Biot–Savart law
In physics, specifically electromagnetism, the Biot–Savart law (/ˈbiːoʊ səˈvɑːr/ or /ˈbjoʊ səˈvɑːr/)[1] 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 law is named after Jean-Baptiste Biot and Félix Savart, who discovered this relationship in 1820.
This allows for straightforward derivation of magnetic field B, while the fundamental vector here is H.[3] The Biot–Savart law[4]: Sec 5-2-1 is used for computing the resultant magnetic flux density B at position r in 3D-space generated by a filamentary current I (for example due to a wire).
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).
whose magnitude is the length of the differential element of the wire in the direction of conventional current,
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 (
along the center line of the loop, the magnetic field vector at that point is:
[4]: Sec 5-2, Eqn (25) Loops such as the one described appear in devices like the Helmholtz coil, the solenoid, and the 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.
[7] 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:
In the special case of a uniform constant current I, the magnetic field
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:[8]
[12] 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 Biot–Savart law is also used in aerodynamic theory to calculate the velocity induced by vortex lines.
In Maxwell's 1861 paper 'On Physical Lines of Force',[13] 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.
By analogy, the magnetic equation is an inductive current involving spin.
In aerodynamics the induced air currents form solenoidal rings around a vortex axis.
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.
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
Since the divergence of a curl is always zero, this establishes Gauss's law for magnetism.
(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[14]
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.
[15] Another approach involves a general solution of the inhomogeneous wave equation for the vector potential in the case of constant currents.
[17] 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.
