Biot-Savart Law

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

[itex] d\mathbf{B} = K_m \frac{I d\mathbf{l} \times \mathbf{\hat r}}{r^2} [itex]

where

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

For a particle with charge [itex]q[itex] moving at a constant velocity [itex]\mathbf{v}[itex], the magnetic field produced is

[itex] \mathbf{B} = K_m \frac{ q \mathbf{v} \times \mathbf{r}}{r^3} [itex]

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

[itex] \mathbf B = K_m I \int \frac{d\mathbf l \times \mathbf{\hat r}}{r^2}[itex]

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

[itex]v = \frac{\Gamma}{4\pi d}[itex]

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:

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

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

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