Magnetic flux

Magnetic flux, represented by the Greek letter Φ (phi), is a measure of quantity of magnetism, taking account of the strength and the extent of a magnetic field. The SI unit of magnetic flux is the weber (in derived units: volt-seconds), and the unit of magnetic field intensity is the weber per square meter, or tesla.

Description

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. 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. This law is a consequence of the empirical observation that magnetic monopoles do not exist or are not measurable.

The magnetic flux is defined as the integral of the magnetic field over an area:

$\Phi _{m}=\int \!\!\!\int \mathbf {B} \cdot d\mathbf {S} \,$

where

$\Phi _{m}\$ is the magnetic flux

B is the magnetic field density

S is the area.

We know from Gauss's law for magnetism that

$\nabla \cdot \mathbf {B} =0.\,$

The volume integral of this equation, in combination with the divergence theorem, provides the following result:

$\int \!\!\!\int \!\!\!\int _{V}\nabla \cdot \mathbf {B} \,d\tau =\oint \!\!\!\oint _{\partial V}\mathbf {B} \cdot d\mathbf {S} =0.$

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

$\nabla \cdot \mathbf {E} ={\rho \over \epsilon _{0}},$

where

E is the electric field intensity,

$\rho$ is the free electric charge density, (not including dipole charges bound in a material),

$\epsilon _{0}$ 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:

${\mathcal {E}}=\oint \mathbf {E} \cdot d\mathbf {s} =-{d\Phi _{m} \over dt}.$

This is the principle behind an electrical generator.

See also

• Magnetic field
• Maxwell's equations (sometimes called the Maxwell equations) are the set of four equations, attributed to James Clerk Maxwell, that describe the behavior of both the electric and magnetic fields, as well as their interactions with matter.
• Gauss's law gives the relation between the electric flux flowing out a closed surface and the electric charge enclosed in the surface.
• Magnetic monopole is a hypothetical particle that may be loosely described as "a magnet with only one pole".
• Magnetic flux quantum is the quantum of magnetic flux passing through a superconductor.
• Karl Friedrich Gauss developed a fruitful collaboration with the physics professor Wilhelm Weber; it led to new knowledge in the field of magnetism.
• James Clerk Maxwell demonstrated that electric and magnetic forces are two complementary aspects of electromagnetism.

External articles

Patents

• Vicci, U.S. patent 6,720,855, Magnetic-flux conduits