Biot–Savart law

   

In physics, specifically electromagnetism, the Biot–Savart law 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. It is valid in the magnetostatic approximation and consistent with both Ampère's circuital law and Gauss's law for magnetism. When magnetostatics does not apply, the Biot–Savart law should be replaced by Jefimenko's equations. The law is named after Jean-Baptiste Biot and Félix Savart, who discovered this relationship in 1820.

The equation in SI units teslas (T) is:

\[ \mathbf{B(r)} = \frac{μ_0}{4π} \int_C \frac{ I d\mathbf{l} \times \mathbf{r'}}{|\mathbf{r'}|^3} \]

Alternatively:

\[ \mathbf{B(r)} = \frac{μ_0}{4π} \int_C \frac{ I d\mathbf{l} \times \hat{r'} }{|\mathbf{r'}|^2} \]

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 (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.

Source: Wikipedia