Biot-Savart law

In physics, more particularly in electrodynamics, the law first formulated by Jean-Baptiste Biot and Félix Savart describes the magnetic field caused by a direct electric current in a wire. Biot and Savart interpreted their measurements by an integral relation. Laplace gave a differential form of their result, which now often is also referred to as the Biot-Savart law, or sometimes as the Biot-Savart-Laplace law. By integrating Laplace's equation over an infinitely long wire, the original integral form of Biot and Savart is obtained.

Magnetic field dB at point r due to infinitesimal piece ds (red) of wire (blue) transporting electric current i.

Laplace's formula

The infinitesimal magnetic induction dB at point r is given by the following formula due to Laplace,

$d{\vec {\mathbf {B} }}=k{\frac {id{\vec {\mathbf {s} }}\times {\vec {\mathbf {r} }}}{|{\vec {\mathbf {r} }}|^{3}}},$

where the magnetic induction is given as a vector product, i.e., is perpendicular to the plane spanned by ds and r. The constant k depends on the units chosen. In rationalized SI units k is the magnetic constant (vacuum permeability) μ₀ = 4π ×10⁻⁷ N/A² (newton divided by ampere squared). If we think of the fact that the vector r has dimension length, we see that this equation is an inverse distance squared law.

(To be continued)