Магнетски флукс

Магнетни флукс или магнетни ток (магнетни флукс или магнетни ток), који се представља грчким словом Φ (фи), је физичка величина која описује магнетно поље у околини покретног наелектрисања. Уколико магнетно поље замишљамо помоћу магнетних линија сила које се шире у простору, тада је флукс број линија која пролази кроз неку затворену контуру.

СИ јединица за магнетни флукс је Wb (вебер), или V s (волт секунда) преко основних јединица, док је јединица која описује индукцију магнетног поља Wb/m² или T (тесла).

Магнетни флукс кроз елемент нормалан у односу на смер магнетне индукције (или магнетног поља) је производ вредности магнетне индукције и елементарне површине. Уопште, магнетни флукс је дефинисан скаларним производом вектора магнетне индукције и вектора елементарне површине. Гаусов закон магнетизма, један од четири Максвелове једначине, говори да је магнетни флукс кроз затворену контуру једнак нули. Овај закон је последица тога што се магнетни дипол не може раставити на елементарне полове, северни и јужни пол.

Опис

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 (see surface integration).

Each point on a surface is associated with a direction, called the surface 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 a vector field, where each point in space is associated with a vector that determines what force a moving charge would experience at that point (see Lorentz force). Since a vector field is quite difficult to visualize at first, in elementary physics one may instead visualize this field with field lines. 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 the net 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).

Магнетни флукс се дефинише као интеграл магнетне индукције кроз неку површину:

$\Phi _{m}=\int \!\!\!\int \mathbf {B} \cdot d\mathbf {S} \,$

где је

$\Phi _{m}\$ магнетни флукс

B је магнетна индукција

S је површина.

Гаусов закон магнетизма казује да

$\nabla \cdot \mathbf {B} =0.\,$

Интеграл по запремини ове једначине, заједно са теоремом дивергенције, даје следећи резултат:

$\int \!\!\!\int \!\!\!\int _{V}\nabla \cdot \mathbf {B} \,d\tau =\oint \!\!\!\oint _{\partial V}\mathbf {B} \cdot d\mathbf {S} =0.$

Другим речима, магнетни флукс кроз било коју затворену контуру мора бити једнак нули, јер се магнет не може поделити на северни и јужни пол.

Насупрот томе, Гаусов закон за електрично поље, још једна од Максвелових једначина, је:

$\nabla \cdot \mathbf {E} ={\rho \over \epsilon _{0}},$

где је

E јачина електричног поља,

$\rho$ је густина слободних наелектрисања (не укључује наелектрисања везана за материјал),

$\epsilon _{0}$ је пермитивност вакуума.

Ова једначина наговештава постојање електричних монопола, позитивног и негативног наелектрисања.

Смер вектора магнетног поља $\mathbf {B}$ је по дефиницији од јужног ка северном полу унутар магнета, док ван магнета линије силе иду од северног пола ка јужном полу.

Промена магнетног флукса кроз навојак проводника ће индуковати електромоторну силу, а тиме и електричну струју кроз навојак (ако је струјно коло затворено). Ова једначина је дата Фарадејевим законом електромагнетне индукције:

${\mathcal {E}}=\oint \mathbf {E} \cdot d\mathbf {l} =-{d\Phi _{m} \over dt}.$

На овоме се заснива принцип рада електричног генератора.

Магнетни флукс кроз затворену површину

Some examples of closed surfaces (left) and open surfaces (right). Left: Surface of a sphere, surface of a torus, surface of a cube. Right: Disk surface, square surface, surface of a hemisphere. (The surface is blue, the boundary is red.)

Главни чланак: Gauss's law for magnetism

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:

$\Phi _{B}=\,\!$$\scriptstyle S$$\mathbf {B} \cdot d\mathbf {S} =0$

for any closed surface S.

Магнетни ток кроз отворену површину

For an open surface Σ, the electromotive force along the surface boundary, ∂Σ, is a combination of the boundary's motion, with velocity v, through a magnetic field B (illustrated by the generic F field in the diagram) and the induced electric field caused by the changing magnetic field.

While the magnetic flux through a closed surface is always zero, the magnetic flux through an open 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.

Промена магнетног флукса

Главни чланак: Faraday's law of induction

Area defined by an electric coil with three turns.

For example, a change in the magnetic flux passing through a loop of conductive wire will cause an electromotive force, and therefore an electric current, in the loop. The relationship is given by Faraday's law:

${\mathcal {E}}=\oint _{\partial \Sigma }\left(\mathbf {E} +\mathbf {v\times B} \right)\cdot d{\boldsymbol {\ell }}=-{d\Phi _{B} \over dt},$

where

${\mathcal {E}}$ is the electromotive force (EMF),

Φ_B is the magnetic flux through the open surface Σ,

∂Σ is the boundary of the open surface Σ; 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 an infinitesimal vector element of the contour ∂Σ,

v is the velocity of the boundary ∂Σ,

E is the electric field,

B is the magnetic field.

The two equations for the EMF are, firstly, the work per unit charge done against the Lorentz force in moving a test charge around the (possibly moving) surface boundary ∂Σ and, secondly, as the change of magnetic flux through the open surface Σ. This equation is the principle behind an electrical generator.

Поређење са електричним флуксом

Главни чланци: Електрични флукс и Гаусов закон

By way of contrast, Gauss's law for electric fields, another of Maxwell's equations, is

$\Phi _{E}=\,\!$$\scriptstyle S$$\mathbf {E} \cdot d\mathbf {S} ={\frac {Q}{\varepsilon _{0}}}\,\!$

where

E is the electric field,

S is any closed surface,

Q is the total electric charge inside the surface S,

ε₀ is the electric constant (a universal constant, also called the "permittivity of free space").

