Biot–Savart Law
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In
physics Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...
, specifically
electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge via electromagnetic fields. The electromagnetic force is one of the four fundamental forces of nature. It is the dominant force in the interacti ...
, the Biot–Savart law ( or ) is an equation describing the
magnetic field A magnetic field (sometimes called B-field) is a physical field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular ...
generated by a constant
electric current An electric current is a flow of charged particles, such as electrons or ions, moving through an electrical conductor or space. It is defined as the net rate of flow of electric charge through a surface. The moving particles are called charge c ...
. It relates the magnetic field to the magnitude, direction, length, and proximity of the electric current. The Biot–Savart law is fundamental to
magnetostatics Magnetostatics is the study of magnetic fields in systems where the electric currents, currents are steady current, steady (not changing with time). It is the magnetic analogue of electrostatics, where the electric charge, charges are stationary ...
. It is valid in the magnetostatic approximation and consistent with both
Ampère's circuital law In classical electromagnetism, Ampère's circuital law, often simply called Ampère's law, and sometimes Oersted's law, relates the circulation of a magnetic field around a closed loop to the electric current passing through the loop. James ...
and
Gauss's law for magnetism In physics, Gauss's law for magnetism is one of the four Maxwell's equations that underlie classical electrodynamics. It states that the magnetic field has divergence equal to zero, in other words, that it is a solenoidal vector field. It is ...
. When magnetostatics does not apply, the Biot–Savart law should be replaced by Jefimenko's equations. The law is named after
Jean-Baptiste Biot Jean-Baptiste Biot (; ; 21 April 1774 – 3 February 1862) was a French people, French physicist, astronomer, and mathematician who co-discovered the Biot–Savart law of magnetostatics with Félix Savart, established the reality of meteorites, ma ...
and
Félix Savart Félix Savart (; ; 30 June 1791, Mézières – 16 March 1841, Paris) was a French physicist and mathematician who is primarily known for the Biot–Savart law of electromagnetism, which he discovered together with his colleague Jean-Baptist ...
, 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.


Electric currents (along a closed curve/wire)

The Biot–Savart law is used for computing the resultant
magnetic flux density A magnetic field (sometimes called B-field) is a physical field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular ...
B at position r in 3D-space generated by a filamentary current ''I'' (for example due to a wire). A steady (or stationary) current is a continual flow of 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 In mathematics, a line integral is an integral where the function (mathematics), function to be integrated is evaluated along a curve. The terms ''path integral'', ''curve integral'', and ''curvilinear integral'' are also used; ''contour integr ...
, being evaluated over the path ''C'' in which the electric currents flow (e.g. the wire). The equation in SI units teslas (T) is where d\boldsymbol \ell is a vector along the path C whose magnitude is the length of the differential element of the wire in the direction of '' conventional current'', \boldsymbol \ell is a point on path C, and \mathbf = \mathbf - \boldsymbol \ell is the full
displacement vector In geometry and mechanics, a displacement is a vector whose length is the shortest distance from the initial to the final position of a point P undergoing motion. It quantifies both the distance and direction of the net or total motion along ...
from the wire element (d\boldsymbol \ell) at point \boldsymbol \ell to the point at which the field is being computed (\mathbf), and ''μ''0 is the
magnetic constant The vacuum magnetic permeability (variously ''vacuum permeability'', ''permeability of free space'', ''permeability of vacuum'', ''magnetic constant'') is the magnetic permeability in a classical vacuum. It is a physical constant, conventionall ...
. Alternatively: \mathbf(\mathbf) = \frac\int_C \frac where \mathbf is the
unit vector In mathematics, a unit vector in a normed vector space is a Vector (mathematics and physics), vector (often a vector (geometry), spatial vector) of Norm (mathematics), length 1. A unit vector is often denoted by a lowercase letter with a circumfle ...
of \mathbf. The symbols in boldface denote vector quantities. The integral is usually around a
closed curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (ge ...
, 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 The ampere ( , ; symbol: A), often shortened to amp,SI supports only the use of symbols and deprecates the use of abbreviations for units. is the unit of electric current in the International System of Units (SI). One ampere is equal to 1 c ...
—until 20 May 2019). To apply the equation, the point in space where the magnetic field is to be calculated is arbitrarily chosen (\mathbf r). 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 The superposition principle, also known as superposition property, states that, for all linear systems, the net response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually. So th ...
for magnetic fields, i.e. the fact that the magnetic field is a
vector sum In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Euclidean vectors can be added and scal ...
of the field created by each infinitesimal section of the wire individually. For example, consider the magnetic field of a loop of radius R carrying a current I. For a point at a distance x along the center line of the loop, the magnetic field vector at that point is:\mathbf B = \hat\mathbf x,where \hat is the unit vector of along the center-line of the loop (and the loop is taken to be centered at the origin). Loops such as the one described appear in devices like the
Helmholtz coil A Helmholtz coil is a device for producing a region of nearly uniform magnetic field, named after the German physicist Hermann von Helmholtz. It consists of two electromagnets on the same axis, carrying an equal electric current in the same direc ...
, the
solenoid upright=1.20, An illustration of a solenoid upright=1.20, Magnetic field created by a seven-loop solenoid (cross-sectional view) described using field lines A solenoid () is a type of electromagnet formed by a helix, helical coil of wire whos ...
, and the Magsail spacecraft propulsion system. Calculation of the magnetic field at points off the center line requires more complex mathematics involving
elliptic integrals In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied by Giulio Fagnano and Leonhard Euler (). Their name originates from their originally arising in ...
that require numerical solution or approximations.


