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| {{Electromagnetism|cTopic=Magnetostatics}}
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| In [[physics]], particularly [[electromagnetism]], the '''Biot–Savart law''' ({{IPAc-en|ˈ|b|iː|oʊ|_|s|ə|ˈ|v|ɑr}} or {{IPAc-en|ˈ|b|j|oʊ|_|s|ə|ˈ|v|ɑr}})<ref>[http://dictionary.reference.com/browse/biot+savart+law?qsrc=2446]</ref>
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| is an equation describing the [[magnetic field]] generated by an [[electric current]]. It relates the magnetic field to the magnitude, direction, length, and proximity of the electric current. The law is valid in the [[Magnetostatics|magnetostatic approximation]], and is consistent with both [[Ampère's circuital law]] and [[Gauss's law for magnetism]].<ref>{{cite book | author=Jackson, John David | title=Classical Electrodynamics | edition=3rd ed. | location=New York | publisher=Wiley | year=1999 | isbn=0-471-30932-X | page = Chapter 5 | nopp=true}}</ref> It is named for [[Jean-Baptiste Biot]] and [[Félix Savart]] who discovered this relationship in 1820. | |
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| ==Equation==
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| ===Electric currents (along closed curve)===
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| The Biot–Savart law is used for computing the resultant [[magnetic field]] '''B''' at position '''r''' generated by a ''steady'' [[electric current|current]] ''I'' (for example due to a wire): a continual flow of [[electric charge|charges]] which is constant in time and the charge neither accumulates nor depletes at any point. The law is a physical example of a [[line integral]]: evaluated over the path ''C'' the electric currents flow. The equation in [[SI]] units is<ref>Electromagnetism (2nd Edition), I.S. Grant, W.R. Phillips, Manchester Physics, John Wiley & Sons, 2008, ISBN 978-0-471-92712-9</ref>
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| {{Equation box 1
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| |indent =:
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| |equation = <math> \mathbf{B} = \frac{\mu_0}{4\pi} \int_C \frac{I d\mathbf{l} \times \mathbf{r}}{|\mathbf{r}|^3}</math>
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| |cellpadding
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| |border colour = #50C878
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| |background colour = #ECFCF4}}
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| where '''r''' is the full [[displacement vector]] from the wire element to the point at which the field is being computed and <math>\mathbf{\hat{r}}</math> is the [[unit vector]] of '''r'''. Using <math>\mathbf{\hat{r}}</math>, the equation can be equivalently written as:
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| :<math> \mathbf{B} = \frac{\mu_0}{4\pi}\int_C \frac{I d\mathbf{l} \times \mathbf{\hat r}}{|\mathbf{r}|^2}</math> | |
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| where ''d'''''l''' is a vector whose magnitude is the length of the [[infinitesimal|differential]] element of the wire, in the direction of ''[[conventional current]]'', and μ<sub>0</sub> is the [[magnetic constant]]. The symbols in boldface denote [[vector (geometry)|vector quantities]].
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| The integral is usually around a [[closed curve]], since electric currents can only flow around closed paths. An infinitely long wire (as used in the definition of the SI unit of electric current - the [[Ampere]]) is a counter-example.
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| To apply the equation, the point in space where the magnetic field is to be calculated is arbitrarily chosen. Holding that point fixed, the line integral over the path of the electric currents is calculated 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.<ref>The superposition principle holds for the electric and magnetic fields because they are the solution to a set of [[linear differential equation]]s, namely [[Maxwell's equations]], where the current is one of the "source terms".</ref>
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| ===Electric currents (throughout conductor volume)===
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| 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 [[SI]] units) is:
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| {{Equation box 1
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| |indent =:
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| |equation = <math> \mathbf{B} = \frac{\mu_0}{4\pi}\iiint_V\ \frac{(\mathbf{J}\, dV) \times \mathbf{r}}{r^3}</math>
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| |cellpadding= 6
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| |border colour = #0073CF
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| |background colour=#F5FFFA}}
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| or equivalently
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| :<math> \mathbf{B} = \frac{\mu_0}{4\pi}\iiint_V \ \frac{(\mathbf{J}\, dV) \times \mathbf{\hat r}}{r^2} </math>
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| where ''dV'' is the [[Differential (infinitesimal)|differential element]] of [[volume]] and '''J''' is the [[current density]] vector in that volume.
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| In this case the integral is over the volume of the conductor.
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| The Biot–Savart law is fundamental to [[magnetostatics]], playing a similar role to [[Coulomb's law]] in [[electrostatics]]. When magnetostatics does not apply, the Biot–Savart law should be replaced by [[Jefimenko's equations]].
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| ===Constant uniform current===
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| In the special case of a steady constant current ''I'', the magnetic field '''B''' is
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| :<math> \mathbf B = \frac{\mu_0}{4\pi} I \int_C \frac{d\mathbf l \times \mathbf{\hat r}}{r^2}</math>
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| i.e. the current can be taken out the integral.
