Biot-Savart law
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The Biot-Savart Law is an equation in electromagnetism that describes the magnetic field vector B in terms of the magnitude and direction of the source electric current, the distance from the source electric current, and the magnetic permeability weighting factor.
The significance of the Biot-Savart Law is that it is an inverse square law solution to Ampère's Law. It is also a solution to the vorticity equation curl A = B, i.e., A can be regarded as the magnetic vector potential of B. It therefore provides the B field solution to Maxwell's equations much as the Lorentz force provides the E field solution.
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[edit] Introduction
The Biot-Savart law is used to compute the magnetic field generated by a steady current, i.e. a continual flow of charges, for example through a wire, which is constant in time and in which charge is neither building up nor depleting at any point. The equation is as follows:
- <math> d\mathbf{B} = \frac{\mu_0}{4\pi} \frac{I d\mathbf{l} \times \mathbf{\hat r}}{r^2} </math>
(in SI units), where
- <math>\scriptstyle{I}</math> is the current,
- <math>\scriptstyle{d\mathbf{l}}</math> is a vector, whose magnitude is the length of the differential element of the wire, and whose direction is the direction of conventional current,
- <math>\scriptstyle{d\mathbf{B}}</math> is the differential contribution to the magnetic field resulting from this differential element of wire,
- <math>\scriptstyle{\mu_0}</math> is the magnetic constant,
- <math>\scriptstyle{\mathbf{\hat r}}</math> is the unit displacement vector from the wire element to the point at which the field is being computed,
- <math>\scriptstyle{r}</math> is the distance from the wire element to the point at which the field is being computed, and
- <math>\scriptstyle{\times}</math> denotes cross product.
To apply the equation, you choose a point in space at which you want to compute the magnetic field. Holding that point fixed, you integrate over the path of the current(s) to find the total magnetic field at that point. The application of this law implicitly relies on the superposition principle for magnetic fields, i.e. the fact that the magnetic field is a vector sum of the field created by each infinitesimal section of the wire individually.[1]
The formulation given above works well when the current can be approximated as running through an infinitely-narrow wire. If the flow of current has some thickness, the proper formulation of the Biot-Savart law (again in SI units) is:
- <math> d\mathbf{B} = \frac{\mu_0}{4\pi} \frac{(\mathbf{J}\, dV) \times \mathbf{\hat r}}{r^2} </math>
where
- <math>\scriptstyle{dV}</math> is the differential element of volume and
- <math>\scriptstyle{\mathbf{J}}</math> is the current density vector in that volume.
The Biot-Savart law is fundamental to magnetostatics, playing a similar role to Coulomb's law in electrostatics.
[edit] Forms
[edit] General
In the magnetostatic approximation, the magnetic field can be determined if the current density j is known:
- <math>\mathbf{B}= K_m\int{\frac{\mathbf{j} \times \mathbf{\hat r}}{r^2}dV}</math>
where
- <math>\mathbf{\hat{r}} = { \mathbf{r} \over r } </math> is the unit vector in the direction of r.
- <math> \ dV</math> = is the differential element of volume.
[edit] Constant uniform current
In the special case of a constant, uniform current I, the magnetic field B is
- <math> \mathbf B = K_m I \int \frac{d\mathbf l \times \mathbf{\hat r}}{r^2}</math>
[edit] Point charge at constant velocity
In the special case of a charged point particle <math>q\mathbf{}</math> moving at a constant, non-relativistic velocity <math>\mathbf{v}</math>, then the magnetic field is[2]:
- <math> \mathbf{B} = K_m \frac{ q \mathbf{v} \times \mathbf{\hat{r}}}{r^2} </math>
This equation is also sometimes called the Biot-Savart law, due to its closely analogous form to the "standard" Biot-Savart law given above.
[edit] Microscopic Scale
On the microscopic scale, the Biot-Savart law becomes,
- <math> \mathbf{H} = \epsilon \mathbf{v} \times \mathbf{E} </math>
where the solution to <math>\mathbf{E}</math> is the Coulomb force, and where,
- <math>\mathbf{B} = \mu \mathbf{H}</math>
and hence,
- <math>
\mathbf{B} = \mathbf{v}\times \frac{1}{c^2}\mathbf{E} </math>
[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 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 <math>\mathbf{H}</math> was directly equated with pure vorticity (spin), whereas <math>\mathbf{B}</math> 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
<math>\mathbf{B} = \mu \mathbf{H}</math>
was essentially a rotational analogy to the linear electric current relationship,
(2) Electric Convection Current
<math>\mathbf{J} = \rho \mathbf{v}</math>
where ρ is electric charge density. <math>\mathbf{B}</math> was seen as a kind of magnetic current of vortices aligned in their axial planes, with <math>\mathbf{H}</math> 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 <math>\mathbf{B}</math> 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 <math>\mathbf{B}</math> in electromagnetism.
In electromagnetism the <math>\mathbf{B}</math> 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 <math>\mathbf{B}</math> lines in isolation, we see exactly the aerodynamic scenario in so much as that <math>\mathbf{B}</math> is the vortex axis and <math>\mathbf{H}</math> 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
- <math>v = \frac{\Gamma}{2\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}{4 \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.
[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.
- ^ The superposition principle holds for the electric and magnetic fields because they are the solution to a set of linear differential equations, namely Maxwell's equations, where the current is one of the "source terms".
- ^ See Griffiths, Example 10.4
[edit] External links
- Electromagnetism, B. Crowell, Fullerton College
- MISN-0-125 The Ampere-Laplace-Biot-Savart Law (PDF file) by Orilla McHarris and Peter Signell for Project PHYSNET.ca:Llei de Biot-Savart
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