- Biot–Savart law
The Biot–Savart Law is an equation in electromagnetism that describes the
magnetic fieldB generated by an electric current. The vector fieldB depends on the magnitude, direction, length, and proximity of the electric current, and also on a fundamental constant called the magnetic constant. The law is valid in the magnetostatic approximation, and results in a B field consistent with both Ampère's circuital lawand Gauss's law for magnetism. [cite book | author=Jackson, John David | title=Classical Electrodynamics | edition=3rd ed. | location=New York | publisher=Wiley | year=1999 | id=ISBN 0-471-30932-X | page = Chapter 5]
The Biot–Savart law is used to compute the
magnetic fieldgenerated 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:
SIunits), where: is the current,: is a vector, whose magnitude is the length of the differential element of the wire, and whose direction is the direction of conventional current,: is the differential contribution to the magnetic field resulting from this differential element of wire,: is the magnetic constant,: is the displacement unit vectorin the direction pointing from the wire element towards the point at which the field is being computed,: and is the distance from the wire element to the point at which the field is being computed,:the symbols in boldface denote vector quantities.
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 principlefor magnetic fields, i.e. the fact that the magnetic field is a vector sumof the field created by each infinitesimal section of the wire individually. [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".]
Another form of the equation is::
Here, r is the full displacement vector instead of a unit vector and it has r3 in the denominator to compensate. This equation is applied identically to the one above.
The formulations given above work well when the current can be approximated as running through an infinitely-narrow wire. If the current has some thickness, the proper formulation of the Biot-Savart law (again in
where: is the differential element of
volumeand: is the current densityvector in that volume.
The Biot-Savart law is fundamental to
magnetostatics, playing a similar role to Coulomb's lawin electrostatics.
In the magnetostatic approximation, the magnetic field can be determined if the current density j is known:
where is the differential element of volume.
Constant uniform current
In the special case of a constant, uniform current I, the magnetic field B is
Point charge at constant velocity
In the case of a charged point particle q moving at a constant, non-relativistic velocity v, then
Maxwell's equationsgive the following expression for the magnetic field:See Griffiths, Example 10.4]
This equation is also sometimes called the "Biot-Savart law,"fact|date=April 2008 due to its closely analogous form to the "standard" Biot-Savart law given above. Note that the law is only approximate, with its accuracy decreasing as the particle's velocity approaches "c"; this happens because the situation is not perfectly approximated by
This expression can also be rewritten as
where E is the electric field which the charge would create if it were stationary (as given by Coulomb's law), i.e.:.
The exact, relativistic expression is as follows:::where is the vector pointing from the current (non-retarded) position of the particle to the point at which the field is being measured, and "θ" is the angle between the velocity vector and .
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.
The Biot-Savart law is also used to calculate the velocity induced by vortex lines in
aerodynamicapplication, the roles of vorticity and current are reversed as when compared to the magnetic application.
In Maxwell's 1861 paper ' [http://vacuum-physics.com/Maxwell/maxwell_oplf.pdf 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
: is the strength of the vortex: 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.
The Biot-Savart law, Ampère's circuital law, and Gauss's law for magnetism
Here is a demonstration that the magnetic field B as computed from the Biot-Savart law will always satisfy
Ampere's circuital lawand Gauss's law for magnetism. [See Jackson, page 178–9 or Griffiths p. 222–4. The presentation in Griffiths is particularly thorough, with all the details spelled out.] Click "show" in the box below for an outline of the proof.
:Since the divergence of a curl is always zero, this establishes
Gauss's law for magnetism. Next, taking the curl of both sides, using the formula for the curl of a curl (see the article Curl (mathematics)), and again using the fact that J does not depend on the unprimed coordinates, we eventually get the result:
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Biot-Savart law — /bee oh seuh vahr , byoh /, Physics. the law that the magnetic induction near a long, straight conductor, as wire, varies inversely as the distance from the conductor and directly as the intensity of the current in the conductor. [named after J.… … Universalium
biot-savart law — |bēˌōsə|vär , |byōsə noun Usage: usually capitalized B&S Etymology: after Jean B. Biot died 1862 French mathematician and Félix Savart died 1841 French physician & physicist : a statement in electromagnetism: the magnetic intensity at any point… … Useful english dictionary
Biot-Savart's law — Biot ir Savart o dėsnis statusas T sritis automatika atitikmenys: angl. Biot Savart s law vok. Biot Savartsches Gesetz, n rus. закон Био Савара, m pranc. loi de Biot Savart, f ryšiai: sinonimas – Bio ir Savaro dėsnis … Automatikos terminų žodynas
Biot-Savart’s law — Bio ir Savaro dėsnis statusas T sritis fizika atitikmenys: angl. Biot Savart’s law vok. Biot Savartsches Gesetz, n rus. закон Био Савара, m pranc. loi de Biot et Savart, f … Fizikos terminų žodynas
loi de Biot-Savart — Biot ir Savart o dėsnis statusas T sritis automatika atitikmenys: angl. Biot Savart s law vok. Biot Savartsches Gesetz, n rus. закон Био Савара, m pranc. loi de Biot Savart, f ryšiai: sinonimas – Bio ir Savaro dėsnis … Automatikos terminų žodynas
Biot, Jean-Baptiste — ▪ French physicist born April 21, 1774, Paris, France died Feb. 3, 1862, Paris French physicist who helped formulate the Biot Savart law, which concerns magnetic fields, and laid the basis for saccharimetry, a useful technique of analyzing sugar … Universalium
Biot ir Savart'o dėsnis — statusas T sritis automatika atitikmenys: angl. Biot Savart s law vok. Biot Savartsches Gesetz, n rus. закон Био Савара, m pranc. loi de Biot Savart, f ryšiai: sinonimas – Bio ir Savaro dėsnis … Automatikos terminų žodynas
Biot-Savartsches Gesetz — Biot ir Savart o dėsnis statusas T sritis automatika atitikmenys: angl. Biot Savart s law vok. Biot Savartsches Gesetz, n rus. закон Био Савара, m pranc. loi de Biot Savart, f ryšiai: sinonimas – Bio ir Savaro dėsnis … Automatikos terminų žodynas
Biot, Jean-Baptiste — (1774 1862) physicist, mathematician, astronomer Born in Paris, Jean Baptiste Biot became professor of physics at the collège de france in 1800 and was elected to membership in the Academy of Sciences in 1803. In that year he discovered the … France. A reference guide from Renaissance to the Present
Biot-Savartsches Gesetz — Bio ir Savaro dėsnis statusas T sritis fizika atitikmenys: angl. Biot Savart’s law vok. Biot Savartsches Gesetz, n rus. закон Био Савара, m pranc. loi de Biot et Savart, f … Fizikos terminų žodynas