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Magnetic flux

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Magnetic flux (most often denoted as Φm), is the amount of magnetic field (also called "magnetic flux density") passing through a surface (such as a conducting coil). The SI unit of magnetic flux is the weber (Wb) (in derived units: volt-seconds). The CGS unit is the maxwell. For example, it is used by electrical engineers trying to design systems with electromagnets or designing dynamos. Physicists designing particle accelerators also calculate it.


Figure 1: The definition of surface integral relies on splitting the surface into small surface elements. Each element is associated with a vector dS of magnitude equal to the area of the element and with direction at a right angle to the element and pointing outward.
Figure 2: A vector field of normals to a surface

The magnetic flux through a given surface is proportional to the number of magnetic B field lines that pass through the surface. This is the net number, that is 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 an area that is at a 90° angle (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 a:

[math]\displaystyle \Phi_m = \mathbf{B} \cdot \mathbf{a} = Ba \cos \theta[/math]    (uniform B with flat area only)

where θ is the angle between B and a vector a that is perpendicular (normal) to the surface. So, the angle between the magnetic field and the surface will reduce the measured flux.

Scientists use calculus to handle real world examples of flux through surfaces that have strange shapes. 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):

[math]\Phi_m = \iint\limits_S \mathbf{B} \cdot d\mathbf S,[/math]

where [math]\textstyle \Phi_m \ [/math] is the magnetic flux, B is the magnetic field,

S is the surface (area),[math]\cdot[/math] denotes dot product, and dS is an infinitesimal vector, whose magnitude is the area of a differential element of S, and whose direction is the surface normal. (See surface integral for more details.)

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

[math]\Phi_m = \oint\limits_{\Sigma} \mathbf{A} \cdot d\boldsymbol{\ell}[/math]

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

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.)

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:

[math]\Phi_m=\int \!\!\! \int \mathbf{B} \cdot d\mathbf S = 0,[/math]

for any closed surface S.

Magnetic flux through an open surface

Figure 3: A vector field F ( r, t ) defined throughout space, and a surface Σ bounded by curve ∂Σ moving with velocity v over which the field is integrated.

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

[math]\mathcal{E} = \oint_{\partial \Sigma (t)}\left( \mathbf{E}( \mathbf{r},\ t) +\mathbf{ v \times B}(\mathbf{r},\ t)\right) \cdot d\boldsymbol{\ell} = -{d\Phi_m \over dt},[/math]

where (see Figure 3):

[math]\mathcal{E}[/math] is the EMF,
Φm is the flux through a surface with an opening bounded by a curve ∂Σ(t),
∂Σ(t) is a closed contour that can change with time; the EMF is found around this contour, and the contour is a boundary of the surface over which Φm is found,
d is an infinitesimal vector element of the contour ∂Σ(t),
v is the velocity of the segment d,
E is the electric field,
B is the magnetic field.

The EMF is determined in this equation in two ways: first, as the work per unit charge done against the Lorentz force in moving a test charge around the (possibly moving) closed curve ∂Σ(t), and second, as the magnetic flux through the open surface Σ(t).

This equation is the principle behind an electrical generator.

Comparison with electric flux

The flux visualized. The rings show the surface boundaries. The red arrows stand for the magnetic field or the flow of electrons. The number of arrows that pass through each ring is the flux.

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

[math]\Phi_E = \int \!\!\!\int_S \mathbf{E}\cdot d\mathbf{S} = {Q \over \epsilon_0},[/math]


E is the electric field,
S is any closed surface,
Q is the total electric charge inside the surface S,
[math] \epsilon_0 [/math] is the electric constant (a universal constant, also called the "permittivity of free space").

Note that 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.

Magnetic Circuits

Conventional Magnetic Circuits

Phasor Magnetic Circuits

Related Concepts

Gyrator-capacitor model variables

Other pages

  • Magnetic field
  • Maxwell's equations are the set of four equations, attributed to James Clerk Maxwell, that describe the behavior of both the electric and magnetic fields, as well as their interactions with matter.
  • Gauss's law gives the relation between the electric flux flowing out a closed surface and the electric charge enclosed in the surface.
  • Magnetic circuit is a method using an analogy with electric circuits to calculate the flux of complex systems of magnetic components.
  • Magnetic monopole is a hypothetical particle that may be loosely described as "a magnet with only one pole".
  • Magnetic flux quantum is the quantum of magnetic flux passing through a superconductor.
  • Carl Friedrich Gauss developed a fruitful collaboration with the physics professor Wilhelm Weber; it led to new knowledge in the field of magnetism.
  • James Clerk Maxwell demonstrated that electric and magnetic forces are two complementary aspects of electromagnetism.


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