The magnetic interaction is described in terms of a vector field, where each point in space is associated with a vector that determines what force a moving charge would experience at that point. Since a vector field is quite difficult to visualize at first, in elementary physics one may instead visualize this field with field lines. The magnetic flux through some surface, in this simplified picture, is proportional to the number of field lines passing through that surface. 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. 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 areaS is where B is the magnitude of the magnetic field having the unit of Wb/m2, S is the area of the surface, and θ is the angle between the magnetic field lines and the normal 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: A generic surface, S, can then be broken into infinitesimal elements and the total magnetic flux through the surface is then the surface integral From the definition of the magnetic vector potentialA and the fundamental theorem of the curl the magnetic flux may also be defined as: where the line integral is taken over the boundary of the surface S, which is denoted ∂S.
While 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. 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, and therefore an electric current, in the loop. The relationship is given by Faraday's law: where The two equations for the EMF are, firstly, the work per unit charge done against the Lorentz force in moving a test charge around the 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.
By way of contrast, Gauss's law for electric fields, another of Maxwell's equations, is where 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.
