View source for Biot–Savart law

{{Use American English|date = February 2019}}
{{Short description|Important law of classical magnetism}}
{{Electromagnetism|cTopic=Magnetostatics}}
In [[physics]], specifically [[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>[https://www.dictionary.com/browse/biot-savart-law?s=ts "Biot–Savart law"]. ''[[Random House Webster's Unabridged Dictionary]]''.</ref> is an equation describing the [[magnetic field]] generated by a constant [[electric current]]. It relates the magnetic field to the magnitude, direction, length, and proximity of the electric current. 

The Biot–Savart law is fundamental to [[magnetostatics]]. It is valid in the [[Magnetostatics|magnetostatic approximation]] and 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 | location=New York | publisher=Wiley | year=1999 | isbn=0-471-30932-X | page = Chapter 5 | no-pp=true}}</ref> When magnetostatics does not apply, the Biot–Savart law should be replaced by [[Jefimenko's equations]]. The law is named after [[Jean-Baptiste Biot]] and [[Félix Savart]], who discovered this relationship in 1820.

==Equation==
In the following equations, it is assumed that the medium is not magnetic (e.g., vacuum). This allows for straightforward derivation of magnetic field '''B''', while the fundamental vector here is '''H'''.<ref>{{Cite book |last=Landau |first=L. D. |title=The Classical Theory of Fields: Volume 2 |last2=Lifshitz |first2=E. M. |publisher=Butterworth-Heinemann |year=1980 |isbn=978-0750627689 |edition=4th |language=en}}</ref>

===Electric currents (along a closed curve/wire)===
[[File:Magnetic field element (Biot-Savart Law) PRIME.svg|thumb|180px|Shown are the directions of
<math>I d\boldsymbol\ell</math>, <math>\mathbf{\hat r'}</math>, and the value of <math>|\mathbf{r'}|</math> ]]
The Biot–Savart law<ref name=":0" />{{Rp|location=Sec 5-2-1}} is used for computing the resultant [[magnetic field|magnetic flux density]] '''B''' at position '''r''' in 3D-space generated by a filamentary [[electric current|current]] ''I'' (for example due to a wire). A steady (or stationary) current is a continual flow of [[electric charge|charges]] which does not change with time and the charge neither accumulates nor depletes at any point. The law is a physical example of a [[line integral]], being evaluated over the path ''C'' in which the electric currents flow (e.g. the wire). The equation in [[SI]] units [[Tesla (unit)|teslas (T)]] is<ref>Electromagnetism (2nd Edition), I.S. Grant, W.R. Phillips, Manchester Physics, John Wiley & Sons, 2008, {{ISBN|978-0-471-92712-9}}</ref>

{{Equation box 1
|indent =:
|equation = <math> \mathbf{B}(\mathbf{r}) = \frac{\mu_0}{4\pi} \int_C \frac{I \, d\boldsymbol \ell\times\mathbf{r'}}{|\mathbf{r'}|^3}</math>
|cellpadding
|border
|border colour = #50C878
|background colour = #ECFCF4}}

where <math>d\boldsymbol \ell</math> is a vector along the path <math>C</math> whose magnitude is the length of the [[infinitesimal|differential]] element of the wire in the direction of ''[[conventional current]]'', <math>\boldsymbol \ell</math> is a point on path <math>C</math>, and  <math>\mathbf{r'} = \mathbf{r} - \boldsymbol \ell</math> is the full [[displacement vector]] from the wire element (<math>d\boldsymbol \ell</math>) at point <math>\boldsymbol \ell</math> to the point at which the field is being computed (<math>\mathbf{r}</math>), and ''μ''<sub>0</sub> is the [[magnetic constant]]. Alternatively:
<math display="block"> \mathbf{B}(\mathbf{r}) = \frac{\mu_0}{4\pi}\int_C \frac{I \, d\boldsymbol \ell\times\mathbf{\hat r'}}{|\mathbf{r'}|^2}</math>
where <math>\mathbf{\hat r'}</math> is the [[unit vector]] of <math>\mathbf{r'}</math>. The symbols in boldface denote [[vector (geometry)|vector quantities]].

