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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 sight and touch, and is ...
or substance. Flux is a concept in
applied mathematics Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathemati ...
and
vector calculus Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for the broader subjec ...
which has many applications to
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 ...
. For
transport phenomena In engineering, physics, and chemistry, the study of transport phenomena concerns the exchange of mass, energy, charge, momentum and angular momentum between observed and studied systems. While it draws from fields as diverse as continuum mecha ...
, flux 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 ...
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 differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for the broader subjec ...
flux is a scalar quantity, defined as the surface integral 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 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 the power of the ...
: ''fluxus'' means "flow", and ''fluere'' is "to flow". As ''
fluxion A fluxion is the instantaneous rate of change, or gradient, of a fluent (a time-varying quantity, or function) at a given point. Fluxions were introduced by Isaac Newton to describe his form of a time derivative (a derivative with respect to ti ...
'', this term was introduced into differential calculus by
Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, theologian, and author (described in his time as a " natural philosopher"), widely recognised as one of the grea ...
. The concept of heat flux was a key contribution of Joseph Fourier, 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 li ...
, 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, or ...
, 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, momentum and angular momentum between observed and studied systems. While it draws from fields as diverse as continuum mecha ...
(
heat transfer Heat transfer is a discipline of thermal engineering that concerns the generation, use, conversion, and exchange of thermal energy (heat) between physical systems. Heat transfer is classified into various mechanisms, such as thermal conduction, ...
,
mass transfer Mass transfer is the net movement of mass from one location (usually meaning stream, phase, fraction or component) to another. Mass transfer occurs in many processes, such as absorption, evaporation, drying, precipitation, membrane filtration ...
and fluid dynamics), 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 mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordin ...
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 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 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. It is typically used as an architectural ornament to the highest point of a building. The word ''vane'' comes from the Old English word , m ...
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 as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alge ...
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 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 sight and touch, and is ...
''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 and vector calculus, an area vector is a vector combining an area quantity with a direction, thus representing an ''oriented area'' in three dimensions. Every bounded surface in three dimensions can be associated with ...
, 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 (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 ...
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 across a unit area (N·s·m−2·s−1). ( Newton's law of viscosity) # Heat flux, 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 ...
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 and were derived by Adolf Fick in 1855. They can be used to solve for the diffusion coefficient, . Fick's first law can be used to derive his second law which in turn is identical to the diffusion equ ...
, 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 and were derived by Adolf Fick in 1855. They can be used to solve for the diffusion coefficient, . Fick's first law can be used to derive his second law which in turn is identical to the diffusion e ...
) #
Volumetric flux In fluid dynamics, the volumetric flux is the rate of volume flow across a unit area (m3·s−1·m−2). Volumetric flux has dimensions of volume/(time*area). The density of a particular property in a fluid's volume, multiplied with the volumetri ...
, 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). Th ...
flow across a unit area (m3·m−2·s−1). ( Darcy's law of groundwater flow) #
Mass flux In physics and engineering, mass flux is the rate of mass flow. Its SI units are kg m−2 s−1. The common symbols are ''j'', ''J'', ''q'', ''Q'', ''φ'', or Φ (Greek lower or capital Phi), sometimes with subscript ''m'' to indicate mass is th ...
, the rate of
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 ...
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, the amount of energy transferred in the form of photons at a certain distance from the source per unit area per second (J·m−2·s−1). Used in astronomy to determine the
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
spectral class In astronomy, stellar classification is the classification of stars based on their spectral characteristics. Electromagnetic radiation from the star is analyzed by splitting it with a prism or diffraction grating into a spectrum exhibiting the ...
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, the rate of transfer of
energy In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of hea ...
through a unit area (J·m−2·s−1). The radiative flux and heat flux are specific cases of energy flux. #
Particle flux In the physical sciences, a particle (or corpuscule in older texts) is a small localized object which can be described by several physical or chemical properties, such as volume, density, or mass. They vary greatly in size or quantity, from su ...
, 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 t ...
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 or more generally continuum mechanics, incompressible flow ( isochoric flow) refers to a flow in which the material density is constant within a fluid parcel—an infinitesimal volume that moves with the flow velocity. A ...
, 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 ''T'' of a system remains constant: Δ''T'' = 0. This typically occurs when a system is in contact with an outside thermal reservoir, and ...
, isobaric system is defined in
Fick's law of diffusion Fick's laws of diffusion describe diffusion and were derived by Adolf Fick in 1855. They can be used to solve for the diffusion coefficient, . Fick's first law can be used to derive his second law which in turn is identical to the diffusion e ...
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. \nabla or ∇. The name comes, by reason of the symbol's shape, from the Hellenistic Greek word ...
∇ denotes 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 ...
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 of a constituent divided by the total volume of a mixture. Several types of mathematical description can be distinguished: '' mass concentration'', ''molar concentration'', '' number 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) ** Cross-sectional views in architecture & engineering 3D *Cross section (geology) * Cross section (electronics) * Radar cross section, measure of detectability * Cross section (physics) **Abs ...
\sigma, and the
absolute temperature Thermodynamic temperature is a quantity defined in thermodynamics as distinct from kinetic theory or statistical mechanics. Historically, thermodynamic temperature was defined by Kelvin in terms of a macroscopic relation between thermodynamic w ...
''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 gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constant, ...
''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 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 quantum chemistr ...
, 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 on a system. Knowledge of the quantum state together with the rules for the system's evolution i ...
''ψ''(r, ''t'') have a
probability density In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can ...
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 integrating a function with respect to volume in various coordinate systems such as spherical coordinates and cylindrical coordinates. Thus a volume element is an expression of the form :dV ...
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 and vector calculus, an area vector is a vector combining an area quantity with a direction, thus representing an ''oriented area'' in three dimensions. Every bounded surface in three dimensions can be associated with ...
of the surface ''A'', directed as the surface normal. For the second, n is the outward pointed unit normal vector to the surface. The surface has to be
orientable In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space i ...
, 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. 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 t ...
(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, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
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 pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
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 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 t ...
). If the surface is not closed, it has an oriented curve as boundary. Stokes' theorem states that the flux of the curl of a vector field is the
line integral In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms ''path integral'', ''curve integral'', and ''curvilinear integral'' are also used; ''contour integral'' is used as well, al ...
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 (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 Normal(s) or The Normal(s) may refer to: Film and television * ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson * ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie * ''Norma ...
component of the electric field over a given area. Hence, units of electric flux are, in the MKS system, newtons per coulomb 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 Normal(s) or The Normal(s) may refer to: Film and television * ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson * ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie * ''Norma ...
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): : 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 sta ...
– which states that the flux of the electric field E out of a
closed surface In the part of mathematics referred to as topology, a surface is a two-dimensional manifold. Some surfaces arise as the boundaries of three-dimensional solids; for example, the sphere is the boundary of the solid ball. Other surfaces arise as g ...
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 electrons respe ...
''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. 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 is given by the
constitutive relation In physics and engineering, a constitutive equation or constitutive relation is a relation between two physical quantities (especially kinetic quantities as related to kinematic quantities) that is specific to a material or substance, and app ...
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) having the unit Wb/m2 ( Tesla) is denoted by B, and magnetic flux is defined analogously: : with the same notation above. The quantity arises in Faraday's law of induction, 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 ...
of the
closed curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
\partial A, with
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 ...
equal to the length of the infinitesimal 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 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 when electric current flows through it. An inductor typically consists of an insulated wire wound into a c ...
s and many electric generators.


