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The Biot-Savart law is used for computing the resultant magnetic fieldB at position r in 3D-space generated by a flexible currentI (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, being evaluated over the path C in which the electric currents flow (e.g. the wire). The equation in SI units is
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 (). 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.
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). 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, and is the current density vector in that volume (in SI in units of A/m2).
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.
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 ? is the angle between and .
When v2c2, the electric field and magnetic field can be approximated as
These equations were first derived by Oliver Heaviside 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 shieldings or magnetic susceptibilities, provided that the current density can be obtained from a quantum mechanical calculation or theory.
The figure shows the velocity () induced at a point P by an element of vortex filament () of strength .
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', 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 ? 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 ? 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
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):
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