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OR:

Flux describes any effect that appears to pass or travel (whether it actually moves or not) through a
surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of Visual perception, sight ...
or substance. Flux is a concept in
applied mathematics Applied mathematics is the application of mathematics, mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and Industrial sector, industry. Thus, applied mathematics is ...
and
vector calculus Vector calculus, or vector analysis, is concerned with derivative, differentiation and integral, integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for ...
which has many applications to
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 ...
. For
transport phenomena In engineering, physics, and chemistry, the study of transport phenomena concerns the exchange of mass, energy, charge (physics), charge, momentum and angular momentum between observed and studied Physical system, systems. While it draws from fi ...
, flux is a
vector Vector most often refers to: *Euclidean vector 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 ...
quantity, describing the magnitude and direction of the flow of a substance or property. In
vector calculus Vector calculus, or vector analysis, is concerned with derivative, differentiation and integral, integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for ...
flux is a scalar quantity, defined as the
surface integral In mathematics, particularly multivariable calculus, a surface integral is a generalization of multiple integrals to Integral, integration over surface (differential geometry), surfaces. It can be thought of as the double integral analogue of th ...
of the perpendicular component of a vector field over a surface.

# Terminology

The word ''flux'' comes from
Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through ...
: ''fluxus'' means "flow", and ''fluere'' is "to flow". As '' fluxion'', this term was introduced into
differential calculus In mathematics, differential calculus is a subfield of calculus that studies the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus—the study of the area beneath a curve. ...
by
Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, Theology, theologian, and author (described in his time as a "natural philosophy, natural philosopher"), widely ...
. The concept of heat flux was a key contribution of
Joseph Fourier Jean-Baptiste Joseph Fourier (; ; 21 March 1768 – 16 May 1830) was a French people, French mathematician and physicist born in Auxerre and best known for initiating the investigation of Fourier series, which eventually developed into Fourier an ...
, in the analysis of heat transfer phenomena. His seminal treatise ''Théorie analytique de la chaleur'' (''The Analytical Theory of Heat''), defines ''fluxion'' as a central quantity and proceeds to derive the now well-known expressions of flux in terms of temperature differences across a slab, and then more generally in terms of temperature gradients or differentials of temperature, across other geometries. One could argue, based on the work of
James Clerk Maxwell James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and scientist responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism and light ...
, that the transport definition precedes the definition of flux used in electromagnetism. The specific quote from Maxwell is: According to the transport definition, flux may be a single vector, or it may be a vector field / function of position. In the latter case flux can readily be integrated over a surface. By contrast, according to the electromagnetism definition, flux ''is'' the integral over a surface; it makes no sense to integrate a second-definition flux for one would be integrating over a surface twice. Thus, Maxwell's quote only makes sense if "flux" is being used according to the transport definition (and furthermore is a vector field rather than single vector). This is ironic because Maxwell was one of the major developers of what we now call "electric flux" and "magnetic flux" according to the electromagnetism definition. Their names in accordance with the quote (and transport definition) would be "surface integral of electric flux" and "surface integral of magnetic flux", in which case "electric flux" would instead be defined as "electric field" and "magnetic flux" defined as "magnetic field". This implies that Maxwell conceived of these fields as flows/fluxes of some sort. Given a flux according to the electromagnetism definition, the corresponding flux density, if that term is used, refers to its derivative along the surface that was integrated. By the
Fundamental theorem of calculus The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each time) with the concept of integrating a function (calculating the area under its graph, ...
, the corresponding flux density is a flux according to the transport definition. Given a current such as electric current—charge per time, current density would also be a flux according to the transport definition—charge per time per area. Due to the conflicting definitions of ''flux'', and the interchangeability of ''flux'', ''flow'', and ''current'' in nontechnical English, all of the terms used in this paragraph are sometimes used interchangeably and ambiguously. Concrete fluxes in the rest of this article will be used in accordance to their broad acceptance in the literature, regardless of which definition of flux the term corresponds to.

