# Biot-Savart Law

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## Biot-Savart law

A law of physics which states that the magnetic flux density (magnetic induction) near a long, straight conductor is directly proportional to the current in the conductor and inversely proportional to the distance from the conductor. The field near a straight conductor can be found by application of Ampère's law. The magnetic flux density near a long, straight conductor is at every point perpendicular to the plane determined by the point and the line of the conductor. Therefore, the lines of induction are circles with their centers at the conductor. Furthermore, each line of induction is a closed line. This observation concerning flux about a straight conductor may be generalized to include lines of induction due to a conductor of any shape by the statement that every line of induction forms a closed path.

## Biot-Savart Law

a law that determines the strength of the magnetic field created by an electric current. The Biot-Savart law was discovered by the French scientists J. B. Biot and F. Savart in 1820 and given a general formulation by P. Laplace. According to this law, a small segment of a conductor Δl along which a current of strength / is flowing creates—at a given point *M* in space, located at a distance *r* from the segment Δl (Δl << *r*)—a magnetic field of strength

Here *θ* is the angle between the direction of the current in segment / and the radius vector r, drawn from the segment to the observation point *M,* and *k* is the coefficient of proportionality, which depends on the choice of the system of units. In the centimeter-gram-second (cgs) system (Gauss), *k=1/c,* where *c* = 3 x 10^{10} cm/sec (the velocity of light in a vacuum); in the International System of Units (SI), *k = l/4π.*

The magnetic field strength *ΔH* is perpendicular to the plane *P,* which contains *Δl* and *r,* and its direction is determined by the auger rule: if the shaft of the auger (with a right-handed rifling) is turned from *Δl* toward *r,* then the auger advances in the direction of *ΔH.*

The total magnetic field strength *ΔH* created by a conductor with a current at point *M* is equal to the vector sum of the quantities *ΔH,* which are determined by all the elements Δ/ of the conductor. In particular, the strength *H* of a magnetic field at a distance *d* from a long (much greater than *d*), straight conducting wire along which a current of strength / is flowing is *H = k·2I/d*; in the center of a circle with radius *R* along which a current of strength / is flowing, *H* = *k. 2πI/R*; and on its axis, at a point remote from the plane of the circle at a distance *d ≪ R, H = k·2*π *R ^{2}I/d3.* On the axis of a solenoid of

*n*turns,

*H = k·4πn/.*

The Biot-Savart law can also be considered as a law that determines the magnetic induction Δ*B* ; in the cgs system, this requires that the expression for *ΔH* be multiplied by the magnetic permeability of the medium *μ,* whereas in the SI system, in addition, it must be multiplied by the magnetic permeability of a vacuum, *μ _{o}* = 4π10

^{-7}henry/m.

G. IA. MIAKISHEV