Biot–Savart law[zt]

Aerodynamics applications

The Biot-Savart law is also used to calculate the velocity induced by vortex lines in aerodynamic theory.

In the aerodynamic application, the roles of vorticity and current are reversed as when compared to the magnetic application.

In Maxwell's 1861 paper 'On Physical Lines of Force', magnetic field strength H was directly equated with pure vorticity (spin), whereas B was a weighted vorticity that was weighted for the density of the vortex sea. Maxwell considered magnetic permeability μ to be a measure of the density of the vortex sea. Hence the relationship,

(1) Magnetic Induction Current

was essentially a rotational analogy to the linear electric current relationship,

(2) Electric Convection Current

where ρ is electric charge density. B was seen as a kind of magnetic current of vortices aligned in their axial planes, with H being the circumferential velocity of the vortices.

The electric current equation can be viewed as a convective current of electric charge that involves linear motion. By analogy, the magnetic equation is an inductive current involving spin. There is no linear motion in the inductive current along the direction of the B vector. The magnetic inductive current represents lines of force. In particular, it represents lines of inverse square law force.

In aerodynamics the induced air currents are forming solenoidal rings around a vortex axis that is playing the role that electric current plays in magnetism. This puts the air currents of aerodynamics into the equivalent role of the magnetic induction vector B in electromagnetism.

In electromagnetism the B lines form solenoidal rings around the source electric current, whereas in aerodynamics, the air currents form solenoidal rings around the source vortex axis.

Hence in electromagnetism, the vortex plays the role of 'effect' whereas in aerodynamics, the vortex plays the role of 'cause'. Yet when we look at the B lines in isolation, we see exactly the aerodynamic scenario in so much as that B is the vortex axis and H is the circumferential velocity as in Maxwell's 1861 paper.

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

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:

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

track back:http://en.wikipedia.org/wiki/Biot-Savart_law