# Flux as flow rate per unit area

In
transport phenomena In engineering, physics, and chemistry, the study of transport phenomena concerns the exchange of mass, energy, charge (physics), charge, momentum and angular momentum between observed and studied Physical system, systems. While it draws from fi ...
(
heat transfer Heat transfer is a discipline of thermal engineering that concerns the generation, use, conversion, and exchange of thermal energy (heat In thermodynamics, heat is defined as the form of energy crossing the boundary of a thermodynamic s ...
,
mass transfer Mass transfer is the net movement of mass from one location (usually meaning stream, Phase (matter), phase, fraction or component) to another. Mass transfer occurs in many processes, such as Absorption (chemistry), absorption, evaporation, dryi ...
and
fluid dynamics In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) and ...
), flux is defined as the ''rate of flow of a property per unit area,'' which has the
dimensions In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
uantity imesup>−1· reasup>−1. The area is of the surface the property is flowing "through" or "across". For example, the amount of water that flows through a cross section of a river each second divided by the area of that cross section, or the amount of sunlight energy that lands on a patch of ground each second divided by the area of the patch, are kinds of flux.

## General mathematical definition (transport)

Here are 3 definitions in increasing order of complexity. Each is a special case of the following. In all cases the frequent symbol ''j'', (or ''J'') is used for flux, ''q'' for the
physical quantity A physical quantity is a physical property of a material or system that can be quantified by measurement. A physical quantity can be expressed as a ''value'', which is the algebraic multiplication of a ' Numerical value ' and a ' Unit '. For examp ...
that flows, ''t'' for time, and ''A'' for area. These identifiers will be written in bold when and only when they are vectors. First, flux as a (single) scalar: $j = \frac$ where: $I = \lim_\frac = \frac$ In this case the surface in which flux is being measured is fixed, and has area ''A''. The surface is assumed to be flat, and the flow is assumed to be everywhere constant with respect to position, and perpendicular to the surface. Second, flux as a
scalar field In mathematics and physics, a scalar field is a function (mathematics), function associating a single number to every point (geometry), point in a space (mathematics), space – possibly physical space. The scalar may either be a pure Scalar ( ...
defined along a surface, i.e. a function of points on the surface: $j(\mathbf) = \frac(\mathbf)$ $I(A,\mathbf) = \frac(A,\mathbf)$ As before, the surface is assumed to be flat, and the flow is assumed to be everywhere perpendicular to it. However the flow need not be constant. ''q'' is now a function of p, a point on the surface, and ''A'', an area. Rather than measure the total flow through the surface, q measures the flow through the disk with area ''A'' centered at ''p'' along the surface. Finally, flux as a vector field: $\mathbf(\mathbf) = \frac(\mathbf)$ $\mathbf(A,\mathbf) = \underset\, \mathbf_ \frac(A,\mathbf, \mathbf)$ In this case, there is no fixed surface we are measuring over. ''q'' is a function of a point, an area, and a direction (given by a unit vector, $\mathbf$), and measures the flow through the disk of area A perpendicular to that unit vector. ''I'' is defined picking the unit vector that maximizes the flow around the point, because the true flow is maximized across the disk that is perpendicular to it. The unit vector thus uniquely maximizes the function when it points in the "true direction" of the flow. trictly speaking, this is an abuse of notation because the "arg max" cannot directly compare vectors; we take the vector with the biggest norm instead.]

