Magnetic flux, represented by the Greek letter Φ (phi), is a measure of quantity of magnetism, taking into account the strength and the extent of a magnetic field. The SI unit of magnetic flux is the weber (in derived units: voltseconds), and the unit of magnetic field is the weber per square meter, or tesla.
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The flux through an element of area perpendicular to the direction of magnetic field is given by the product of the magnetic field and the area element.
More generally, the magnetic flux at any angle to a surface is defined by a scalar product of the magnetic field and the area element vector.
The direction of the magnetic field vector B is by definition from the south to the north pole of a magnet (within the magnet). Outside of the magnet, the field lines will go from north to south.
The magnetic flux through a surface is proportional to the number of magnetic 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.
Quantitatively, 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
In the special case where the surface S is a planar surface with area A, and the magnetic field is constant with magnitude B, the formula simplifies to
where θ is the angle between B and the
surface normal to S.
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.
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 without boundaries, such as the surface of a sphere or a cube, but not like the surface of a disk.) 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.
While the magnetic flux through a closed surface is always zero, the magnetic flux through an open surface 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):
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 thorough the open surface Σ(t).
This equation is the principle behind an electrical generator.
By way of contrast, Gauss's law for electric fields, another of Maxwell's equations, is
where
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

Phasor Magnetic Circuits

Related Concepts 
GyratorCapacitor Model variables