The integral is usually around a [[closed curve]], since stationary electric currents can only flow around closed paths when they are bounded. However, the law also applies to infinitely long wires (this concept was used in the definition of the SI unit of electric current—the [[Ampere]]—until 20 May 2019).

To apply the equation, the point in space where the magnetic field is to be calculated is arbitrarily chosen (<math>\mathbf r</math>). Holding that point fixed, the line integral over the path of the electric current 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>

For example, consider the magnetic field of a loop of radius <math>R</math> carrying a current <math>I.</math> For a point a distance <math>x</math> along the center line of the loop, the magnetic field vector at that point is:<math display="block">\mathbf B(\hat{\mathbf x})=\frac{\mu_0IR^2}{2(x^2+R^2)^{3/2}}\hat{\mathbf x},</math>where <math>\hat{\mathbf x}</math> is the unit vector of along the center-line of the loop (and the loop is taken to be centered at the origin).<ref name=":0">{{Cite web |last=Zhan |first=Marcus |date=2003 |title=Electromagnetic Field Theory: A Problem Solving Approach |url=https://ocw.mit.edu/courses/res-6-002-electromagnetic-field-theory-a-problem-solving-approach-spring-2008/pages/textbook-contents/ |access-date=July 3, 2022 |website=cow.mit.edu |page=}}</ref>{{Rp|location=Sec 5-2, Eqn (25)}} Loops such as the one described appear in devices like the [[Helmholtz coil]], the [[solenoid]], and the [[Magnetic sail#Magsail (MS)|Magsail]] spacecraft propulsion system. Calculation of the magnetic field at points off the center line requires more complex mathematics involving [[elliptic integrals]] that require numerical solution or approximations.<ref name=":14">{{Cite journal |last=Freeland |first=R.M. |date=2015 |title=Mathematics of Magsail |url=https://bis-space.com/shop/product/mathematics-of-magsails/ |journal=Journal of the British Interplanetary Society |volume=68 |pages=306–323 |via=bis-space.com}}</ref>

===Electric current density (throughout conductor volume)===

The formulations given above work well when the current can be approximated as running through an infinitely-narrow wire. If the conductor has some thickness, the proper formulation of the Biot–Savart law (again in [[SI]] units) is:

{{Equation box 1
|indent =:
|equation = <math> \mathbf B (\mathbf r) = \frac{\mu_0}{4\pi}\iiint_V\ \frac{(\mathbf{J}\,dV)\times\mathbf r'}{|\mathbf r'|^3}</math>
|cellpadding= 6
|border
|border colour = #0073CF
|background colour=#F5FFFA}}

where <math>\mathbf{r'}</math> is the vector from dV to the observation point <math>\mathbf{r}</math>, <math>dV</math> is the [[volume element]], and <math>\mathbf{J}</math> is the [[current density]] vector in that volume (in SI in units of A/m<sup>2</sup>).

In terms of unit vector <math>\mathbf{\hat r'}</math>
<math display="block"> \mathbf B (\mathbf r) = \frac{\mu_0}{4\pi}\iiint_V\ dV \frac{\mathbf J\times\mathbf{\hat r'}}{|\mathbf r'|^2} </math>

===Constant uniform current===

In the special case of a uniform constant current ''I'', the magnetic field <math>\mathbf{B}</math> is
<math display="block"> \mathbf{B}(\mathbf{r}) = \frac{\mu_0}{4\pi} I \int_C \frac{d\boldsymbol \ell \times \mathbf{r'}}{|\mathbf{r'}|^3}</math>
i.e., the current can be taken out of the integral.