### Properties

These direct definitions, especially the last, are rather unwieldy. For example, the argmax construction is artificial from the perspective of empirical measurements, when with a
Weathervane A wind vane, weather vane, or weathercock is an instrument used for showing the direction of the wind Wind is the natural movement of atmosphere of Earth, air or other gases relative to a planetary surface, planet's surface. Winds occur ...
or similar one can easily deduce the direction of flux at a point. Rather than defining the vector flux directly, it is often more intuitive to state some properties about it. Furthermore, from these properties the flux can uniquely be determined anyway. If the flux j passes through the area at an angle θ to the area normal $\mathbf$, then $\mathbf\cdot\mathbf= j\cos\theta$ where · is the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar (mathematics), scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidea ...
of the unit vectors. That is, the component of flux passing through the surface (i.e. normal to it) is ''j'' cos ''θ'', while the component of flux passing tangential to the area is ''j'' sin ''θ'', but there is ''no'' flux actually passing ''through'' the area in the tangential direction. The ''only'' component of flux passing normal to the area is the cosine component. For vector flux, the
surface integral In mathematics, particularly multivariable calculus, a surface integral is a generalization of multiple integrals to Integral, integration over surface (differential geometry), surfaces. It can be thought of as the double integral analogue of th ...
of j over a
surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of Visual perception, sight ...
''S'', gives the proper flowing per unit of time through the surface. $\frac = \iint_S \mathbf\cdot\mathbf\, dA = \iint_S \mathbf\cdot d\mathbf$ A (and its infinitesimal) is the
vector area In 3-dimensional geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in t ...
, combination of the magnitude of the area through which the property passes, ''A'', and a
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 ...
normal to the area, $\mathbf$. The relation is $\mathbf = A \mathbf$. Unlike in the second set of equations, the surface here need not be flat. Finally, we can integrate again over the time duration ''t''1 to ''t''2, getting the total amount of the property flowing through the surface in that time (''t''2 − ''t''1): $q = \int_^\iint_S \mathbf\cdot d\mathbf A\, dt$

## Transport fluxes

Eight of the most common forms of flux from the transport phenomena literature are defined as follows: # Momentum flux, the rate of transfer of
momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the Multiplication, product of the mass and velocity of an object. It is a Euclidean vector, vector quantity, possessing a magnitude and a dire ...
across a unit area (N·s·m−2·s−1). ( Newton's law of viscosity) #
Heat flux Heat flux or thermal flux, sometimes also referred to as ''heat flux density'', heat-flow density or ''heat flow rate intensity'' is a flow of energy per unit area per unit time (physics), time. In SI its units are watts per square metre (W/m2). ...
, the rate of
heat In thermodynamics, heat is defined as the form of energy crossing the boundary of a thermodynamic system by virtue of a temperature difference across the boundary. A thermodynamic system does not ''contain'' heat. Nevertheless, the term is al ...
flow across a unit area (J·m−2·s−1). ( Fourier's law of conduction) (This definition of heat flux fits Maxwell's original definition.) #
Diffusion flux Fick's laws of diffusion describe diffusion Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven ...
, the rate of movement of molecules across a unit area (mol·m−2·s−1). (
Fick's law of diffusion Fick's laws of diffusion describe diffusion Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven ...
) #
Volumetric flux In fluid dynamics In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other ga ...
, the rate of
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). ...
flow across a unit area (m3·m−2·s−1). ( Darcy's law of groundwater flow) #
Mass flux In 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 ...
, the rate of
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 ...
flow across a unit area (kg·m−2·s−1). (Either an alternate form of Fick's law that includes the molecular mass, or an alternate form of Darcy's law that includes the density.) #
Radiative flux Radiative flux, also known as radiative flux density or radiation flux (or sometimes power flux density), is the amount of power radiated through a given area, in the form of photon A photon () is an elementary particle that is a quantum of th ...
, the amount of energy transferred in the form of
photons A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are Massless particle, massless ...
at a certain distance from the source per unit area per second (J·m−2·s−1). Used in astronomy to determine the magnitude and
spectral class In astronomy, stellar classification is the classification of stars based on their stellar spectrum, spectral characteristics. Electromagnetic radiation from the star is analyzed by splitting it with a Prism (optics), prism or diffraction grati ...
of a star. Also acts as a generalization of heat flux, which is equal to the radiative flux when restricted to the electromagnetic spectrum. #
Energy flux Energy flux is the rate of transfer of energy In physics, energy (from Ancient Greek: wikt:ἐνέργεια#Ancient_Greek, ἐνέργεια, ''enérgeia'', “activity”) is the physical quantity, quantitative physical property, property ...
, the rate of transfer of
energy In physics, energy (from Ancient Greek: wikt:ἐνέργεια#Ancient_Greek, ἐνέργεια, ''enérgeia'', “activity”) is the physical quantity, quantitative physical property, property that is #Energy transfer, transferred to a phy ...
through a unit area (J·m−2·s−1). The radiative flux and heat flux are specific cases of energy flux. # Particle flux, the rate of transfer of particles through a unit area ( umber of particlesm−2·s−1) These fluxes are vectors at each point in space, and have a definite magnitude and direction. Also, one can take the
divergence In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the ...
of any of these fluxes to determine the accumulation rate of the quantity in a control volume around a given point in space. For
incompressible flow In fluid mechanics Fluid mechanics is the branch of physics concerned with the mechanics of fluids (liquids, gases, and plasma (physics), plasmas) and the forces on them. It has applications in a wide range of disciplines, including mechanic ...
, the divergence of the volume flux is zero.

