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In
physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of eve ...

physics
, specifically
electromagnetism Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electric charge, electrically charged particles. The electromagnetic force is carried by electromagnet ...

electromagnetism
, the magnetic flux through a surface is the
surface integral In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

surface integral
of the normal component of the
magnetic field A magnetic field is a vector field In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space. For instance, a vector field in the plane can be visualised as a collection of arrows with ...

magnetic field
B over that surface. It is usually denoted or . The
SI
SI
unit Unit may refer to: Arts and entertainment * UNIT Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in ...
of magnetic flux is the
weber Weber (, or ; German: ) is a surname of German language, German origin, derived from the noun meaning "weaving, weaver". In some cases, following migration to English-speaking countries, it has been anglicised to the English surname 'Webber' or ev ...
(Wb; in derived units, volt–seconds), and the CGS unit is the maxwell. Magnetic flux is usually measured with a fluxmeter, which contains measuring coils and
electronics The field of electronics is a branch of physics and electrical engineering that deals with the emission, behaviour and effects of electrons The electron is a subatomic particle In physical sciences, subatomic particles are smaller than ...
, that evaluates the change of
voltage Voltage, electric potential difference, electric pressure or electric tension is the difference in electric potential The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is the ...

voltage
in the measuring coils to calculate the measurement of magnetic flux.


Description

The magnetic interaction is described in terms of a
vector field In vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Product ...

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 In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Ph ...

Lorentz force
). Since a vector field is quite difficult to visualize at first, in elementary physics one may instead visualize this field with
field line A field line is a graphical for visualizing s. It consists of a directed line which is to the field at each point along its length. A diagram showing a representative set of neighboring field lines is a common way of depicting a vector field ...
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). 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 areaIn 3-dimensional geometry, for a finite planar surface of scalar area and unit normal , the vector area is defined as the unit normal scaled by the area: :\mathbf = \mathbfS For an orientable is non-orientable In mathematics, orientability ...

vector area
S is :\Phi_B = \mathbf \cdot \mathbf = BS \cos \theta, where ''B'' is the magnitude of the magnetic field (the magnetic flux density) having the unit of Wb/m2 ( tesla), ''S'' is the area of the surface, and ''θ'' is the angle between the magnetic
field line A field line is a graphical for visualizing s. It consists of a directed line which is to the field at each point along its length. A diagram showing a representative set of neighboring field lines is a common way of depicting a vector field ...
s and the
normal (perpendicular)
normal (perpendicular)
to S. For a varying magnetic field, we first consider the magnetic flux through an infinitesimal area element dS, where we may consider the field to be constant: :d\Phi_B = \mathbf \cdot d\mathbf. A generic surface, S, can then be broken into infinitesimal elements and the total magnetic flux through the surface is then the
surface integral In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

surface integral
:\Phi_B = \iint_S \mathbf \cdot d\mathbf S. From the definition of the
magnetic vector potential Magnetic vector potential, A, is the vector quantity in classical electromagnetism Classical electromagnetism or classical electrodynamics is a branch of theoretical physics that studies the interactions between electric charge Electric charg ...
A and the fundamental theorem of the curl the magnetic flux may also be defined as: :\Phi_B = \oint_ \mathbf \cdot d\boldsymbol, where the
line integral In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
is taken over the boundary of the surface ''S'', which is denoted ∂''S''.


Magnetic flux through a closed surface

Gauss's law for magnetism In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through ...
, which is one of the four
Maxwell's equations Maxwell's equations are a set of coupled partial differential equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), ...
, states that the total magnetic flux through a
closed surface with ''x''-, ''y''-, and ''z''-contours shown. In the part of mathematics referred to as topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Gree ...
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 In particle physics Particle physics (also known as high energy physics) is a branch of physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is th ...
s have never been found. In other words, Gauss's law for magnetism is the statement: : for any
closed surface with ''x''-, ''y''-, and ''z''-contours shown. In the part of mathematics referred to as topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Gree ...
''S''.


