Inphysics, specificallyelectromagnetism, themagnetic flux through a surface is thesurface integral of thenormal component of themagnetic fieldB over that surface. It is usually denotedΦ orΦB. TheSIunit of magnetic flux is theweber (Wb; in derived units, volt–seconds or V⋅s), and theCGS unit is themaxwell.[1] Magnetic flux is usually measured with afluxmeter, which contains measuringcoils, and it calculates the magnetic flux from the change ofvoltage on the coils.
The magnetic flux through a surface—when the magnetic field is variable—relies on splitting the surface into small surface elements, over which the magnetic field can be considered to be locally constant. The total flux is then a formal summation of these surface elements (seesurface integration).
Each point on a surface is associated with a direction, called thesurface normal; the magnetic flux through a point is then the component of the magnetic field along this direction.
The magnetic interaction is described in terms of avector field, where each point in space is associated with a vector that determines what force a moving charge would experience at that point (seeLorentz force).[1] Since a vector field is quite difficult to visualize, introductory physics instruction often usesfield lines to visualize this field. The magnetic flux, through some surface, in this simplified picture, is proportional to the number of field lines passing through that surface (in some contexts, the flux may be defined to be precisely the number of field lines passing through that surface; although technically misleading, this distinction is not important). The magnetic flux is thenet 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 (see below for deciding in which direction the field lines carry a positive sign and in which they carry a negative sign).[2]More sophisticated physical models drop the field line analogy and define magnetic flux 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 ofvector areaS iswhereB is the magnitude of the magnetic field (the magnetic flux density) having the unit of Wb/m2 (tesla),S is the area of the surface, andθ is the angle between the magneticfield lines and thenormal (perpendicular) toS. 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 thesurface integralFrom the definition of themagnetic vector potentialA and thefundamental theorem of the curl the magnetic flux may also be defined as:where theline integral is taken over the boundary of the surfaceS, which is denoted∂S.
Some examples ofclosed surfaces (left) andopen surfaces (right). Left: Surface of a sphere, surface of atorus, surface of a cube. Right:Disk surface, square surface, surface of a hemisphere. (The surface is blue, the boundary is red.)
Gauss's law for magnetism, which is one of the fourMaxwell's equations, states that the total magnetic flux through aclosed surface is equal to zero. (A "closed surface" is a surface that completely encloses a volume(s) with no holes.) This law is a consequence of the empirical observation thatmagnetic monopoles have never been found.
In other words, Gauss's law for magnetism is the statement:
For an open surface Σ, theelectromotive force along the surface boundary, ∂Σ, is a combination of the boundary's motion, with velocityv, through a magnetic fieldB (illustrated by the genericF field in the diagram) and the induced electric field caused by the changing magnetic field.
While the magnetic flux through aclosed surface is always zero, the magnetic flux through anopen 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.
For example, a change in the magnetic flux passing through a loop of conductive wire will cause anelectromotive force (emf), and therefore an electric current, in the loop. The relationship is given byFaraday's law:where:
ΦB is the magnetic flux through the open surfaceΣ,
∂Σ is the boundary of the open surfaceΣ; the surface, in general, may be in motion and deforming, and so is generally a function of time. The electromotive force is induced along this boundary.
dℓ is aninfinitesimal vector element of the contour∂Σ,
The two equations for the emf are, firstly, the work per unit charge done against theLorentz force in moving a test charge around the (possibly moving) surface boundary∂Σ and, secondly, as the change of magnetic flux through the open surfaceΣ. This equation is the principle behind anelectrical generator.
Area defined by an electric coil with three turns.
Theflux ofE through a closed surface isnot always zero; this indicates the presence of "electric monopoles", that is, free positive or negativecharges.
