math lessons - 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 $d\mathbf{l}$ is

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

where

$K_m = \frac{\mu_0}{4\pi}$ is the magnetic constant

I is the current, measured in amperes

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

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

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

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

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

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:

$v = \frac{\Gamma}{8 \pi d} [cos(A) - cos(B)]$

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

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