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{{About|magnetic flux|the magnetic fields "B" (magnetic flux density) and "H"|Magnetic field|}}
{{no footnotes|date=July 2016}}
{{Electromagnetism|cTopic=Magnetostatics}}

In [[physics]], specifically [[electromagnetism]], the '''magnetic flux''' (often denoted {{math|Φ}} or {{math|Φ<sub>'''''B'''''</sub>}}) through a surface is the [[surface integral]] of the normal component of the [[magnetic field]] '''B''' passing through that surface. The [[SI]] [[unit of measurement|unit]] of magnetic flux is the [[Weber (unit)|weber]] (Wb) (in derived units: volt <math>\cdot</math> seconds), and the [[Centimetre gram second system of units|CGS]] unit is the [[Maxwell (unit)|maxwell]]. Magnetic flux is usually measured with a fluxmeter, which contains measuring coils and [[electronics]], that evaluates the change of [[voltage]] in the measuring coils to calculate the  measurement of magnetic flux.

==Description==
 {{multiple image
   | align = right
   | direction = vertical
   | image1    = Surface integral illustration.svg
   | caption1  = The magnetic flux through a surface—when the magnetic field is variable—relies on splitting the surface into small surface elements, over which the magnetic field can be considered to be locally constant. The total flux is then a formal summation of these surface elements (see [[surface integral|surface integration]]).
   | width1     = 250
   | image2    = Surface normal.png
   | caption2  = Each point on a surface is associated with a direction, called the [[surface normal]]; the magnetic flux through a point is then the component of the magnetic field along this direction.
   | width2     = 250
  }}
The magnetic interaction is described in terms of a [[vector field]], where each point in space is associated with a vector that determines what force a moving charge would experience at that point (see [[Lorentz force]]). <ref>Purcell, p278,</ref> Since a vector field is quite difficult to visualize at first, in elementary physics one may instead visualize this field with [[field line]]s. The magnetic flux through some surface, in this simplified picture, is proportional to the number of field lines passing through that surface (in some contexts, the flux may be defined to be precisely the number of field lines passing through that surface; although technically misleading, this distinction is not important). The magnetic flux is the ''net'' number of field lines passing through that surface; that is, the number passing through in one direction minus the number passing through in the other direction (see below for deciding in which direction the field lines carry a positive sign and in which they carry a negative sign).<ref>Browne,p235</ref>
In more advanced physics, the field line analogy is dropped and the magnetic flux is properly defined as the surface integral of the normal component of the magnetic field passing through a surface. If the magnetic field is constant, the magnetic flux passing through a surface of [[vector area]] '''S''' is
:<math>
\Phi_B = \mathbf{B} \cdot \mathbf{S} = BS \cos \theta,
</math>
where ''B'' is the magnitude of the magnetic field (the magnetic flux density) having the unit of Wb/m<sup>2</sup> ([[Tesla (unit)|tesla]]), ''S'' is the area of the surface, and ''θ'' is the angle between the magnetic [[field line]]s and the [[surface normal|normal (perpendicular)]] to '''S'''. For a varying magnetic field, we first consider the magnetic flux through an infinitesimal area element d'''S''', where we may consider the field to be constant: 
:<math>
d\Phi_B = \mathbf{B} \cdot d\mathbf{S}.
</math>
A generic surface, '''S''', can then be broken into infinitesimal elements and the total magnetic flux through the surface is then the [[surface integral]]
:<math>
\Phi_B = \iint\limits_S \mathbf{B} \cdot d\mathbf S.
</math>
From the definition of the [[magnetic vector potential]] '''A''' and the [[Kelvin-Stokes theorem|fundamental theorem of the curl]] the magnetic flux may also be defined as:
:<math>\Phi_B = \oint\limits_{\partial S} \mathbf{A} \cdot d\boldsymbol{\ell},</math>
where the [[line integral]] is taken over the boundary of the surface ''S'', which is denoted ∂''S''.
{{-}}

==Magnetic flux through a closed surface==

[[File:SurfacesWithAndWithoutBoundary.svg|right|thumb|200px|Some examples of [[closed surface]]s (left) and [[open surface]]s (right). Left: Surface of a sphere, surface of a [[torus]], surface of a cube. Right: [[Disk (mathematics)|Disk surface]], square surface, surface of a hemisphere. (The surface is blue, the boundary is red.)]]

{{Main|Gauss's law for magnetism}}

[[Gauss's law for magnetism]], which is one of the four [[Maxwell's equations]], states that the total magnetic flux through a [[closed surface]] is equal to zero. (A "closed surface" is a surface that completely encloses a volume(s) with no holes.) This law is a consequence of the empirical observation that [[magnetic monopole]]s have never been found.