Poynting flux

Using this definition, the flux of the Poynting vector 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 through a surface is the electromagnetic
power Power most often refers to: * Power (physics), meaning "rate of doing work" ** Engine power, the power put out by an engine ** Electric power * Power (social and political), the ability to influence people or events ** Abusive power Power may a ...
, or
energy In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of hea ...
per unit
time Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. It is a component quantity of various measurements used to sequence events, ...
, passing through that surface. This is commonly used in analysis of
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 ...
, 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 ...
s per square metre (W/m2).


SI radiometry units


See also

*
AB magnitude The AB magnitude system is an astronomical magnitude system. Unlike many other magnitude systems, it is based on flux measurements that are calibrated in absolute units, namely spectral flux densities. Definition The ''monochromatic'' AB magni ...
*
Explosively pumped flux compression generator An explosively pumped flux compression generator (EPFCG) is a device used to generate a high-power electromagnetic pulse by compressing magnetic flux using high explosive. An EPFCG only ever generates a single pulse as the device is physically d ...
*
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 and scal ...
flux (aka, eddy correlation, eddy flux) *
Fast Flux Test Facility The Fast Flux Test Facility (FFTF) is a 400 MW thermal, liquid sodium cooled, nuclear test reactor owned by the U.S. Department of Energy. It does not generate electricity. It is situated in the ''400 Area'' of the Hanford Site, which is located ...
* Fluence (flux of the first sort for particle beams) * Fluid dynamics *
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 sta ...
* Inverse-square law *
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 ''monoch ...
(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 first-order phase transition. Latent heat can be underst ...
* Luminous flux * Magnetic flux *
Magnetic flux quantum 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 ...
*
Neutron flux The neutron flux, φ, is a 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 neutrons travellin ...
* Poynting flux *
Poynting theorem In electrodynamics, Poynting's theorem is a statement of conservation of energy for electromagnetic fields developed by British physicist John Henry Poynting. It states that in a given volume, the stored energy changes at a rate given by the wor ...
* Radiant flux * Rapid single flux quantum *
Sound energy flux Sound power or acoustic power is the rate at which sound energy is emitted, reflected, transmitted or received, per unit time. It is defined as "through a surface, the product of the sound pressure, and the component of the particle velocity, at ...
*
Volumetric flux In fluid dynamics, the volumetric flux is the rate of volume flow across a unit area (m3·s−1·m−2). Volumetric flux has dimensions of volume/(time*area). The density of a particular property in a fluid's volume, multiplied with the volumetri ...
(flux of the first sort for fluids) * Volumetric flow rate (flux of the second sort for fluids)


Notes

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Further reading

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External links

* {{Wiktionary-inline Physical quantities Vector calculus Rates