### Chemical diffusion

As mentioned above, chemical molar flux of a component A in an
isothermal In thermodynamics, an isothermal process is a type of thermodynamic process in which the temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measurement, mea ...
, isobaric system is defined in
Fick's law of diffusion Fick's laws of diffusion describe diffusion Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven ...
as: $\mathbf_A = -D_ \nabla c_A$ where the
nabla symbol ∇ The nabla symbol The nabla is a triangular symbol resembling an inverted Greek delta:Indeed, it is called ( ανάδελτα) in Modern Greek Modern Greek (, , or , ''Kiní Neoellinikí Glóssa''), generally referred to by speakers ...
∇ denotes 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 ...
operator, ''DAB'' is the diffusion coefficient (m2·s−1) of component A diffusing through component B, ''cA'' is the
concentration In chemistry, concentration is the Abundance (chemistry), abundance of a constituent divided by the total volume of a mixture. Several types of mathematical description can be distinguished: ''mass concentration (chemistry), mass concentration'', ...
( mol/m3) of component A. This flux has units of mol·m−2·s−1, and fits Maxwell's original definition of flux. For dilute gases, kinetic molecular theory relates the diffusion coefficient ''D'' to the particle density ''n'' = ''N''/''V'', the molecular mass ''m'', the collision
cross section Cross section may refer to: * Cross section (geometry) **Multiview orthographic projection#Section, Cross-sectional views in architecture & engineering 3D *Cross section (geology) * Cross section (electronics) * Radar cross section, measure of det ...
$\sigma$, and the
absolute temperature Thermodynamic temperature is a quantity defined in thermodynamics as distinct from Kinetic theory of gases, kinetic theory or statistical mechanics. Historically, thermodynamic temperature was defined by William Thomson, 1st Baron Kelvin, Kelvin ...
''T'' by $D = \frac\sqrt$ where the second factor is the
mean free path In physics, mean free path is the average distance over which a moving particle (such as an atom, a molecule, or a photon) travels before substantially changing its direction or energy (or, in a specific context, other properties), typically as a ...
and the square root (with the
Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a ideal gas, gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas ...
''k'') is the mean velocity of the particles. In turbulent flows, the transport by eddy motion can be expressed as a grossly increased diffusion coefficient.

## Quantum mechanics

In
quantum mechanics Quantum mechanics is a fundamental Scientific theory, theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including qua ...
, particles of mass ''m'' in the
quantum state In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement in quantum mechanics, measurement on a system. Knowledge of the quantum state together with the rul ...
''ψ''(r, ''t'') have a probability density defined as $\rho = \psi^* \psi = , \psi, ^2.$ So the probability of finding a particle in a differential
volume element In mathematics, a volume element provides a means for Integral, integrating a function (mathematics), function with respect to volume in various coordinate systems such as Spherical coordinate system, spherical coordinates and Cylindrical coordinate ...
d3r is $dP = , \psi, ^2 \, d^3\mathbf.$ Then the number of particles passing perpendicularly through unit area of a cross-section per unit time is the probability flux; $\mathbf = \frac \left(\psi \nabla \psi^* - \psi^* \nabla \psi \right).$ This is sometimes referred to as the probability current or current density, or probability flux density.

