The Biot-Savart law is a physical law with applications in both electromagnetics and fluid dynamics. As originally formulated, the law describes the magnetic field set up by a steady current density. More recently, by a simple analogy between magnetostatics and fluid dynamics, the same law has been used to calculate the velocity of air induced by vortex lines in aerodynamic systems.

The Biot-Savart law is fundamental to magnetostatics just as Coulomb's law is to electrostatics. The Biot-Savart law follows from the Lorentz transformations of the electric field of a point-like electric charge, which results in a magnetic field, and is fully consistent with Ampère's law, much as Coulomb's law is consistent with Gauss' law.

In particular, if we define a differential element of current

$I\; d\backslash mathbf\{l\}$

then the corresponding differential element of magnetic field is

$d\backslash mathbf\{B\}\; =\; K\_m\; \backslash frac\{I\; d\backslash mathbf\{l\}\; \backslash times\; \backslash mathbf\{\backslash hat\; r\}\}\{r^2\}$

where

$K\_m\; =\; \backslash frac\{\backslash mu\_0\}\{4\backslash pi\}\; \backslash ,$ is the magnetic constant

$I\backslash mathbf\{\}$ is the current, measured in amperes

$d\backslash mathbf\{l\}$ is the differential length vector of the current element

$\backslash mathbf\{\backslash hat\; r\}$ is the unit displacement vector from the current element to the field point and

$r\backslash mathbf\{\}$ is the distance from the current element to the field point

Forms

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General

In the magnetostatic approximation, the magnetic field can be determined from the current I if all current densities J are known:

$\backslash mathbf\{B\}=\; \backslash frac\{\backslash mu\_\{0\}\}\{4\backslash pi\}\; \backslash int\{\backslash frac\{d\backslash mathbf\{I\}\; \backslash times\; \backslash mathbf\{\backslash hat\; r\}\}\{r^2\}\}$

where

$\backslash mathbf\{\backslash hat\{r\}\}\; =\; \{\; \backslash mathbf\{r\}\; \backslash over\; r\; \}$ is the unit vector in the direction of r.

Constant uniform current

In the special case of a constant, uniform current I, the magnetic field B is

$\backslash mathbf\; B\; =\; K\_m\; I\; \backslash int\; \backslash frac\{d\backslash mathbf\; l\; \backslash times\; \backslash mathbf\{\backslash hat\; r\}\}\{r^2\}$

Point charge at constant velocity

In the special case of a charged point particle $q\backslash mathbf\{\}$ moving at a constant velocity $\backslash mathbf\{v\}$, then the equation above reduces to a magnetic field of the form:

$\backslash mathbf\{B\}\; =\; K\_m\; \backslash frac\{\; q\; \backslash mathbf\{v\}\; \backslash times\; \backslash mathbf\{\backslash hat\{r\}\}\}\{r^2\}$

However, this equation can only be considered to be a nonrelativistic approximation. Strictly speaking, the Biot-Savart law holds only for steady-state currents (zero divergence of the current density or, equivalently, zero time derivative of the charge density at all points in space), and the motion of a single point charge does not constitute such a steady-state current.

Magnetic responses applications

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The Biot-Savart law can be used in the calculation of magnetic responses, e.g. chemical shieldings or magnetic susceptibilities, provided that the current density can be obtained from a quantum mechanical calculation of theory.

The limitations of the law

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It is important to note that this law has limitations. Since it assumes infinite density of the current in infinitely thin conducting wire, the $r^2$ in the divisor of the formula will drive the resulting magnetic field to infinity as we move towards the axis of the conductor.

We know from established experiments that this does not correspond to reality. In fact, with uniform distribution of current across the conductor, the magnetic field on the conductor axis is zero (for a conductor in the shape of a perfect, very long conducting cylinder), as encompassed current is also zero (see Maxwell's equations).

Neglecting quantum effects, the magnetic field falls towards zero from the surface of conductor to its core, instead of rising to infinity. This has a great importance when calculating magnetic field, such as the field of a dense coil of a large magnet.

Aerodynamics applications

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The Biot-Savart law is also used to calculate the velocity induced by vortex lines in aerodynamic theory. (The theory is closely parallel to that of magnetostatics; vorticity corresponds to current, and induced velocity to magnetic field strength.)

For a vortex line of infinite length, the induced velocity at a point is given by

$v\; =\; \backslash frac\{\backslash Gamma\}\{4\backslash pi\; d\}$

where

Γ is the strength of the vortex

d is the perpendicular distance between the point and the vortex line.

This is a limiting case of the formula for vortex segments of finite length:

$v\; =\; \backslash frac\{\backslash Gamma\}\{8\; \backslash pi\; d\}\; \backslash leftA\; +\; \backslash cos\; B\; \backslash right$

where A and B are the (signed) angles between the line and the two ends of the segment.

See also

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People

• Jean Baptiste Biot
• Felix Savart
• Andre-Marie Ampere
• James Clerk Maxwell

Electromagnetism

• Maxwell's equations
• Ampere's Law
• magnetism
• Coulomb's law

Aerodynamics

• vorticity

References

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External links

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• MISN-0-125 The Ampere-Laplace-Biot-Savart Law (PDF file) by Orilla McHarris and Peter Signell for Project PHYSNET.

Magnetostatics | Aerodynamics | Introductory physics | Eponymous laws

Biot-Savart-Gesetz | Ley de Biot-Savart | Loi de Biot et Savart | 비오-사바르의 법칙 | חוק ביו-סבר | ビオ・サバールの法則 | Prawo Biota-Savarta | Lei de Biot-Savart | Biot'n ja Savartin laki | Biot-Savarts lag