# Magnetic river

Physical size
Surname Magnetic river
Formula symbol ${\ displaystyle \ Phi}$
Size and
unit system
unit dimension
SI Wb M · L 2 · I −1 · T −2
Gauss ( cgs ) Mx L 02/03 · M 02/01 · T -1
esE ( cgs ) statWb = dyn -1/2 · s L 1/2 x M 1/2
emE ( cgs ) Mx L 02/03 · M 02/01 · T -1

The magnetic flux (symbol  ) is a scalar physical quantity to describe the magnetic field . It is - analogously to the electric current - the result of a magnetic tension and flows through a magnetic resistance . Since even the vacuum represents such a magnetic resistance, the magnetic flux is not bound to a specific “medium” and is described by field values . ${\ displaystyle \ Phi}$

## General

For example, if you consider a small cylinder made of a material with a given magnetic conductivity , to which a magnetic voltage (determined by its length and the magnetic field strength ) is applied, a current is established proportional to its cross-sectional area. Analogous to the electrical resistance, one defines the magnetic resistance and comes to the following relationship: ${\ displaystyle U _ {\ text {m}}}$ ${\ displaystyle H}$ ${\ displaystyle R _ {\ text {m}}}$

${\ displaystyle \ Phi = {\ frac {U _ {\ text {m}}} {R _ {\ text {m}}}}}$

For example , if the magnetic field strength prevails in the simplest linear, homogeneous case between the pole pieces of a magnet that are spaced apart from one another , the magnetic voltage prevails along the route : ${\ displaystyle d}$${\ displaystyle H}$${\ displaystyle d}$

${\ displaystyle U _ {\ text {m}} = H \ cdot d}$

This magnetic voltage creates the magnetic flux between the pole pieces. Its size depends on the magnetic resistance of the material between the pole pieces (or the empty space). The magnetic resistance is linked to the magnetic conductivity of the material as a material constant or to the magnetic conductivity of the empty space as a natural constant , just as an ohmic resistance is linked to the material constant of the electrical conductivity of the resistance material.

As a rule, one works in the field theory not with the magnetic flux, one can only assign a specific area in the room, but not discrete field points: There is no function , wherein , , denote spatial coordinates; d. H. the magnetic flux is not a scalar field . In the drawing, the magnetic flux is therefore represented as a kind of "tube" (flux tube). ${\ displaystyle \ Phi (x, y, z)}$${\ displaystyle x}$${\ displaystyle y}$${\ displaystyle z}$

In order to avoid these difficulties, the vector size of the magnetic flux density is usually used instead of the magnetic flux . It is related to the magnetic flux through an oriented surface as follows: ${\ displaystyle {\ vec {B}}}$ ${\ displaystyle {\ vec {A}}}$

${\ displaystyle \ Phi = \ int \ limits _ {A} {\ vec {B}} \ cdot \ mathrm {d} {\ vec {A}}}$.

## Special cases

• If the magnetic field is homogeneous and the surface is not curved, the magnetic flux is equal to the scalar product of the magnetic flux density and the surface vector ( normal vector of the surface): ${\ displaystyle {\ vec {A}}}$
${\ displaystyle \ Phi = {\ vec {B}} \ cdot {\ vec {A}}}$
• Since the magnetic field is free of sources ( magnetic monopoles are only hypothetical particles), the lines of magnetic flux density are always self-contained. This is expressed in the Maxwell equations by:
${\ displaystyle \ nabla \ cdot {\ vec {B}} = 0}$
Since, according to Gauss’s integral theorem, the following applies:
${\ displaystyle \ Phi = \ oint \ limits _ {\ partial V} {\ vec {B}} \ cdot \ mathrm {d} {\ vec {A}} = \ int \ limits _ {V} \ nabla \ cdot {\ vec {B}} \; \ mathrm {d} V}$,
the magnetic flux through a closed surface of a space segment is always zero.

## Chained flow, chain flow, induction flow

Area of ​​a coil with three turns

As linked flux (also linkage flux, magnetic flux or flux coil), the total magnetic flux of an inductor or coil denotes that extends in the integration of the magnetic flux density over the surface results, which is formed by the coil, together with their supply lines: ${\ displaystyle B}$${\ displaystyle A_ {v}}$

${\ displaystyle \ Psi = \ int \ limits _ {A_ {v}} {\ vec {B}} \ cdot \ mathrm {d} {\ vec {A_ {v}}}}$

