weber

SI coherent derived unit with special name and symbol

Name	Symbol	Quantity	Base units
weber	Wb	magnetic flux	kg m² s⁻² A^-1
The weber, symbol Wb, is the SI coherent derived unit of magnetic flux. It is the special name for the kg m² s⁻² A^-1. 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.
Definition			h e^-1
$1 \mspace{4mu} \text{Wb} \mspace{6mu} = \dfrac{1.602 \mspace{4mu} 176 \mspace{4mu} 634 \times 10^{-19}}{6.626 \mspace{4mu} 070 \mspace{4mu} 15 \times 10^{-34}} \mspace{6mu} \dfrac{h}{e}\\ \\ \\ 1 \mspace{4mu} \text{Wb} \mspace{6mu} \approx 2.417 \mspace{4mu} 989 \mspace{4mu} 242 \mspace{4mu} 084 \mspace{4mu} 918 \mspace{4mu} 162 \times 10^{14} \mspace{6mu} h \mspace{4mu} e^{-1}$

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 through a surface is the surface integral of the normal component of the magnetic flux density, B, passing through that surface. For a constant magnetic field, the magnetic flux, Φ_B, passing through a surface of vector area S can be calculated as follows:

Using SI coherent units,

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

where:

Φ_B is the magnetic flux in webers, symbol Wb,
B is 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 square 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:

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

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.

$\Phi _{B} = 0$

Magnetic flux quantum

A magnetic flux threading a superconducting ring, or a closed loop in a superconductor, is quantized. The unit of this quantization is the magnetic flux quantum, symbol Φ₀.

The magnetic flux quantum is defined by the ratio of the Planck constant, h, to the elementary charge, e:

$\Phi_{0} = \dfrac{h} {2e}\\ \\ \\ \Phi_{0} = \dfrac{6.626 \mspace{4mu} 070 \mspace{4mu} 15 \times 10^{-34} \mspace{4mu} \text{J} \ \text{s}} {2 \times (1.602 \mspace{4mu} 176 \mspace{4mu} 634 \times 10^{-19} \ \text{C})}\\ \\ \\ \Phi_{0} \approx 2.067 \mspace{4mu} 833 \mspace{4mu} 848 \mspace{4mu} 461 \mspace{4mu} 929 \mspace{4mu} 323 \mspace{4mu} 081 \mspace{4mu} 115 \times 10^{-15} \ \text{Wb}$

The inverse of the magnetic flux quantum, ¹⁄_Φ₀, is called the Josephson constant, symbol K_J. The Josephson constant is the constant of proportionality of the Josephson effect, relating the potential difference, or voltage, across a Josephson junction to the frequency of its irradiation.

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.