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# magnetic flux

flux, magnetic, in physics, term used to describe the total amount of magnetic field in a given region. The term flux was chosen because the power of a magnet seems to "flow" out of the magnet at one pole and return at the other pole in a circulating pattern, as suggested by the patterns formed by iron filings sprinkled on a paper placed over a magnet or a conductor carrying an electric current. These patterns are called lines of induction. Although there is no actual physical flow, the lines of induction suggest the correct mathematical description of magnetism in terms of a field of force. The lines of induction originate on the north pole of the magnet and end on the south pole; their direction at any point is the direction of the magnetic field, and their density (the number of lines passing through a unit area) gives the strength of the field. Near the poles where the lines converge, the field and the force it produces are large; away from the poles where the lines diverge, the field and force are progressively weaker.
Magnetic flux, represented by the Greek letter Φ (phi), is a measure of quantity of magnetism, taking into account 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 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 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 $mathbf\left\{B\right\}$ 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):

$Phi_m = int !!! int_S mathbf\left\{B\right\} cdot dmathbf S,$

where

$Phi_m$ is the magnetic flux
B is the magnetic field,
S is the surface (area),
$cdot$ denotes dot product,
dS is an infinitesimal vector, whose magnitude is the area of a differential element of S, and whose direction is the surface normal. (See surface integral for more details.)

## Magnetic flux through a closed surface

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:

$Phi_m=int !!! int mathbf\left\{B\right\} cdot dmathbf S = 0,$
for any closed surface S.

## Magnetic flux through an open surface

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:

$mathcal\left\{E\right\} = oint_\left\{partial Sigma \left(t\right)\right\}left\left( mathbf\left\{E\right\}\left(mathbf\left\{r\right\}, t\right) +mathbf\left\{ v times B\right\}\left(mathbf\left\{r\right\}, t\right)right\right) cdot dboldsymbol\left\{ell\right\} = -\left\{dPhi_m over dt\right\}.$

where (see Figure 3):

E is the EMF,
Φm is the flux through a surface with an opening bounded by a curve ∂Σ(t),
∂Σ(t) is a closed contour that can change with time; the EMF is found around this contour, and the contour is a boundary of the surface over which Φm is found,
d is an infinitesimal vector element of the contour ∂Σ(t),
v is the velocity of the segment d,
E is the electric field,
B is the magnetic field.

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.

## Comparison with electric flux

By way of contrast, Gauss's law for electric fields, another of Maxwell's equations, is

$Phi_E = int !!!int_S mathbf\left\{E\right\}cdot dmathbf\left\{S\right\} = \left\{Q over epsilon_0\right\},$

where

E is the electric field,
S is any closed surface,
Q is the total electric charge inside the surface S,
$epsilon_0$ is the electric constant (a universal constant, also called the "permittivity of free space").

Note that the flux of E is not always zero; this indicates the presence of electric "monopoles", that is, free positive or negative charges.

• 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.
• Carl 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.

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