Law of Biot and Savart

The law of Biot and Savart (1820) gives the Magnetic field created by a continuous distribution of running S. It constitutes one of the fundamental laws of the Magnétostatique, as well as the Loi of Coulomb for the electrostatic .

Case of a thread-like circuit

A thread-like circuit is a modeling where the electric wire has only one dimension. It is a idealization of a real wire of which the length would be much greater than transverse dimensions of its sectional surface.

Law of Biot & Savart

Let us note $\ mathcal C$ the geometrical curve representing the thread-like circuit, and is $S$ a point of this curve $\ mathcal C$ . One notes $\ vec {DLL}$ the vector tangent elementary displacement with the curve $\ mathcal C$ at the point $S$ . In the vacuum, the circuit traversed by a D.C. current of intensity $I$ creates in any point $M$ space $\ left (M \ notin \ mathcal {C} \ right)$ the magnetic field:

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where $\ mu_0$ is a fundamental constant, called magnetic Perméabilité of the vacuum.

Notice on a notation

It is said sometimes that the element infinitesimal length $\ vec {DLL}$ , located at the point $S$ and traversed by the current $I$ , creates the elementary magnetic field $\ vec {dB}$ located at the point $M$ :

\ vec {dB} (M) \ = \ \ frac {\ mu_0} {4 \ pi} \ \ frac {I \, \ vec {DLL} \ wedge \ vec {SM}}

Other modelings

Surface density of current

In the case of a surface density of current

\ vec {J} _s

existing on surface

\ Sigma

, the magnetic field created is:

$\ vec {B} (M) \ = \ \ frac {\ mu_0} {4 \ pi} \ iint_ {S \ in \ Sigma} \ frac {\ vec {J} _s (S) \ wedge \ vec {SM}}$

Voluminal density of current

In the case of a voluminal density of existing current

\ vec j

in volume

\ mathcal V

, the magnetic field created is:

$\ vec {B} (M) \ = \ \ frac {\ mu_0} {4 \ pi} \ iiint_ {S \ in \ mathcal {V}} \ frac {\ vec {J} (S) \ wedge \ vec {SM}}$

Theorem of Amp

By integrating the law of Biot and Savart on a closed loop

\ unspecified Gamma

(which a priori is not an electrical circuit), one shows the Théorème of Amp:

\ oint_ {M \ in \ Gamma} \ vec {B} (M) \ cdot \ vec {DM} \ = \ \ mu_0 \ interior I_ {}

where $I_ {interior}$ is the algebraic intensity intertwined by the curve $\ Gamma$

The case of a particle charged

By noticing that a specific particle of electric charge $q$ animated a constant speed $\ vec v$ has a density of current: $\ vec {J} \ = \ Q \ \ vec {v}$ , the law of Biot and Savart suggests writing that this load (moving) at the point $S$ creates a magnetic field at the point $M$ :

\ vec {B} (M) \ = \ \ frac {\ mu_0} {4 \ pi} \ frac {Q \ vec {v} \ wedge \ vec {SM}}

Application to aerodynamics

The law of Biot and Savart is used to calculate the speed induced by lines of Vortex in Aérodynamique. Indeed, an analogy with the magnetostatic one is possible if it is admitted that the Vorticité corresponds to the current, and speed induced with the intensity of the magnetic field. For a line of vortex infinite length, induced speed is given by:

v \ = \ \ frac {\ Gamma} {4 \ pi D}

where:

Γ is the intensity of the vortex

D is the perpendicular distance between the point and the line of vortex.

For a line of vortex finite length:

$v \ = \ \ frac {\ Gamma} {8 \ pi D} \ \ left \ \ cos has - \ cos B \ \ right$

where has and B is the angles (directed) between the line and the two ends of the segment.

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