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# Magnetic flux

Magnetic flux, is a measure of quantity of magnetism, taking account of the strength and the extent of a magnetic field. The flux through an element of area perpendicular to the direction of magnetic field is given by the product of the magnetic field density and the area element. More generally, magnetic flux is defined by a scalar product of the magnetic field density and the area element vector. The Maxwell's equations in the absence of magnetic monopoles requires that the magnetic flux through a closed surface be zero.

In symbols, this means:

$\Phi_m \equiv \int \!\!\! \int \vec{B} \cdot \hat{n} \, da$

We know from one of Maxwell's equations

$\vec{\nabla} \cdot \vec{B}=0$

in combination with the divergence theorem that

$\oint \!\!\! \oint_{\partial V} \vec{B} \cdot \hat{n} \, da=\int \!\!\! \int \!\!\! \int_V \vec{\nabla} \cdot \vec{B} \, d\tau = 0$.

In other words, the magnetic flux through any closed surface must be zero. By way of contrast, one of Maxwell's equations for electric fields is

$\vec{\nabla} \cdot \vec{E} = {\rho \over \epsilon_0}$

indicating the presence of electric monopoles. By the same token, there are no magnetic monopoles.

The SI unit of magnetic flux is the weber.

The direction or vector of the magnetic flux 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.

A change of magnetic flux in a spool of electrical conductive wire will cause an electric current in the spool. This is the basis of the production of electricity. When turned around, that is, running a current through a spool, a magnetic flux will be produced in the spool. This is electromagnetism. The relationship is given by Faraday's Law:

$\mathcal{E} = \oint \vec{E} \cdot d\vec{l} = -{d\Phi_m \over dt}$

Related concepts: gauss, henry, tesla, maxwell, oersted, weber (Wb), volt, and B-Field.