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| ===Point charge at constant velocity===
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| In the case of a point [[charged particle]] ''q'' moving at a constant [[velocity]] '''v''', [[Maxwell's equations]] give the following expression for the [[electric field]] and magnetic field:<ref name=Griffiths/>
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| :<math> \mathbf{E} = \frac{q}{4\pi \epsilon_0} \frac{1-v^2/c^2}{(1-v^2\sin^2\theta/c^2)^{3/2}}\frac{\mathbf{\hat r}}{r^2}</math>
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| :<math> \mathbf{B} = \mathbf{v} \times \frac{1}{c^2} \mathbf{E} </math>
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| where '''r̂''' 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 '''v''' and '''r'''.
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| When ''v''<sup>2</sup> ≪ ''c''<sup>2</sup>, the electric field and magnetic field can be approximated as<ref name=Griffiths>{{cite book | author=Griffiths, David J.|title=Introduction to Electrodynamics (3rd ed.)| publisher=Prentice Hall|pages= 222–224, 435–440 |year=1998 |isbn=0-13-805326-X}}</ref>
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| :<math> \mathbf{E} =\frac{q}{4\pi\epsilon_0}\ \frac{\mathbf{\hat r}}{r^2} </math>
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| :<math> \mathbf{B} =\frac{\mu_0 q \mathbf{v}}{4\pi} \times \frac{\mathbf{\hat r}}{r^2} </math>
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| These equations are called the "Biot–Savart law for a point charge"<ref>http://maxwell.ucdavis.edu/~electro/magnetic_field/pointcharge.html</ref> due to its closely analogous form to the "standard" Biot–Savart law given previously. These equations were first derived by [[Oliver Heaviside]] in 1888.
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| ==Magnetic responses applications==
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| The Biot–Savart law can be used in the calculation of magnetic responses even at the atomic or molecular level, e.g. [[chemical shift|chemical shielding]]s or [[magnetic susceptibility|magnetic susceptibilities]], provided that the current density can be obtained from a quantum mechanical calculation or theory.
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| ==Aerodynamics applications==<!-- This section is linked from [[Vorticity]] -->
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| [[File:Vortex filament (Biot-Savart law illustration).png|thumb|right|200px| The figure shows the velocity (<math>dV</math>) induced at a point P by an element of vortex filament (<math>dL</math>) of strength <math>\Gamma</math>.]]
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| The Biot–Savart law is also used in [[aerodynamic]] theory to calculate the velocity induced by [[vortex|vortex lines]].
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| In the [[aerodynamic]] application, the roles of vorticity and current are reversed in comparison to the magnetic application.
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| In Maxwell's 1861 paper 'On Physical Lines of Force',<ref>{{cite web|url=http://commons.wikimedia.org/wiki/File:On_Physical_Lines_of_Force.pdf |title=On Physical Lines of Force |first=J. C. |last=Maxwell |work=Wikimedia commons |accessdate=25 December 2011}}</ref> 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,
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| {{ordered list
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| |1='''Magnetic induction current'''
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| :<math>\mathbf{B} = \mu \mathbf{H}</math>
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| was essentially a rotational analogy to the linear electric current relationship,
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| |2='''Electric convection current'''
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| :<math>\mathbf{J} = \rho \mathbf{v}</math>
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| 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.
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| }}
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| 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.
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| 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.
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| 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.
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| 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.
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| ''In two dimensions'', for a vortex line of infinite length, the induced velocity at a point is given by
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| :<math>v = \frac{\Gamma}{2\pi r}</math>
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| where Γ is the strength of the vortex and ''r'' is the perpendicular distance between the point and the vortex line.
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| This is a limiting case of the formula for vortex segments of finite length:
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| :<math>v = \frac{\Gamma}{4 \pi r} \left[\cos A - \cos B \right]</math>
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| where ''A'' and ''B'' are the (signed) angles between the line and the two ends of the segment.
