In
physics
Physics is the natural science that studies matter, its 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 which r ...
, specifically
electromagnetism
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of a ...
, the Biot–Savart law ( or ) is an equation describing the
magnetic field
A magnetic field is a vector 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 to its own velocity and to ...
generated by a constant
electric current
An electric current is a stream of charged particles, such as electrons or ions, moving through an electrical conductor or space. It is measured as the net rate of flow of electric charge through a surface or into a control volume. The moving pa ...
. 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 currents are steady (not changing with time). It is the magnetic analogue of electrostatics, where the electric charge, charges are stationary. The magnetization need not be st ...
, playing a role similar to that of
Coulomb's law
Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is conventiona ...
in
electrostatics
Electrostatics is a branch of physics that studies electric charges at rest (static electricity).
Since classical times, it has been known that some materials, such as amber, attract lightweight particles after rubbing. The Greek word for amber ...
. When magnetostatics does not apply, the Biot–Savart law should be replaced by
Jefimenko's equations
In electromagnetism, Jefimenko's equations (named after Oleg D. Jefimenko) give the electric field and magnetic field due to a distribution of electric charges and electric current in space, that takes into account the propagation delay ( reta ...
. The law is valid in the
magnetostatic approximation, and consistent with both
Ampère's circuital law 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 ...
. It is named after
Jean-Baptiste Biot
Jean-Baptiste Biot (; ; 21 April 1774 – 3 February 1862) was a French physicist, astronomer, and mathematician who co-discovered the Biot–Savart law of magnetostatics with Félix Savart, established the reality of meteorites, made an early ba ...
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
Electric currents (along a closed curve/wire)
The Biot–Savart law
is used for computing the resultant
magnetic field
A magnetic field is a vector 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 to its own velocity and to ...
B at position r in 3D-space generated by a flexible
current
Currents, Current or The Current may refer to:
Science and technology
* Current (fluid), the flow of a liquid or a gas
** Air current, a flow of air
** Ocean current, a current in the ocean
*** Rip current, a kind of water current
** Current (stre ...
''I'' (for example due to a wire). A steady (or stationary) current is a continual flow of
charges
Charge or charged may refer to:
Arts, entertainment, and media Films
* ''Charge, Zero Emissions/Maximum Speed'', a 2011 documentary
Music
* ''Charge'' (David Ford album)
* ''Charge'' (Machel Montano album)
* '' Charge!!'', an album by The Aqu ...
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 to be integrated is evaluated along a curve. The terms ''path integral'', ''curve integral'', and ''curvilinear integral'' are also used; ''contour integral'' is used as well, alt ...
, being evaluated over the path ''C'' in which the electric currents flow (e.g. the wire). The equation in
SI units is
where
is a vector along the path
whose magnitude is the length of the
differential element of the wire in the direction of ''
conventional current
An electric current is a stream of charged particles, such as electrons or ions, moving through an electrical conductor or space. It is measured as the net rate of flow of electric charge through a surface or into a control volume. The moving par ...
''.
is a point on path
.
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 a ...
from the wire element (
) at point
to the point at which the field is being computed (
), and ''μ''
0 is the
magnetic constant
The vacuum magnetic permeability (variously ''vacuum permeability'', ''permeability of free space'', ''permeability of vacuum''), also known as the magnetic constant, is the magnetic permeability in a classical vacuum. It is a physical constant, ...
. Alternatively:
where
is the
unit vector
In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat").
The term ''direction vecto ...
of
. 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 elect ...
—until 20 May 2019).
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 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 tha ...
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. Vectors can be added to other vectors a ...
of the field created by each infinitesimal section of the wire individually.
An example used in 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 ...
,
solenoids
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 whose ...
and the
Magsail spacecraft propulsion design is the magnetic field at a distance
along the center-line (cl) chosen as the x axis of a loop of radius
carrying a current
as follows:
where
is the unit vector of
from a 1979 physics textbook
and a website with an on-line calculator. Calculation of the off center-line axis magnetic field 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 using commercial mathematical tools or approximations,
code, with further details of the derivation given in.
There is also a 2D version of the Biot–Savart equation, used when the sources are invariant in one direction. In general, the current need not flow only in a plane normal to the invariant direction and it is given by
(
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 ar ...
). The resulting formula is:
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
is the vector from dV to the observation point
,
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
:dV = \ ...
, and
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 ar ...
vector in that volume (in SI in units of A/m
2).
In terms of unit vector
Constant uniform current
In the special case of a uniform constant current ''I'', the magnetic field
is
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. It may be an ion, such as a molecule or atom with a surplus or deficit of electrons relative to protons. It can also be an electron or a proton, or another elementary particle, ...
''q'' moving at a constant
velocity
Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity is a ...
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, and electric circuits.
...
give the following expression for the
electric field
An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field fo ...
and magnetic field:
[
where 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 and .
When , the electric field and magnetic field can be approximated as]
These equations were first derived by Oliver Heaviside
Oliver Heaviside FRS (; 18 May 1850 – 3 February 1925) was an English self-taught mathematician and physicist who invented a new technique for solving differential equations (equivalent to the Laplace transform), independently developed vec ...
in 1888. Some authors call the above equation for 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 shielding
A chemical substance is a form of matter having constant chemical composition and characteristic properties. Some references add that chemical substance cannot be separated into its constituent elements by physical separation methods, i.e., wit ...
s 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, from grc, ἀήρ ''aero'' (air) + grc, δυναμική (dynamics), is the study of the motion of air, particularly when affected by a solid object, such as an airplane wing. It involves topics covered in the field of fluid dyn ...
theory to calculate the velocity induced by vortex lines.
In the aerodynamic
Aerodynamics, from grc, ἀήρ ''aero'' (air) + grc, δυναμική (dynamics), is the study of the motion of air, particularly when affected by a solid object, such as an airplane wing. It involves topics covered in the field of fluid dyn ...
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 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 and traveling along wit ...
(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: was essentially a rotational analogy to the linear electric current relationship,
; 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 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 are mediated by a magnetic field, which refers to the capacity to induce attractive and repulsive phenomena in other entities. Electric currents and the magnetic moments of elementary particles ...
. 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
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):