# Magnetic flux

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Magnetic flux (most often denoted as Φm), is a measure of the amount of magnetic field passing through a given surface (such as a conducting coil). The SI unit of magnetic flux is the weber (in derived units: volt-seconds). The CGS unit is the maxwell.

## Contents

### Description

The magnetic flux through a given 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. (See below for how the positive sign is chosen.) For a uniform magnetic field B passing through a perpendicular area the magnetic flux is given by the product of the magnetic field and the area element. The magnetic flux for a uniform B at any angle to a surface is defined by a dot product of the magnetic field and the area element vector.

where θ is the angle between B and a vector that is perpendicular (normal) to S.

In the general case, 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):

where $\textstyle \Phi_m \$ is the magnetic flux, B is the magnetic field,

From the definition of the magnetic vector potential A and the fundamental theorem of the curl the magnetic flux may also be defined as:

where the closed line integral is over the boundary of the surface and d is an infinitesimal vector element of that contour Σ.

The magnetic flux is usually measured with a fluxmeter. The fluxmeter contains measuring coils and electronics that evaluates the change of voltage in the measuring coils to calculate the magnetic flux.

### 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 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 that magnetic monopoles have never been found.

In other words, Gauss's law for magnetism is the statement:

for any closed surface S.