In other words, Gauss's law for magnetism is the statement:

:{{oiint
| preintegral = <math>\Phi_B=\,\!</math>
| intsubscpt = <math>\scriptstyle S</math>
| integrand = <math>\mathbf{B} \cdot d\mathbf S = 0</math>}}

for any [[closed surface]] ''S''.
==Magnetic flux through an open surface==
[[Image:Vector field on a surface.svg|right|thumb|250px|For an open surface Σ, the [[electromotive force]] along the surface boundary, ∂Σ, is a combination of the boundary's motion, with velocity '''v''', through a magnetic field '''B''' (illustrated by the generic '''F''' field in the diagram) and the induced electric field caused by the changing magnetic field.]]

While the magnetic flux through a [[closed surface]] is always zero, the magnetic flux through an [[open surface]] need not be zero and is an important quantity in electromagnetism.{{cn|reason=yeah, but we need a reference|date=December 2016}}

==Changing magnetic flux==
{{Main|Faraday's law of induction}}
For example, a change in the magnetic flux passing through a loop of conductive wire will cause an [[electromotive force]], and therefore an electric current, in the loop.  The relationship is given by [[Faraday's law of induction|Faraday's law]]:

:<math>\mathcal{E} = \oint_{\partial \Sigma}\left(  \mathbf{E} +\mathbf{ v \times B}\right) \cdot d\boldsymbol{\ell} = -{d\Phi_B \over dt},</math>

where
:<math>\mathcal{E}</math> is the electromotive force ([[electromotive force|EMF]]),
:Φ<sub>''B''</sub> is the magnetic flux through the open surface Σ,
:∂Σ is the boundary of the open surface Σ; the surface, in general, may be in motion and deforming, and so is generally a function of time. The electromotive force is induced along this boundary.
:d'''ℓ''' is an [[infinitesimal]] vector element of the contour ∂Σ,
:'''v''' is the velocity of the boundary ∂Σ,
:'''E''' is the [[electric field]],
:'''B''' is the [[magnetic field]].

The two equations for the EMF are, firstly, the work per unit charge done against the [[Lorentz force]] in moving a test charge around the (possibly moving) surface boundary ∂Σ and, secondly, as the change of magnetic flux through the open surface Σ. This equation is the principle behind an [[electrical generator]].
{{-}}
[[File:Spulenflaeche.jpg|thumb|center|300px|Area defined by an electric coil with three turns.]]
{{-}}

==Comparison with electric flux==

{{Main|Electric flux|Gauss's law}}

By way of contrast, [[Gauss's law|Gauss's law for electric fields]], another of [[Maxwell's equations]], is

:{{oiint
| preintegral = <math>\Phi_E =\,\!</math>
| intsubscpt = <math>\scriptstyle S</math>
| integrand = <math>\mathbf{E}\cdot d\mathbf{S} = \frac{Q}{\epsilon_0}\,\!</math>}}

where
:'''E''' is the [[electric field]],
:''S'' is any [[closed surface]],
:''Q'' is the total [[electric charge]] inside the surface ''S'',
:''ε''<sub>0</sub> is the [[electric constant]] (a universal constant, also called the "[[permittivity]] of free space").

The [[electric flux|flux of '''E''']] through a closed surface is ''not'' always zero; this indicates the presence of "electric monopoles", that is, free positive or negative [[electric charge|charges]].

{{MagneticCircuitSegments}}

==See also==
{{Portal|Electronics}}
*[[Magnetic circuit]] is a closed path in which magnetic flux flows
*[[Magnetic flux quantum]] is the quantum of magnetic flux passing through a superconductor
*[[Flux linkage]], an extension of the concept of magnetic flux.

==References==
* Purcell, Edward and Morin, David; ''Electricity and Magnetism 3rd Edition''; Cambridge University Press, New York, 2013. {{ISBN|978-1-107-01402-2}}.
* Browne, Michael; ''Physics for Engineering and Science 2nd Edition'', McGraw-Hill/Schaum; 2008; {{ISBN|978-0-07-161399-6}}.

{{Reflist}}

==External articles==
*Vicci, {{US patent|6720855}}, ''Magnetic-flux conduits''
*[http://demonstrations.wolfram.com/MagneticFluxThroughALoopOfWire/ Magnetic Flux through a Loop of Wire] by Ernest Lee, [[Wolfram Demonstrations Project]].
*[http://www.sengpielaudio.com/calculator-magneticflux.htm Conversion Magnetic flux Φ in nWb per meter track width to flux level in dB - Tape Operating Levels and Tape Alignment Levels]
* [[Wiktionary:magnetic flux]]

[[Category:Physical quantities]]
[[Category:Magnetism]]