# Flux as a surface integral

## General mathematical definition (surface integral)

As a mathematical concept, flux is represented by the surface integral of a vector field, :$\Phi_F=\iint_A\mathbf\cdot\mathrm\mathbf$ :$\Phi_F=\iint_A\mathbf\cdot\mathbf\,\mathrmA$ where F is a vector field, and d''A'' is the
vector area In 3-dimensional geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in t ...
of the surface ''A'', directed as the
surface normal In geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field ...
. For the second, n is the outward pointed unit normal vector to the surface. The surface has to be
orientable 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 ...
, i.e. two sides can be distinguished: the surface does not fold back onto itself. Also, the surface has to be actually oriented, i.e. we use a convention as to flowing which way is counted positive; flowing backward is then counted negative. The surface normal is usually directed by the
right-hand rule In mathematics and physics, the right-hand rule is a common mnemonic for understanding orientation (vector space), orientation of Cartesian coordinate system, axes in three-dimensional space. It is also a convenient method for quickly finding t ...
. Conversely, one can consider the flux the more fundamental quantity and call the vector field the flux density. Often a vector field is drawn by curves (field lines) following the "flow"; the magnitude of the vector field is then the line density, and the flux through a surface is the number of lines. Lines originate from areas of positive
divergence In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the ...
(sources) and end at areas of negative divergence (sinks). See also the image at right: the number of red arrows passing through a unit area is the flux density, the
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (ge ...
encircling the red arrows denotes the boundary of the surface, and the orientation of the arrows with respect to the surface denotes the sign of the
inner product In mathematics, an inner product space (or, rarely, a Hausdorff space, Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation (mathematics), operation called an inner product. The inner product of two ve ...
of the vector field with the surface normals. If the surface encloses a 3D region, usually the surface is oriented such that the influx is counted positive; the opposite is the outflux. The
divergence theorem In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a theorem which relates the ''flux'' of a vector field through a closed surface (mathematics), surface to the ''divergence'' o ...
states that the net outflux through a closed surface, in other words the net outflux from a 3D region, is found by adding the local net outflow from each point in the region (which is expressed by the
divergence In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the ...
). If the surface is not closed, it has an oriented curve as boundary.
Stokes' theorem Stokes's theorem, also known as the Kelvin–Stokes theoremNagayoshi Iwahori, et al.:"Bi-Bun-Seki-Bun-Gaku" :ja:裳華房, Sho-Ka-Bou(jp) 1983/12Written in Japanese)Atsuo Fujimoto;"Vector-Kai-Seki Gendai su-gaku rekucha zu. C(1)" :ja:培風館, ...
states that the flux of 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 a vector field is the
line integral In mathematics, a line integral is an integral where the function (mathematics), function to be integrated is evaluated along a curve. The terms ''path integral'', ''curve integral'', and ''curvilinear integral'' are also used; ''contour integra ...
of the vector field over this boundary. This path integral is also called circulation, especially in fluid dynamics. Thus the curl is the circulation density. We can apply the flux and these theorems to many disciplines in which we see currents, forces, etc., applied through areas.

## Electromagnetism

### Electric flux

An electric "charge," such as a single proton in space, has a magnitude defined in coulombs. Such a charge has an electric field surrounding it. In pictorial form, the electric field from a positive point charge can be visualized as a dot radiating
electric field lines Electricity is the set of physical phenomena A phenomenon ( : phenomena) is an observable event. The term came into its modern philosophical Philosophy (from , ) is the systematized study of general and fundamental questions, ...
(sometimes also called "lines of force"). Conceptually, electric flux can be thought of as "the number of field lines" passing through a given area. Mathematically, electric flux is the integral of the normal component of the electric field over a given area. Hence, units of electric flux are, in the MKS system,
newtons The newton (symbol: N) is the unit of force in the International System of Units, International System of Units (SI). It is defined as 1 kg⋅m/s, the force which gives a mass of 1 kilogram an acceleration of 1 metre per second per second. It i ...
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 ...
times meters squared, or N m2/C. (Electric flux density is the electric flux per unit area, and is a measure of strength of the normal component of the electric field averaged over the area of integration. Its units are N/C, the same as the electric field in MKS units.) Two forms of
electric flux In electromagnetism, electric flux is the measure of the electric field through a given surface, although an electric field in itself cannot flow. The electric field E can exert a force on an electric charge at any point in space. The electric fi ...
are used, one for the E-field: : and one for the D-field (called the
electric displacement In 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 ...
): : This quantity arises in
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 ...
– which states that the flux of the
electric field An electric field (sometimes E-field) is the field (physics), physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the ...
E out of a
closed surface In the part of mathematics referred to as topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous func ...
is proportional to the
electric charge 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 electron ...
''QA'' enclosed in the surface (independent of how that charge is distributed), the integral form is: : where ''ε''0 is the
permittivity of free space 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 ...
. If one considers the flux of the electric field vector, E, for a tube near a point charge in the field of the charge but not containing it with sides formed by lines tangent to the field, the flux for the sides is zero and there is an equal and opposite flux at both ends of the tube. This is a consequence of Gauss's Law applied to an inverse square field. The flux for any cross-sectional surface of the tube will be the same. The total flux for any surface surrounding a charge ''q'' is ''q''/''ε''0. In free space the
electric displacement In 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 given by the constitutive relation D = ''ε''0 E, so for any bounding surface the D-field flux equals the charge ''QA'' within it. Here the expression "flux of" indicates a mathematical operation and, as can be seen, the result is not necessarily a "flow", since nothing actually flows along electric field lines.