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: where \mathbf is the vector from dV to the observation point \mathbf, dV is the
volume element In mathematics, a volume element provides a means for integrating a function with respect to volume in various coordinate systems such as spherical coordinates and cylindrical coordinates. Thus a volume element is an expression of the form \ma ...
, and \mathbf is the
current density In electromagnetism, current density is the amount of charge per unit time that flows through a unit area of a chosen cross section. The current density vector is defined as a vector whose magnitude is the electric current per cross-sectional ...
vector in that volume (in SI in units of A/m2). In terms of unit vector \mathbf \mathbf B (\mathbf r) = \frac\iiint_V\ dV \frac


Constant uniform current

In the special case of a uniform constant current ''I'', the magnetic field \mathbf is \mathbf(\mathbf) = \frac I \int_C \frac i.e., the current can be taken out of the integral.


Point charge at constant velocity

In the case of a point
charged particle In physics, a charged particle is a particle with an electric charge. For example, some elementary particles, like the electron or quarks are charged. Some composite particles like protons are charged particles. An ion, such as a molecule or atom ...
''q'' moving at a constant
velocity Velocity is a measurement of speed in a certain direction of motion. It is a fundamental concept in kinematics, the branch of classical mechanics that describes the motion of physical objects. Velocity is a vector (geometry), vector Physical q ...
v,
Maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, Electrical network, electr ...
give the following expression for the
electric field An electric field (sometimes called E-field) is a field (physics), physical field that surrounds electrically charged particles such as electrons. In classical electromagnetism, the electric field of a single charge (or group of charges) descri ...
and magnetic field: \begin \mathbf &= \frac \frac \frac \\ ex \mathbf &= \mathbf \times \mathbf \\ ex \mathbf &= \frac \mathbf \times \mathbf \end where * \mathbf \hat r' is the unit vector pointing from the current (non-retarded) position of the particle to the point at which the field is being measured, * \beta = v/c is the speed in units of c and * is the angle between \mathbf v and \mathbf r'. Alternatively, these can be derived by considering
Lorentz transformation In physics, the Lorentz transformations are a six-parameter family of Linear transformation, linear coordinate transformation, transformations from a Frame of Reference, coordinate frame in spacetime to another frame that moves at a constant vel ...
of Coulomb's force (in
four-force In the special theory of relativity, four-force is a four-vector that replaces the classical force. In special relativity The four-force is defined as the rate of change in the four-momentum of a particle with respect to the particle's proper t ...
form) in the source charge's inertial frame. When , the electric field and magnetic field can be approximated as \mathbf = \frac\ \frac \mathbf = q These equations were first derived by
Oliver Heaviside Oliver Heaviside ( ; 18 May 1850 – 3 February 1925) was an English mathematician and physicist who invented a new technique for solving differential equations (equivalent to the Laplace transform), independently developed vector calculus, an ...
in 1888. Some authors call the above equation for \mathbf 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.


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.


Aerodynamics applications

The Biot–Savart law is also used in
aerodynamic Aerodynamics () is the study of the motion of atmosphere of Earth, air, particularly when affected by a solid object, such as an airplane wing. It involves topics covered in the field of fluid dynamics and its subfield of gas dynamics, and is an ...
theory to calculate the velocity induced by vortex lines. In the
aerodynamic Aerodynamics () is the study of the motion of atmosphere of Earth, air, particularly when affected by a solid object, such as an airplane wing. It involves topics covered in the field of fluid dynamics and its subfield of gas dynamics, and is an ...
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', magnetic field strength H was directly equated with pure
vorticity In continuum mechanics, vorticity is a pseudovector (or axial vector) field that describes the local spinning motion of a continuum near some point (the tendency of something to rotate), as would be seen by an observer located at that point an ...
(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: \mathbf = \mu \mathbf was essentially a rotational analogy to the linear electric current relationship, ; Electric convection current: \mathbf = \rho \mathbf , 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 Magnetism is the class of physical attributes that occur through a magnetic field, which allows objects to attract or repel each other. Because both electric currents and magnetic moments of elementary particles give rise to a magnetic field, ...
. 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 v = \frac where 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): v = \frac \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