Any oriented surface that is bordered by the short-circuited coil can be used as the integration surface. Because there are no magnetic monopoly charges, the calculation of the flux depends exclusively on the edge line, not on the exact shape of the surface. The adjacent picture shows a possible coil area using the example of a coil with three turns. In a conventional coil arrangement, the surface is penetrated by the magnetic field lines in the coil core- times when the field in the core is approximately homogeneous. Then we get: ${\ displaystyle A_ {v}}$ ${\ displaystyle N}$

${\ displaystyle \ Psi \ approx N \ cdot \ Phi _ {\ text {w}}}$,

wherein the magnetic flux is through a turn or the cross-sectional area of ​​the magnetic core. ${\ displaystyle \ Phi _ {\ text {w}}}$

The linked flux can be clearly described in the following form: The induced voltage in a single turn results from the change in the magnetic flux enclosed by the single turn . If, as in the case of a coil, another turn is connected in series with the first , an induced voltage of the same size results in this turn too, provided that both turns comprise the same flux. Both induced voltages add up due to the series connection of the windings. With turns for the entire coil there is thus an induced voltage proportional to the change in . This total voltage is applied to the terminals of the coil; thus the linked magnetic flux and not the simple magnetic flux must be taken into account for the current-voltage relationship and the inductance of the coil. ${\ displaystyle \ Phi _ {\ text {w}}}$${\ displaystyle N}$${\ displaystyle \ Psi}$${\ displaystyle \ Psi}$${\ displaystyle \ Phi _ {\ text {w}}}$

In the electrotechnical literature it has become widely accepted to denote the magnetic flux in the magnetic core with and the magnetic flux through the area spanned by the coil . The choice of the different letter should not lead to the obvious mistake that the linked flux is a new physical quantity different from the usual magnetic flux. From a physical point of view, the linked flux of a coil is nothing else than the ordinary magnetic flux that results in the special case of a coil surface. The choice of the new letter, however, is useful in distinguishing the coil flux from the magnetic flux that penetrates the cross section of the coil core. ${\ displaystyle \ Phi}$${\ displaystyle \ Psi}$

## Unit of measurement

The unit of measurement of the magnetic flux in the SI system of units is Weber , the unit symbol  Wb:

${\ displaystyle [\ Phi] = \ mathrm {T \ cdot m ^ {2}} = \ mathrm {V \ cdot s} = \ mathrm {Wb}}$

With

• the symbol for the unit Tesla${\ displaystyle \ mathrm {T}}$
• the symbol for the unit volt .${\ displaystyle \ mathrm {V}}$

## illustration

Magnetic flux along the axis of a long, thin solenoid.

While it is relatively easy for the electrical flow and the electrical charge behind it in C (or As) to develop a descriptive idea of ​​it, namely that of a correspondingly large number of electrons, capable of e.g. For example, maintaining a current of 1 A for one second is much more difficult with the magnetic flux, measured in Wb (or Vs). ${\ displaystyle Q}$

One of the options is to use the term in some (older) physics textbooks, the time sum of the voltage or the voltage time sum , based on the term impulse, also called voltage impulse :

${\ displaystyle \ Phi = - \ int U _ {\ text {ind}} \ cdot \ mathrm {d} t}$

If you record the induction voltage in a conductor loop as a function of time, it becomes apparent that the area below the voltage curve always remains the same with the same strength of the excitation field, regardless of how fast or slow the change in flux takes place. Accordingly, one of the definitions of the magnetic flux based on the term of the stress time sum is as follows: ${\ displaystyle \ Phi}$

"The force flow through a surface is 1 Weber [...] if a voltage time sum of 1 V · s is induced in a circuit surrounding it when the force flow through the surface disappears."

To put it more colloquially: A magnetic flux of 1 Weber (or 1 Vs) is the "amount of magnetism" that is able to maintain a voltage of 1 V for one second when it disappears in the circuit surrounding it. (See also voltage time area )

## Quantum theory

When considering quantum phenomena (e.g. Aharonov-Bohm effect , quantum Hall effect ) is the magnetic flux quantum

${\ displaystyle \ Phi _ {0} = {\ frac {h} {e}}}$,

thus the quotient of Planck's quantum of action and the elementary charge , an appropriate quantity. ${\ displaystyle h}$ ${\ displaystyle e}$

Experiments on superconductors showed that the magnetic flux through a superconducting ring is a multiple of

${\ displaystyle \ Phi _ {0} = {\ frac {h} {2e}}}$

is (→ see flux quantization ) , which implies that the charge carriers involved carry two elementary charges. This is a clear confirmation of the BCS theory , according to which superconductivity is mediated by electron pairs ( Cooper pairs ).

## literature

• Karl Küpfmüller, Gerhard Kohn: Theoretical electrical engineering and electronics . 14th edition. Springer, 1993, ISBN 3-540-56500-0 .