weber

SI coherent derived unit with special name and symbol

Name

Symbol

Derived quantity

Expressed in terms of SI base units

weber

magnetic flux

kg m² s⁻² A^-1

Definition

The weber, symbol Wb, is the SI coherent derived unit of magnetic flux.

One weber is the magnetic flux that, linking a circuit of one turn, would produce in it an electromotive force of one volt if it were reduced to zero at a uniform rate in one second.

The weber is named after the German physicist Wilhelm Eduard Weber (1804 – 1891).

The weber may be defined in terms of Faraday’s law, which relates a changing magnetic flux through a loop to the electric field around the loop. A change in flux of one weber per second will induce an electromotive force of one volt (produce an electric potential difference of one volt across two open-circuited terminals).

A flux density of one Wb/m² (one weber per square metre) is equal to one tesla.

Magnetic flux

The magnetic flux, symbol Φ or Φ_B, through a surface is the surface integral of the normal component of the magnetic field B passing through that surface.

For a constant magnetic field, the magnetic flux passing through a surface of vector area S is

$\Phi _{B} = \mathbf {B} \mathbf {S} = BS \cos \theta$

where:

B is the magnitude of the magnetic field, the magnetic flux density, measured in webers per square metre, symbol Wb/m², or teslas, symbol T,
S is the area of the surface, measured in metres, symbol m²,
θ is the angle between the magnetic field lines and the normal (perpendicular) to the surface S.

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:

$d \Phi _{B} = \mathbf {B} d \mathbf {S}$

A generic surface, S, can then be broken into infinitesimal elements and the total magnetic flux through the surface is then the surface integral

$\Phi _{B} = \iint \limits _{S}\mathbf {B} \cdot d\mathbf {S}$

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

$\Phi _{B} = \oint \limits _{\partial S}\mathbf {A} d{\boldsymbol {\ell }}$

where the line integral is taken over the boundary of the surface S, which is denoted ∂S.

Magnetic flux through a closed surface

Gauss’s law for magnetism states that the total magnetic flux through a closed surface is equal to zero.

$\Phi _{B} = \mathbf {B} d \mathbf {S} = 0$

Measurement

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