### Magnetic flux

The magnetic flux density (
magnetic field 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 t ...
) having the unit Wb/m2 ( Tesla) is denoted by B, and
magnetic flux In physics, specifically electromagnetism, the magnetic flux through a surface is the surface integral of the normal component of the magnetic field B over that surface. It is usually denoted or . The SI unit of measurement, unit of magnetic ...
is defined analogously: : with the same notation above. The quantity arises in
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 ...
, where the magnetic flux is time-dependent either because the boundary is time-dependent or magnetic field is time-dependent. In integral form: :$- \frac = \oint_ \mathbf \cdot d \boldsymbol$ where ''d'' is an infinitesimal vector
line element In geometry, the line element or length element can be informally thought of as a line segment associated with an infinitesimal displacement vector in a metric space. The length of the line element, which may be thought of as a differential arc l ...
of the
closed curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (ge ...
$\partial A$, with magnitude equal to the length of the
infinitesimal In mathematics, an infinitesimal number is a quantity that is closer to 0, zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century New Latin, Modern Latin coinage ''infinitesimus'', which ori ...
line element, and direction given by the tangent to the curve $\partial A$, with the sign determined by the integration direction. The time-rate of change of the magnetic flux through a loop of wire is minus the
electromotive force In electromagnetism and electronics, electromotive force (also electromotance, abbreviated emf, denoted \mathcal or ) is an energy transfer to an electric circuit per unit of electric charge, measured in volts. Devices called electrical ''Transd ...
created in that wire. The direction is such that if current is allowed to pass through the wire, the electromotive force will cause a current which "opposes" the change in magnetic field by itself producing a magnetic field opposite to the change. This is the basis for
inductor An inductor, also called a coil, choke, or reactor, is a passive two-terminal electrical component that stores energy in a magnetic field A magnetic field is a vector field that describes the magnetic influence on moving electric charges ...
s and many
electric generator In electricity generation, a generator is a device that converts motive power (mechanical energy) or fuel-based power (chemical energy) into electric power for use in an external electrical circuit, circuit. Sources of mechanical energy include s ...
s.

### Poynting flux

Using this definition, the flux of the
Poynting vector In 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 scienc ...
S over a specified surface is the rate at which electromagnetic energy flows through that surface, defined like before: : The flux of the
Poynting vector In 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 scienc ...
through a surface is the electromagnetic
power Power most often refers to: * Power (physics) In physics, power is the amount of energy transferred or converted per unit time. In the International System of Units, the unit of power is the watt, equal to one joule per second. In older work ...
, or
energy In physics, energy (from Ancient Greek: wikt:ἐνέργεια#Ancient_Greek, ἐνέργεια, ''enérgeia'', “activity”) is the physical quantity, quantitative physical property, property that is #Energy transfer, transferred to a phy ...
per unit
time Time is the continued sequence of existence and event (philosophy), events that occurs in an apparently irreversible process, irreversible succession from the past, through the present, into the future. It is a component quantity of various me ...
, passing through that surface. This is commonly used in analysis of
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 ...
, but has application to other electromagnetic systems as well. Confusingly, the Poynting vector is sometimes called the ''power flux'', which is an example of the first usage of flux, above. p.357 It has units of
watt The watt (symbol: W) is the unit of power or radiant flux in the International System of Units (SI), equal to 1 joule per second or 1 kg⋅m2⋅s−3. It is used to quantify the rate of energy transfer. The watt is named after James W ...
s per
square metre The square metre (American and British English spelling differences#-re, -er, international spelling as used by the International Bureau of Weights and Measures) or square meter (American and British English spelling differences#-re, -er, Americ ...
(W/m2).