In a
magnetostatic Magnetostatics is the study of magnetic fields in systems where the currents are steady (not changing with time). It is the magnetic analogue of electrostatics, where the charges are stationary. The magnetization need not be static; the equat ...
situation, the magnetic field B as calculated from the Biot–Savart law will always satisfy
Gauss's law for magnetism In physics, Gauss's law for magnetism is one of the four Maxwell's equations that underlie classical electrodynamics. It states that the magnetic field has divergence equal to zero, in other words, that it is a solenoidal vector field. It is ...
and
Ampère's circuital law In classical electromagnetism, Ampère's circuital law, often simply called Ampère's law, and sometimes Oersted's law, relates the circulation of a magnetic field around a closed loop to the electric current passing through the loop. James ...
:See Jackson, page 178–79 or Griffiths p. 222–24. The presentation in Griffiths is particularly thorough, with all the details spelled out. 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 In physics, Gauss's law for magnetism is one of the four Maxwell's equations that underlie classical electrodynamics. It states that the magnetic field has divergence equal to zero, in other words, that it is a solenoidal vector field. It is ...
and the 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 ''The Feynman Lectures on Physics'' is a physics textbook based on a great number of lectures by Richard Feynman, a Nobel laureate who has sometimes been called "The Great Explainer". The lectures were presented before undergraduate students ...
, at first, the similarity of expressions for the
electric potential Electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as electric potential energy per unit of electric charge. More precisely, electric potential is the amount of work (physic ...
outside the static distribution of charges and the
magnetic vector potential In classical electromagnetism, magnetic vector potential (often denoted A) is the vector quantity defined so that its curl is equal to the magnetic field, B: \nabla \times \mathbf = \mathbf. Together with the electric potential ''φ'', the ma ...
outside the system of continuously distributed currents is emphasized, and then the magnetic field is calculated through the curl from the vector potential. Another approach involves a general solution of the inhomogeneous wave equation for the vector potential in the case of constant currents. The magnetic field can also be calculated as a consequence of the
Lorentz transformations In physics, the Lorentz transformations are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. The respective inverse transformation ...
for the electromagnetic force acting from one charged particle on another particle. 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. Two other ways of deriving the Biot–Savart law include: 1) Lorentz transformation of the
electromagnetic tensor In electromagnetism, the electromagnetic tensor or electromagnetic field tensor (sometimes called the field strength tensor, Faraday tensor or Maxwell bivector) is a mathematical object that describes the electromagnetic field in spacetime. Th ...
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 In electrodynamics, the retarded potentials are the electromagnetic potentials for the electromagnetic field generated by time-varying electric current or charge distributions in the past. The fields propagate at the speed of light ''c'', so t ...
s.


See also


People

*
André-Marie Ampère André-Marie Ampère (, ; ; 20 January 177510 June 1836) was a French physicist and mathematician who was one of the founders of the science of classical electromagnetism, which he referred to as ''electrodynamics''. He is also the inventor of ...
*
James Clerk Maxwell James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish physicist and mathematician who was responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism an ...
*
Pierre-Simon Laplace Pierre-Simon, Marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French polymath, a scholar whose work has been instrumental in the fields of physics, astronomy, mathematics, engineering, statistics, and philosophy. He summariz ...


Electromagnetism

* Darwin Lagrangian


Notes


References

* *


Further reading

* Electricity and Modern Physics (2nd Edition), G.A.G. Bennet, Edward Arnold (UK), 1974, * Essential Principles of Physics, P.M. Whelan, M.J. Hodgeson, 2nd Edition, 1978, John Murray, * The Cambridge Handbook of Physics Formulas, G. Woan, Cambridge University Press, 2010, . * Physics for Scientists and Engineers - with Modern Physics (6th Edition), P. A. Tipler, G. Mosca, Freeman, 2008, * 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 * McGraw Hill Encyclopaedia of Physics (2nd Edition), C.B. Parker, 1994,


External links

* * MISN-0-125
The Ampère–Laplace–Biot–Savart Law
' by Orilla McHarris and Peter Signell fo
Project PHYSNET
{{DEFAULTSORT:Biot-Savart law Aerodynamics Electromagnetism Eponymous laws of physics