Biot-Savart law

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The Biot-Savart law, also called Laplace's law, is a physical law with applications in electromagnetics and other areas requiring vector field analysis, such as vortical fluid-flow. As originally formulated, the law describes the magnetic field set up by a steady current density. Mathematically, the Biot-Savart law provides an inverse to the curl operation; the result is unique up to gauge transformation. As such it has numerous additional applications.

Contents

[edit] Introduction

The Biot-Savart law is fundamental to magnetostatics just as Coulomb's law is to electrostatics. The Biot-Savart law follows from the Lorentz transformations of the electric field of a point-like electric charge, which results in a magnetic field, and is fully consistent with Ampère's law, much as Coulomb's law is consistent with Gauss' law.

In particular, if we define a differential element of current

I d\mathbf{l}

then the corresponding differential element of magnetic field is

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

where

K_m = \frac{\mu_0}{4\pi} \,, where μ0 is the magnetic constant
I\mathbf{} is the current, measured in amperes
d\mathbf{l} is the differential length vector of the current element
\mathbf{\hat r} is the unit displacement vector from the current element to the field point and
r\mathbf{} is the distance from the current element to the field point

[edit] Forms

[edit] General

In the magnetostatic approximation, the magnetic field can be determined if the current density j is known:

\mathbf{B}= K_m\int{\frac{\mathbf{j} \times \mathbf{\hat r}}{r^2}dv}

where

\mathbf{\hat{r}} = { \mathbf{r} \over r } is the unit vector in the direction of r.
dv = is the differential unit of volume.

[edit] Constant uniform current

In the special case of a constant, uniform current I, the magnetic field B is

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

[edit] Point charge at constant velocity

In the special case of a charged point particle q\mathbf{} moving at a constant velocity \mathbf{v}, then the equation above reduces to a magnetic field of the form:

\mathbf{B} = K_m \frac{ q \mathbf{v} \times \mathbf{\hat{r}}}{r^2}

However, this equation can only be considered to be a nonrelativistic approximation. Strictly speaking, the Biot-Savart law holds only for steady-state currents (zero divergence of the current density or, equivalently, zero time derivative of the charge density at all points in space), and the motion of a single point charge does not constitute such a steady-state current.

[edit] Magnetic responses applications

The Biot-Savart law can be used in the calculation of magnetic responses even at the atomic or molecular level, e.g. chemical shieldings or magnetic susceptibilities, provided that the current density can be obtained from a quantum mechanical calculation or theory.

[edit] Aerodynamics applications

The figure shows the velocity induced at a point P (dV) by a vortex filament of strength Γ.
The figure shows the velocity induced at a point P (dV) by a vortex filament of strength Γ.

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

v = \frac{\Gamma}{2\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}{4 \pi d} \left[\cos A + \cos B \right]

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

[edit] See also

[edit] People

[edit] Electromagnetism

[edit] Aerodynamics

[edit] References

  • Griffiths, David J. (1998). Introduction to Electrodynamics (3rd ed.). Prentice Hall. ISBN 0-13-805326-X. 

[edit] External links

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