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.