The flux of E through a closed surface is not always zero; this indicates the presence of "electric monopoles", that is, free positive or negative charges.

Квант магнетног флукса

CODATA values		Units
Φ0	6985206783384799999♠2,067833848...×10−15	Wb
KJ	7014483597848400000♠483597,8484...×109	Hz/V
KJ-90	7014483597900000000♠483597,9×109	Hz/V

The magnetic flux, represented by the symbol Φ, threading some contour or loop is defined as the magnetic field B multiplied by the loop area S, i.e. Φ = B ⋅ S. Both B and S can be arbitrary, meaning Φ can be as well. However, if one deals with the superconducting loop or a hole in a bulk superconductor, the magnetic flux threading such a hole/loop is actually quantized. The (superconducting) magnetic flux quantum Φ₀ = h/(2e) ≈ 6985206783384799999♠2,067833848...×10⁻¹⁵ Wb is a combination of fundamental physical constants: the Planck constant h and the electron charge e. Its value is, therefore, the same for any superconductor. The phenomenon of flux quantization was discovered experimentally by B. S. Deaver and W. M. Fairbank and, independently, by R. Doll and M. Näbauer, in 1961. The quantization of magnetic flux is closely related to the Little–Parks effect, but was predicted earlier by Fritz London in 1948 using a phenomenological model.

The inverse of the flux quantum, 1/Φ₀, is called the Josephson constant, and is denoted K_J. It is the constant of proportionality of the Josephson effect, relating the potential difference across a Josephson junction to the frequency of the irradiation. The Josephson effect is very widely used to provide a standard for high-precision measurements of potential difference, which (from 1990 to 2019) were related to a fixed, conventional value of the Josephson constant, denoted K_J-90. With the 2019 redefinition of SI base units, the Josephson constant has an exact value of K_J = 7005483597848416980♠483597,84841698... GHz⋅V⁻¹, which replaces the conventional value K_J-90.

The following physical equations use SI units. In CGS units, a factor of c would appear.

The superconducting properties in each point of the superconductor are described by the complex quantum mechanical wave function Ψ(r,t) — the superconducting order parameter. As any complex function Ψ can be written as Ψ = Ψ₀e^iθ, where Ψ₀ is the amplitude and θ is the phase. Changing the phase θ by 2πn will not change Ψ and, correspondingly, will not change any physical properties. However, in the superconductor of non-trivial topology, e.g. superconductor with the hole or superconducting loop/cylinder, the phase θ may continuously change from some value θ₀ to the value θ₀ + 2πn as one goes around the hole/loop and comes to the same starting point. If this is so, then one has n magnetic flux quanta trapped in the hole/loop, as shown below:

Per minimal coupling, the probability current of cooper pairs in the superconductor is:

$\mathbf {J} ={\frac {1}{2m}}\left[\left(\Psi ^{*}(-i\hbar \nabla )\Psi -\Psi (-i\hbar \nabla )\Psi ^{*}\right)-2q\mathbf {A} |\Psi |^{2}\right]\,\!.$

Here, the wave function is the Ginzburg–Landau order parameter:

$\Psi (\mathbf {r} )={\sqrt {\rho (\mathbf {r} )}}\,e^{i\theta (\mathbf {r} )}.$

Plugging into the expression of probability current, one obtains:

$\mathbf {J} ={\frac {\hbar }{m}}(\nabla {\theta }-{\frac {q}{\hbar }}\mathbf {A} )\rho .$

While inside the body of the superconductor, the current density J is zero; Therefore:

$\nabla {\theta }={\frac {q}{\hbar }}\mathbf {A} .$

Integrating around the hole/loop using Stokes' theorem and $\nabla \times \mathbf {A} =B$ gives:

$\Phi _{B}=\oint \mathbf {A} \cdot d\mathbf {l} ={\frac {\hbar }{q}}\oint \nabla {\theta }\cdot d\mathbf {l} .$

Now, because the order parameter must return to the same value when the integral goes back to the same point, we have:

$\Phi _{B}={\frac {\hbar }{q}}2c\pi =c{\frac {h}{2e}}.$

Види још

• Густина магнетног флукса
• Магнетно поље
• Максвелове једначине
• Гаусови закони

Литература

• Griffiths, David J. (1998). Introduction to Electrodynamics (3rd изд.). Prentice Hall. ISBN 978-0-13-805326-0. 
• Jackson, John D. (1998). Classical Electrodynamics (3rd изд.). Wiley. ISBN 978-0-471-30932-1. 
• Tipler, Paul (2004). Physics for Scientists and Engineers: Electricity, Magnetism, Light, and Elementary Modern Physics (5th изд.). W. H. Freeman. ISBN 978-0-7167-0810-0. 
• L. O. Chua, "Memristor – The Missing Circuit Element", IEEE Trans. Circuit Theory, vol. CT_18, no. 5, pp. 507–519, 1971.

Спољашње везе

• US 6720855, Vicci, "Magnetic-flux conduits", issued 2003 
• Magnetic Flux through a Loop of Wire by Ernest Lee, Wolfram Demonstrations Project.
• Conversion Magnetic flux Φ in nWb per meter track width to flux level in dB – Tape Operating Levels and Tape Alignment Levels