===Point charge at constant velocity===

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/>
<math display="block">\begin{align}
  \mathbf{E} &= \frac{q}{4\pi \epsilon_0} \frac{1 - \frac{v^2}{c^2}}{\left(1 - \frac{v^2}{c^2}\sin^2\theta\right)^\frac{3}{2}} \frac{\mathbf{\hat r'}}{|\mathbf r'|^2} \\
  \mathbf{H} &= \mathbf{v} \times \mathbf{D} \\ 
  \mathbf{B} &= \frac{1}{c^2} \mathbf{v} \times \mathbf{E}
\end{align}</math>
where <math>\mathbf \hat r'</math> is the unit 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 <math>\mathbf v</math> and <math>\mathbf r'</math>. Alternatively, these can be derived by considering [[Lorentz transformation]] of [[Coulomb's law|Coulomb's force]] (in [[four-force]] form) in the source charge's inertial frame.<ref>{{Cite book |last1=Rosser |first1=W. G. V. |url=https://link.springer.com/book/10.1007/978-1-4899-6559-2 |title=Classical Electromagnetism via Relativity |year=1968 |isbn=978-1-4899-6258-4 |pages=29–42 |language=en |doi=10.1007/978-1-4899-6559-2}}</ref>

When {{math|''v''<sup>2</sup> ≪ ''c''<sup>2</sup>}}, the electric field and magnetic field can be approximated as<ref name=Griffiths>{{cite book|last=Griffiths | first = David J.| title=Introduction to Electrodynamics | edition = 3rd | publisher=Prentice Hall| pages=[https://archive.org/details/introductiontoel00grif_0/page/222 222–224, 435–440]| year=1998| isbn=0-13-805326-X| url=https://archive.org/details/introductiontoel00grif_0/page/222}}</ref>
<math display="block"> \mathbf{E} = \frac{q}{4\pi\varepsilon_0}\ \frac{\mathbf{\hat r'}}{|\mathbf r'|^2} </math>
<math display="block"> \mathbf{B} =\frac{\mu_0 q}{4\pi} \mathbf{v} \times \frac{\mathbf{\hat r'}}{|\mathbf r'|^2} </math>

These equations were first derived by [[Oliver Heaviside]] in 1888.  Some authors<ref>{{cite book |last1=Knight |first1=Randall |title=Physics for Scientists and Engineers |date=2017 |publisher=Pearson Higher Ed. |page=800 |edition=4th}}</ref><ref>{{cite web |url=http://maxwell.ucdavis.edu/~electro/magnetic_field/pointcharge.html |title=Magnetic Field from a Moving Point Charge |access-date=2009-09-30 |url-status=dead |archive-url=https://web.archive.org/web/20090619185915/http://maxwell.ucdavis.edu/~electro/magnetic_field/pointcharge.html |archive-date=2009-06-19 }}</ref> call the above equation for <math>\mathbf{B}</math> the "Biot–Savart law for a point charge" due to its close resemblance to the standard Biot–Savart law.  However, this language is misleading as the Biot–Savart law applies only to steady currents and a point charge moving in space does not constitute a steady current.<ref>See the cautionary footnote in Griffiths p. 219 or the discussion in Jackson p. 175–176.</ref>

==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 shift|chemical shielding]]s or [[magnetic susceptibility|magnetic susceptibilities]], provided that the current density can be obtained from a quantum mechanical calculation or theory.

==Aerodynamics applications==
<!-- This section is linked from [[Vorticity]] -->
[[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>.]]

The Biot–Savart law is also used in [[aerodynamic]] theory to calculate the velocity induced by [[vortex|vortex lines]].

In the [[aerodynamic]] application, the roles of vorticity and current are reversed in comparison to the magnetic application.

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 |access-date=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,
; Magnetic induction current: <math display="block">\mathbf{B} = \mu \mathbf{H}</math> was essentially a rotational analogy to the linear electric current relationship,
; Electric convection current: <math display="block">\mathbf{J} = \rho \mathbf{v} ,</math> 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 form solenoidal rings around a vortex axis. Analogy can be made that the vortex axis is playing the role that electric current plays in [[magnetism]]. This puts the air currents of aerodynamics (fluid velocity field) 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 (velocity) 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 insomuch as '''B''' is the vortex axis and '''H''' is the circumferential velocity as in Maxwell's 1861 paper.

