Biot-Savart Law

From Academic Kids

The Biot-Savart Law describes the magnetic field set up by a steadily flowing line current: the field produced by a current element <math>d\mathbf{l}<math> is

<math> d\mathbf{B} = K_m \frac{I d\mathbf{l} \times \mathbf{\hat r}}{r^2} <math>

where

<math>K_m = \frac{\mu_0}{4\pi}<math> is the magnetic constant
I is the current, measured in amperes
<math>\mathbf{\hat r}<math> is the unit displacement vector from the element to the field point

For a particle with charge <math>q<math> moving at a constant velocity <math>\mathbf{v}<math>, the magnetic field produced is

<math> \mathbf{B} = K_m \frac{ q \mathbf{v} \times \mathbf{r}}{r^3} <math>


Hence, integrating, the field produced by current flowing in a loop is

<math> \mathbf B = K_m I \int \frac{d\mathbf l \times \mathbf{\hat r}}{r^2}<math>

The Biot-Savart law is fundamental to magnetostatics just as Coulomb's law is to electrostatics. It is equivalent to Ampère's law.

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

<math>v = \frac{\Gamma}{4\pi d}<math>

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:

<math>v = \frac{\Gamma}{8 \pi d} \left[\cos A - \cos B \right]<math>

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

See also

ja:ビオ・サバールの法則 he:חוק ביו-סבר pl:Prawo Biota-Savarta

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