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In symbols, this means:

:\Phi_m \equiv \int \!\!\! \int \mathbf{B} \cdot d\mathbf S\,

:where \Phi_m \ is the magnetic flux and B is the magnetic flux density.

We know from Gauss's law for magnetism that

:
abla \cdot \mathbf{B}=0.\,

This equation, in combination with the Divergence Theorem , provides the following result:

:\oint \!\!\! \oint_{\partial V} \mathbf{B} \cdot d\mathbf{S}=\int \!\!\! \int \!\!\! \int_V
abla \cdot \mathbf{B} \, d au = 0.

In other words, the magnetic flux through any closed surface must be zero; there are no free "magnetic charges".

By way of contrast, Gauss's Law For Electric Fields , another of Maxwell's Equations , is

:
abla \cdot \mathbf{E} = { ho \over \epsilon_0},

where E is the electric field intensity, ho is the free electric charge density, (not including dipole charges bound in a material), and \epsilon_0 is the Permittivity of free space.

Note that this indicates the presence of electric monopoles, that is, free positive or negative charges.

The direction of the magnetic-flux-density vector \mathbf{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.

A change of magnetic flux through a loop of conductive wire will cause an emf, and therefore an electric current, in the loop. The relationship is given by Faraday's Law :

: \mathcal{E} = \oint \mathbf{E} \cdot d\mathbf{s} = -{d\Phi_m \over dt}.

and is the principle behind an electric generator.


SEE ALSO