 Magnetic flux

This article is about magnetic flux. For the magnetic field "B" (magnetic flux per area), see magnetic flux density. For the magnetic field "H", see Hfield.
Electromagnetism Electricity · Magnetism Ampère's law · Electric current · Magnetic field · Magnetization · Magnetic flux · Biot–Savart law · Magnetic dipole moment · Gauss's law for magnetismLorentz force law · emf · Electromagnetic induction · Faraday’s law · Lenz's law · Displacement current · Maxwell's equations · EM field · Electromagnetic radiation · Liénard–Wiechert potential · Maxwell tensor · Eddy currentMagnetic flux (most often denoted as Φ_{m}), is a measure of the amount of magnetic B field (also called "magnetic flux density") passing through a given surface (such as a conducting coil). The SI unit of magnetic flux is the weber (Wb) (in derived units: voltseconds). The CGS unit is the maxwell.
Contents
Description
The magnetic flux through a given surface is proportional to the number of magnetic B field lines that pass through the surface. This is the net number, i.e. the number passing through in one direction, minus the number passing through in the other direction. (See below for how the positive sign is chosen.) For a uniform magnetic field B passing through a perpendicular area the magnetic flux is given by the product of the magnetic field and the area element. The magnetic flux for a uniform B at any angle to a surface is defined by a dot product of the magnetic field and the area element vector a.
 (uniform B with flat area only)
where θ is the angle between B and a vector a that is perpendicular (normal) to the surface.
In the general case, the magnetic flux through a surface S is defined as the integral of the magnetic field over the area of the surface (See Figures 1 and 2):
where is the magnetic flux, B is the magnetic field,
 S is the surface (area), denotes dot product, and dS is an infinitesimal vector, whose magnitude is the area of a differential element of S, and whose direction is the surface normal. (See surface integral for more details.)
From the definition of the magnetic vector potential A and the fundamental theorem of the curl the magnetic flux may also be defined as:
where the closed line integral is over the boundary of the surface and dℓ is an infinitesimal vector element of that contour Σ.
The magnetic flux is usually measured with a fluxmeter. The fluxmeter contains measuring coils and electronics that evaluates the change of voltage in the measuring coils to calculate the magnetic flux.
Magnetic flux through a closed surface
Main article: Gauss's law for magnetismGauss'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 monopoles have never been found.
In other words, Gauss's law for magnetism is the statement:
for any closed surface S.
Magnetic flux through an open surface
Main article: Faraday's law of inductionWhile 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. 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:
where (see Figure 3):
 is the EMF,
 Φ_{m} is the flux through a surface with an opening bounded by a curve ∂Σ(t),
 ∂Σ(t) is a closed contour that can change with time; the EMF is found around this contour, and the contour is a boundary of the surface over which Φ_{m} is found,
 dℓ is an infinitesimal vector element of the contour ∂Σ(t),
 v is the velocity of the segment dℓ,
 E is the electric field,
 B is the magnetic field.
The EMF is determined in this equation in two ways: first, as the work per unit charge done against the Lorentz force in moving a test charge around the (possibly moving) closed curve ∂Σ(t), and second, as the magnetic flux through the open surface Σ(t).
This equation is the principle behind an electrical generator.
Comparison with electric flux
Main articles: Electric flux and Gauss's lawBy way of contrast, Gauss's law for electric fields, another of Maxwell's equations, is
where
 E is the electric field,
 S is any closed surface,
 Q is the total electric charge inside the surface S,
 ε_{0} is the electric constant (a universal constant, also called the "permittivity of free space").
Note that 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.
Magnetic Circuits
Conventional Magnetic Circuits Magnetomotive force
 Magnetic flux Φ
 Magnetic reluctance
Phasor Magnetic Circuits Complex reluctance Z_{μ}
Related Concepts
Gyratorcapacitor model variables Magnetic impedance z_{M}
 Effective resistance r_{M}
 Magnetic inductivity L_{M}
 Magnetic capacitivity C_{M}
See also
 Magnetic field
 Maxwell's equations (sometimes called the Maxwell equations) are the set of four equations, attributed to James Clerk Maxwell, that describe the behavior of both the electric and magnetic fields, as well as their interactions with matter.
 Gauss's law gives the relation between the electric flux flowing out a closed surface and the electric charge enclosed in the surface.
 Magnetic circuit is a method using an analogy with electric circuits to calculate the flux of complex systems of magnetic components.
 Magnetic monopole is a hypothetical particle that may be loosely described as "a magnet with only one pole".
 Magnetic flux quantum is the quantum of magnetic flux passing through a superconductor.
 Carl Friedrich Gauss developed a fruitful collaboration with the physics professor Wilhelm Weber; it led to new knowledge in the field of magnetism.
 James Clerk Maxwell demonstrated that electric and magnetic forces are two complementary aspects of electromagnetism.
External articles
 Patents
 Vicci, U.S. Patent 6,720,855, Magneticflux conduits
 Magnetic Flux through a Loop of Wire by Ernest Lee, Wolfram Demonstrations Project.
 Conversion Magnetic flux Φ in nWb per meter track width to flux level in dB  Tape Operating Levels and Tape Alignment Levels
Categories: Physical quantities
 Magnetism
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