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| ==The Biot–Savart law, Ampère's circuital law, and Gauss's law for magnetism==
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| {{see also|Curl (mathematics)|vector calculus identities}}
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| The magnetic field '''B''' as calculated from the Biot–Savart law will always satisfy [[Ampère's circuital law]] and [[Gauss's law for magnetism]].<ref name=GaussAmpereProof>See Jackson, page 178–79 or Griffiths p. 222–24. The presentation in Griffiths is particularly thorough, with all the details spelled out.</ref>
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| :{| class="toccolours collapsible collapsed" width="80%" style="text-align:left"
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| !Outline of proof that a magnetic field calculated by the Biot–Savart law will always satisfy [[Gauss's law for magnetism]] and [[Ampère's circuital law|Ampère's law]].<ref name=GaussAmpereProof/>
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| |-
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| |Starting with the Biot–Savart law:
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| :<math>\mathbf{B}(\mathbf{r}) = \frac{\mu_0}{4\pi} \iiint_V d^3r' \mathbf{J}(\mathbf{r}')\times \frac{\mathbf{r}-\mathbf{r}'}{|\mathbf{r}-\mathbf{r}'|^3}</math>
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| Substituting the relation
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| :<math>\frac{\mathbf{r}-\mathbf{r}'}{|\mathbf{r}-\mathbf{r}'|^3} = -\nabla\left(\frac{1}{|\mathbf{r}-\mathbf{r}'|}\right)</math>
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| and using the [[product rule]] for curls, as well as the fact that '''J''' does not depend on the unprimed coordinates, this equation can be rewritten as<ref name=GaussAmpereProof/>
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| :<math>\mathbf{B}(\mathbf{r}) = \frac{\mu_0}{4\pi} \nabla\times\iiint_V d^3r' \frac{\mathbf{J}(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|}</math>
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| 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 [[Vector calculus identities#Curl of the curl|curl of a curl]], and again using the fact that '''J''' does not depend on the unprimed coordinates, we eventually get the result<ref name=GaussAmpereProof/>
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| :<math>\nabla\times\mathbf{B} = \frac{\mu_0}{4\pi}\nabla\iiint_V d^3r' \mathbf{J}(\mathbf{r}')\cdot\nabla\left(\frac{1}{|\mathbf{r}-\mathbf{r}'|}\right) - \frac{\mu_0}{4\pi}\iiint_V d^3r' \mathbf{J}(\mathbf{r}')\nabla^2\left(\frac{1}{|\mathbf{r}-\mathbf{r}'|}\right)</math>
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| Finally, plugging in the relations<ref name=GaussAmpereProof/>
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| :<math>\nabla\left(\frac{1}{|\mathbf{r}-\mathbf{r}'|}\right) = -\nabla' \left(\frac{1}{|\mathbf{r}-\mathbf{r}'|}\right),</math>
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| :<math>\nabla^2\left(\frac{1}{|\mathbf{r}-\mathbf{r}'|}\right) = -4\pi \delta(\mathbf{r}-\mathbf{r}')</math>
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| (where δ is the [[Dirac delta function]]), using the fact that the divergence of '''J''' is zero (due to the assumption of [[magnetostatics]]), and performing an [[integration by parts]], the result turns out to be<ref name=GaussAmpereProof/>
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| :<math>\nabla\times \mathbf{B} = \mu_0 \mathbf{J}</math>
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| i.e. [[Ampère's circuital law|Ampère's law]] (without Maxwell's correction, the [[displacement current]].).
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| |}
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| ==See also==
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| ===People===
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| * [[Jean-Baptiste Biot]]
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| * [[Félix Savart]]
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| * [[André-Marie Ampère]]
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| * [[James Clerk Maxwell]]
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| ===Electromagnetism===
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| * [[Maxwell's equations]]
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| * [[Ampère's law]]
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| * [[Magnetism]]
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| * [[Coulomb's law]]
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| * [[Darwin Lagrangian]]
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| ==Notes==
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| <references/>
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| ==References==
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| * {{cite book | last=Griffiths | first=David J. | title=Introduction to Electrodynamics | edition=3rd | publisher=Prentice Hall | year=1998 | isbn=0-13-805326-X}}
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| * {{cite book | last=Feynman | first=Richard | title=The Feynman Lectures on Physics| edition=2nd | publisher=Addison-Wesley | year=2005 | isbn=978-0-8053-9045-2}}
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| ==Further reading==
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| * Electricity and Modern Physics (2nd Edition), G.A.G. Bennet, Edward Arnold (UK), 1974, ISBN 0-7131-2459-8
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| * Essential Principles of Physics, P.M. Whelan, M.J. Hodgeson, 2nd Edition, 1978, John Murray, ISBN 0-7195-3382-1
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| * The Cambridge Handbook of Physics Formulas, G. Woan, Cambridge University Press, 2010, ISBN 978-0-521-57507-2.
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| * Physics for Scientists and Engineers - with Modern Physics (6th Edition), P. A. Tipler, G. Mosca, Freeman, 2008, ISBN 0-7167-8964-7
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| * Encyclopaedia of Physics (2nd Edition), R.G. Lerner, G.L. Trigg, VHC publishers, 1991, ISBN (Verlagsgesellschaft) 3-527-26954-1, ISBN (VHC Inc.) 0-89573-752-3
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| * McGraw Hill Encyclopaedia of Physics (2nd Edition), C.B. Parker, 1994, ISBN 0-07-051400-3
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| ==External links==
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| * [http://www.lightandmatter.com/html_books/0sn/ch11/ch11.html Electromagnetism], B. Crowell, Fullerton College
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| * <small>MISN-0-125</small> ''[http://physnet2.pa.msu.edu/home/modules/pdf_modules/m125.pdf The Ampère–Laplace–Biot–Savart Law]'' by Orilla McHarris and Peter Signell for [http://www.physnet.org Project PHYSNET].
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| {{DEFAULTSORT:Biot-Savart law}}
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| [[Category:Aerodynamics]]
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| [[Category:Electromagnetism]]
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