''In two dimensions'', for a vortex line of infinite length, the induced velocity at a point is given by
<math display="block">v = \frac{\Gamma}{2\pi r}</math>
where {{math|Γ}} is the strength of the vortex and ''r'' is the perpendicular distance between the point and the vortex line. This is similar to the magnetic field produced on a plane by an infinitely long straight thin wire normal to the plane.

This is a limiting case of the formula for vortex segments of finite length (similar to a finite wire):
<math display="block">v = \frac{\Gamma}{4 \pi r} \left[\cos A - \cos B \right]</math>
where ''A'' and ''B'' are the (signed) angles between the point and the two ends of the segment.

==The Biot–Savart law, Ampère's circuital law, and Gauss's law for magnetism==

{{see also|Curl (mathematics)|vector calculus identities}}

In a [[magnetostatic]] situation, the magnetic field '''B''' as calculated from the Biot–Savart law will always satisfy [[Gauss's law for magnetism]] and [[Ampère's circuital law|Ampère's law]]:<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>

{{math proof|Outline of proof<ref name=GaussAmpereProof />
|proof=
Starting with the Biot–Savart law:
<math display="block">\mathbf B(\mathbf r) = \frac{\mu_0}{4\pi} \iiint_V d^3l \mathbf J (\mathbf l) \times \frac{\mathbf r - \mathbf l}{|\mathbf r - \mathbf l|^3}</math>

Substituting the relation
<math display="block">\frac{\mathbf r - \mathbf l}{|\mathbf r - \mathbf l|^3} = -\nabla\left(\frac{1}{|\mathbf r - \mathbf l|}\right)</math>
and using the [[Vector calculus identities#Product rule|product rule for curls]], as well as the fact that '''J''' does not depend on <math>\mathbf r</math>, this equation can be rewritten as<ref name=GaussAmpereProof/>
<math display="block">\mathbf B (\mathbf r) = \frac{\mu_0}{4\pi} \nabla\times\iiint_V d^3l \frac{\mathbf J (\mathbf l)}{|\mathbf r - \mathbf l|}</math>

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 curl|curl of a curl]], and again using the fact that '''J''' does not depend on <math>\mathbf r</math>, we eventually get the result<ref name=GaussAmpereProof/>
<math display="block">\nabla\times\mathbf B = \frac{\mu_0}{4\pi}\nabla\iiint_V d^3l \mathbf J (\mathbf l)\cdot\nabla\left(\frac{1}{|\mathbf r - \mathbf l|}\right) = \frac{\mu_0}{4\pi}\iiint_V d^3l \mathbf J (\mathbf l)\nabla^2\left(\frac{1}{|\mathbf r - \mathbf l|}\right)</math>

Finally, plugging in the relations<ref name=GaussAmpereProof/>
<math display="block">\begin{align}
    \nabla\left(\frac{1}{|\mathbf r - \mathbf l|}\right) &= -\nabla_l \left(\frac{1}{|\mathbf r - \mathbf l|}\right), \\
  \nabla^2\left(\frac{1}{|\mathbf r - \mathbf l|}\right) &= -4\pi \delta(\mathbf r - \mathbf l)
\end{align}</math>
(where {{math|''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/>
<math display="block">\nabla\times \mathbf B = \mu_0 \mathbf J</math>
i.e. [[Ampère's circuital law|Ampère's law]]. (Due to the assumption of [[magnetostatics]], <math>\partial \mathbf E / \partial t = \mathbf 0</math>, so there is no extra [[Ampère's circuital law#Extending the original law: the Maxwell–Ampère equation|displacement current term]] in Ampère's law.)
}}

In a ''non''-magnetostatic situation, the Biot–Savart law ceases to be true (it is superseded by [[Jefimenko's equations]]), while [[Gauss's law for magnetism]] and the [[Ampère's circuital law|Maxwell–Ampère law]] are still true.

