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, magnetic flux is defined by a scalar product of the magnetic field and the area element vector.
The direction of the magnetic field vector 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):
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 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 do not exist or are not measurable.
In other words, Gauss's law for magnetism is the statement:
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.
Note that the flux of E is not always zero; this indicates the presence of electric "monopoles", that is, free positive or negative charges.
US Patent Issued to Kabushiki Kaisha Toshiba on Jan. 1 for "Magnetic Flux Measuring Device and Magnetic Flux Measuring Method for Stationary Induction Electrical Apparatus, and Synchronous Switching Control Gear for Circuit Breaker" (Japanese Inventor)
Jan 05, 2013; ALEXANDRIA, Va., Jan. 5 -- United States Patent no. 8,344,557, issued on Jan. 1, was assigned to Kabushiki Kaisha Toshiba...
WIPO PUBLISHES PATENT OF TOSHIBA FOR "METHOD OF ESTIMATING THE RESIDUAL MAGNETIC FLUX OF TRANSFORMER AND RESIDUAL MAGNETIC FLUX ESTIMATION DEVICE" (JAPANESE INVENTORS)
Feb 08, 2012; GENEVA, Feb. 8 -- Publication No. WO/2012/014425 was published on Feb. 02. Title of the invention: "METHOD OF ESTIMATING THE...