Magnetic flux through an open surface

While the magnetic flux through a
closed surface with ''x''-, ''y''-, and ''z''-contours shown. In the part of mathematics referred to as topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Gree ...
is always zero, the magnetic flux through an
open surface with ''x''-, ''y''-, and ''z''-contours shown. In the part of mathematics referred to as topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Gree ...
need not be zero and is an important quantity in electromagnetism. When determining the total magnetic flux through a surface only the boundary of the surface needs to be defined, the actual shape of the surface is irrelevant and the integral over any surface sharing the same boundary will be equal. This is a direct consequence of the closed surface flux being zero.


Changing magnetic flux

For example, a change in the magnetic flux passing through a loop of conductive wire will cause an
electromotive force In electromagnetism Electromagnetism is a branch of physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the ...
, and therefore an electric current, in the loop. The relationship is given by Faraday's law: :\mathcal = \oint_\left( \mathbf +\mathbf\right) \cdot d\boldsymbol = -, where *\mathcal is the electromotive force ( EMF), *Φ''B'' 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. *''dℓ'' is an
infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. They do not exist in the standard real number system, but do exist in many other number systems, such a ...
vector element of the contour ∂Σ, *v is the velocity of the boundary ∂Σ, *E is the
electric field An electric field (sometimes E-field) is the physical field that surrounds electrically-charged particle In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' ' ...

electric field
, *B is the
magnetic field A magnetic field is a vector field In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space. For instance, a vector field in the plane can be visualised as a collection of arrows with ...

magnetic field
. The two equations for the EMF are, firstly, the work per unit charge done against the
Lorentz force In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Ph ...

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 In electricity generation Electricity generation is the process of generating electric power from sources of primary energy. For electric utility, utilities in the electric power industry, it is the stage prior to its Electricity delivery, delive ...
.


Comparison with electric flux

By way of contrast,
Gauss's law In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. ...
for electric fields, another of
Maxwell's equations Maxwell's equations are a set of coupled partial differential equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), ...
, is : where *E is the
electric field An electric field (sometimes E-field) is the physical field that surrounds electrically-charged particle In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' ' ...

electric field
, *''S'' is any
closed surface with ''x''-, ''y''-, and ''z''-contours shown. In the part of mathematics referred to as topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Gree ...
, *''Q'' is the total
electric charge Electric charge is the physical property of matter that causes it to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons respectively). Like c ...
inside the surface ''S'', *''ε''0 is the
electric constant Vacuum permittivity, commonly denoted (pronounced as "epsilon nought" or "epsilon zero") is the value of the absolute dielectric permittivity of classical vacuum. Alternatively it may be referred to as the permittivity of free space, the elec ...
(a universal constant, also called the "
permittivity In electromagnetism Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electric charge, electrically charged particles. The electromagnetic force is car ...
of free space"). The flux of E through a closed surface is ''not'' always zero; this indicates the presence of "electric monopoles", that is, free positive or negative charges.


See also

*
Magnetic circuit A magnetic circuit is made up of one or more closed loop paths containing a magnetic flux In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and ...
is a closed path in which magnetic flux flows *
Magnetic flux quantum The magnetic flux In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion an ...
is the quantum of magnetic flux passing through a superconductor *
Flux linkage In circuit theory, flux linkage is a property of a two-terminal element. It is an extension rather than an equivalent of magnetic flux and is defined as a time integral :\lambda = \int \mathcal \,dt, where \mathcal is the voltage across the devi ...
, an extension of the concept of magnetic flux.


References


External articles

* *
Magnetic Flux through a Loop of Wire
by Ernest Lee,
Wolfram Demonstrations Project File:Legal cases tree (Wolfram Demonstrations Project).jpeg, 150px, Legal structures. The Wolfram Demonstrations Project is an organized, open-source collection of small (or medium-size) interactive programs called Demonstrations, which are mea ...
.
Conversion Magnetic flux Φ in nWb per meter track width to flux level in dB – Tape Operating Levels and Tape Alignment Levels
* wikt:magnetic flux {{Authority control Physical quantities Magnetism