* AB magnitude *
Explosively pumped flux compression generator An explosively pumped flux compression generator (EPFCG) is a device used to generate a high-power electromagnetic pulse An electromagnetic pulse (EMP), also a transient electromagnetic disturbance (TED), is a brief burst of electromagnetic e ...
*
Eddy covariance The eddy covariance (also known as eddy correlation and eddy flux) is a key atmospheric measurement technique to measure and calculate vertical turbulent fluxes within atmospheric boundary layers. The method analyses high-frequency wind Win ...
flux (aka, eddy correlation, eddy flux) *
Fast Flux Test Facility The Fast Flux Test Facility (FFTF) is a 400 MW thermal, liquid sodium Sodium is a chemical element with the Symbol (chemistry), symbol Na (from Latin ''natrium'') and atomic number 11. It is a soft, silvery-white, highly reactive me ...
*
Fluence In radiometry, radiant exposure or fluence is the radiant energy ''received'' by a ''surface'' per unit area, or equivalently the irradiance of a ''surface,'' integrated over time of irradiation, and spectral exposure is the radiant exposure per un ...
(flux of the first sort for particle beams) *
Fluid dynamics In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) and ...
*
Flux footprint Flux footprint (also known as atmospheric flux footprint or footprint) is an upwind area where the atmospheric flux measured by an instrument is generated. Specifically, the term flux footprint describes an upwind area "seen" by the instruments meas ...
* Flux pinning *
Flux quantization The magnetic flux, represented by the symbol , threading some contour or loop is defined as the magnetic field multiplied by the loop area , i.e. . Both and can be arbitrary, meaning can be as well. However, if one deals with the superconducti ...
*
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 ...
*
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 ...
*
Jansky The jansky (symbol Jy, plural ''janskys'') is a non-SI unit of spectral flux density, or spectral irradiance, used especially in radio astronomy. It is equivalent to 10−26 watts per square metre per hertz. The ''flux density'' or ''monochr ...
(non SI unit of spectral flux density) *
Latent heat flux Latent heat (also known as latent energy or heat of transformation) is energy released or absorbed, by a body or a thermodynamic system, during a constant-temperature process — usually a Phase transition#Modern classifications, first-order phase ...
*
Luminous flux In photometry, luminous flux or luminous power is the measure of the perceived power of light Light or visible light is electromagnetic radiation that can be visual perception, perceived by the human eye. Visible light is usually defi ...
*
Magnetic flux In physics, specifically electromagnetism, the magnetic flux through a surface is the surface integral of the normal component of the magnetic field B over that surface. It is usually denoted or . The SI unit of measurement, unit of magnetic ...
* Magnetic flux quantum *
Neutron flux The neutron flux, φ, is a scalar (physics), scalar quantity used in nuclear physics and nuclear reactor physics. It is the total length travelled by all free neutrons per unit time and volume. Equivalently, it can be defined as the number of ne ...
* Poynting flux * Poynting theorem *
Radiant flux In radiometry, radiant flux or radiant power is the radiant energy emitted, reflected, transmitted, or received per unit time, and spectral flux or spectral power is the radiant flux per unit frequency or wavelength, depending on whether the Spec ...
* Rapid single flux quantum * Sound energy flux *
Volumetric flux In fluid dynamics In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other ga ...
(flux of the first sort for fluids) *
Volumetric flow rate In physics and engineering, in particular fluid dynamics, the volumetric flow rate (also known as volume flow rate, or volume velocity) is the volume of fluid which passes per unit time; usually it is represented by the symbol (sometimes ). I ...
(flux of the second sort for fluids)

* *