==Theoretical background==
Initially, the Biot–Savart law was discovered experimentally, then this law was derived in different ways theoretically. In [[The Feynman Lectures on Physics]], at first, the similarity of expressions for the [[electric potential]] outside the static distribution of charges and the [[magnetic vector potential]] outside the system of continuously distributed currents is emphasized, and then the magnetic field is calculated through the curl from the vector potential.<ref>[https://feynmanlectures.caltech.edu/II_14.html The Feynman Lectures on Physics Vol. II Ch. 14: The Magnetic Field in Various Situations]</ref> Another approach involves a general solution of the [[Inhomogeneous electromagnetic wave equation#A and %CF%86 potential fields| inhomogeneous wave equation]] for the vector potential in the case of constant currents.<ref> David Tong. Lectures on Electromagnetism. University of Cambridge, Part IB and Part II Mathematical Tripos (2015). http://www.damtp.cam.ac.uk/user/tong/em.html. </ref> The magnetic field can also be calculated as a consequence of the [[Lorentz transformations]] for the electromagnetic force acting from one charged particle on another particle.<ref> Daniel Zile and James Overdui. Derivation of the Biot-Savart Law from Coulomb’s Law and Implications for Gravity. APS April Meeting 2014, abstract id. D1.033. https://doi.org/10.1103/BAPS.2014.APRIL.D1.33. </ref> Two other ways of deriving the Biot–Savart law include: 1) Lorentz transformation of the [[electromagnetic tensor]] components from a moving frame of reference, where there is only an electric field of some distribution of charges, into a stationary frame of reference, in which these charges move. 2) the use of the method of [[retarded potential]]s.

==See also==

===People===
* [[André-Marie Ampère]]
* [[James Clerk Maxwell]]
* [[Pierre-Simon Laplace]]

===Electromagnetism===
* [[Darwin Lagrangian]]

==Notes==
<references/>

==References==
* {{cite book | last=Griffiths | first=David J. | title=Introduction to Electrodynamics | edition=3rd | publisher=Prentice Hall | year=1998 | isbn=0-13-805326-X | url=https://archive.org/details/introductiontoel00grif_0 }}
* {{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 | url = https://feynmanlectures.caltech.edu}}

==Further reading==

* Electricity and Modern Physics (2nd Edition), G.A.G. Bennet, Edward Arnold (UK), 1974, {{ISBN|0-7131-2459-8}}
* Essential Principles of Physics, P.M. Whelan, M.J. Hodgeson, 2nd Edition, 1978, John Murray, {{ISBN|0-7195-3382-1}}
* The Cambridge Handbook of Physics Formulas, G. Woan, Cambridge University Press, 2010, {{ISBN|978-0-521-57507-2}}.
* Physics for Scientists and Engineers - with Modern Physics (6th Edition), P. A. Tipler, G. Mosca, Freeman, 2008, {{ISBN|0-7167-8964-7}}
* Encyclopaedia of Physics (2nd Edition), [[Rita G. Lerner|R.G. Lerner]], G.L. Trigg, VHC publishers, 1991, ISBN (Verlagsgesellschaft) 3-527-26954-1, ISBN (VHC Inc.) 0-89573-752-3
* McGraw Hill Encyclopaedia of Physics (2nd Edition), C.B. Parker, 1994, {{ISBN|0-07-051400-3}}

==External links==
*{{Commons category inline}}
* <small>MISN-0-125</small> ''[http://www.physnet.org/modules/pdf_modules/m125.pdf The Ampère–Laplace–Biot–Savart Law]'' by Orilla McHarris and Peter Signell for [http://www.physnet.org Project PHYSNET].


{{DEFAULTSORT:Biot-Savart law}}
[[Category:Aerodynamics]]
[[Category:Electromagnetism]]
[[Category:Eponymous laws of physics]]