square of the distance from the source of that physical quantity. The fundamental cause for this can be understoo ...
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 (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 (SI). It is named after the Italian physicist Alessandro Volta (1745–1827). Defin ...
per
meter The metre (British spelling) or meter (American spelling; see spelling differences) (from the French unit , from the Greek noun , "measure"), symbol m, is the primary unit of length in the International System of Units (SI), though its pref ...
(V/m); in terms of the SI base units it is kg⋅m⋅s−3⋅A−1.


Superposition principle

Due to the
linearity Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
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. ...
, 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. 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 \\ pt&= \hat \mathbf_1 + \hat \mathbf_2 + \hat \mathbf_3 + \cdots \\ pt&= \sum_^N \hat \mathbf_k \end where \mathbf is the
unit vector In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat"). The term ''direction v ...
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(\mathbf) (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 SI system in ...
in coulombs per cubic meter). By considering the charge \rho(\mathbf')dV in each small volume of space dV at point \mathbf' as a point charge, the resulting electric field, d\mathbf(\mathbf), at point \mathbf can be calculated as d\mathbf(\mathbf) = \frac\frac \hat \mathbf' where \hat \mathbf' is the unit vector pointing from \mathbf' 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(\mathbf) 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(\mathbf) 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 energy needed to move a unit of electric charge from a reference point to the specific point in ...
, 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 (energy transferred) per unit mass that would be needed to move an object to that location from a fixed reference location. It is analogous to the electric ...
. 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 between two points. In a static electric field, it corresponds to the work needed per unit of charge to m ...
(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, , 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 differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
of the electric potential and \frac is the partial derivative of A with respect to time. Faraday's law of induction can be recovered by taking the curl 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 electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no ...
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(\mathbf) = q\delta(\mathbf - \mathbf_0), where the Dirac delta function (in three dimensions) is used. Conversely, a charge distribution can be approximated by many small point charges.


Electrostatic fields

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 that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is convention ...
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 distan ...
: \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 Central is an adjective usually referring to being in the 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 continent, also known as ...
,
conservative Conservatism is a cultural, social, and political philosophy that seeks to promote and to preserve traditional institutions, practices, and values. The central tenets of conservatism may vary in relation to the culture and civilization in ...
and obey an inverse-square law.


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 static electric field, it corresponds to the work needed per unit of charge to ...
(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 between two points. In a static electric field, it corresponds to the work needed per unit of charge to m ...
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, \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. That is, both
electric currents 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 pa ...
(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 induct ...
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. 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: \mathbf = q\mathbf + q\mathbf \times \mathbf


Energy in the electric field

The total energy per unit volume stored by the electromagnetic field 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), is a measure of the electric polarizability of a dielectric. A material with high permittivity polarizes more in ...
of the medium in which the field exists, \mu its magnetic permeability, 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 ...
– the volume density of electric dipole moments, 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 free and bound charge within materials. "D" stands for "displacement", as in ...
. 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), is a measure of the electric polarizability of a dielectric. A material with high permittivity polarizes more in ...
of the material, ''ε''. For linear, homogeneous, isotropic 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. ...
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 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 that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is convention ...
and assigning electric field and magnetic field by their definition given by the form of Lorentz force. 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 that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is convention ...
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. ...
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 space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The laws o ...
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 theory of relativity, is the upper limit ...
.
Maxwell's laws 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. T ...
are found to confirm to this view since the general solutions of fields are given in terms of retarded time which indicate that
electromagnetic In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions o ...
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 theory of relativity, is the upper limit ...
. 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 pa ...
, 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 theory of relativity, is the upper limit ...
while the electric field lines far away from this will continue to point radially towards an assumed moving charge. This virtual particle will never be outside the range of propagation of the disturbance in electromagnetic field, since charged particles are restricted to have speeds slower than that of light, which makes it impossible to construct a
gaussian surface A Gaussian surface is a closed surface in three-dimensional space through which the flux of a vector field is calculated; usually the gravitational field, electric field, or magnetic field. It is an arbitrary closed surface (the boundary of ...
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 sta ...
. 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 fi ...
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 theory of relativity, is the upper limit ...
. In general, any accelerating point charge radiates electromagnetic waves 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 The Liénard–Wiechert potentials describe the classical electromagnetic effect of a moving electric point charge in terms of a vector potential and a scalar potential in the Lorenz gauge. Stemming directly from Maxwell's equations, these descr ...
. 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. ...
, 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. ...
. 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 In electromagnetism, electromagnetic waves in vacuum travel at the speed of light ''c'', according to Maxwell's Equations. The retarded time is the time when the field began to propagate from the point where it was emitted to an observer. The term ...
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 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 in special relativit ...
. The retarded time is given as solution of: t_r=\mathbf-\frac The uniqueness of solution for for given \mathbf, \mathbf and r_s(t) is valid for charged particles moving slower than speed of light.
Electromagnetic radiation In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visible) li ...
of accelerating charges is known to be caused by the acceleration dependent term in the electric field from which relativistic correction for
Larmor formula In electrodynamics, the Larmor formula is used to calculate the total power radiated by a nonrelativistic point charge as it accelerates. It was first derived by J. J. Larmor in 1897, in the context of the wave theory of light. When any charged ...
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. ...
, 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 \frac \left -\left(\frac-1\right)^\right\hat *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.


See also

*
Classical electromagnetism Classical electromagnetism or classical electrodynamics is a branch of theoretical physics that studies the interactions between electric charges and currents using an extension of the classical Newtonian model; It is, therefore, a classical fie ...
* Relativistic Electromagnetism *
Electricity Electricity is the set of physical 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 electromagnetism, as describ ...
* History of electromagnetic theory *
Optical field An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is the classical c ...
* Magnetism * Teltron tube *
Teledeltos Teledeltos paper is an electrically conductive paper. It is formed by a coating of carbon on one side of a sheet of paper, giving one black and one white side. Western Union developed Teledeltos paper in the late 1940s (several decades after it was ...
, a conductive paper that may be used as a simple analog computer for modelling fields


References

* *


External links


Electric field in "Electricity and Magnetism", R Nave
Hyperphysics ''HyperPhysics'' is an educational website about physics topics. The information architecture of the website is based on HyperCard, the platform on which the material was originally developed, and a thesaurus organization, with thousands of contr ...
,
Georgia State University Georgia State University (Georgia State, State, or GSU) is a public research university in Atlanta, Georgia. Founded in 1913, it is one of the University System of Georgia's four research universities. It is also the largest institution of hig ...

Frank Wolfs's lectures
at
University of Rochester The University of Rochester (U of R, UR, or U of Rochester) is a private university, private research university in Rochester, New York. The university grants Undergraduate education, undergraduate and graduate degrees, including Doctorate, do ...
, chapters 23 and 24
Fields
– a chapter from an online textbook {{DEFAULTSORT:Electric Field Electrostatics Physical quantities Electromagnetism