Biot–Savart law

In physics, specifically electromagnetism, the Biot–Savart Law (/ˈbiːoʊ səˈvɑːr/ or /ˈbjoʊ səˈvɑːr/) 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, playing a role similar to that of Coulomb's law in electrostatics. When magnetostatics does not apply, the Biot–Savart law should be replaced by Jefimenko's equations. The law is valid in the magnetostatic approximation, and is consistent with both Ampère's circuital law and Gauss's law for magnetism. It is named after Jean-Baptiste Biot and Félix Savart, who discovered this relationship in 1820. The Biot–Savart law is used for computing the resultant magnetic field B at position r in 3D-space generated by a flexible 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, being evaluated over the path C in which the electric currents flow (e.g. the wire). The equation in SI units is where d ℓ {displaystyle d{oldsymbol {ell }}} is a vector along the path C {displaystyle C} whose magnitude is the length of the differential element of the wire in the direction of conventional current. ℓ {displaystyle {oldsymbol {ell }}} is a point on curve C {displaystyle C} . r ′ = r − ℓ {displaystyle mathbf {r'} =mathbf {r} -{oldsymbol {ell }}} is the full displacement vector from the wire element ( d ℓ {displaystyle d{oldsymbol {ell }}} ) at point ℓ {displaystyle {oldsymbol {ell }}} to the point at which the field is being computed ( r {displaystyle mathbf {r} } ), and μ0 is the magnetic constant. Alternatively: where r ^ ′ {displaystyle mathbf {{hat {r}}'} } is the unit vector of r ′ {displaystyle mathbf {r'} } . The symbols in boldface denote 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 (as used in the definition of the SI unit of electric current - the Ampere). To apply the equation, the point in space where the magnetic field is to be calculated is arbitrarily chosen ( r {displaystyle 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 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 J {displaystyle mathbf {J} } (current density). The resulting formula is: