Biot–Savart law

The Biot–Savart law gives the magnetic field at a point produced by an arbitrary distribution of steady currents. In differential form, an infinitesimal current element I dℓ at displacement r from the field point contributes dB = (μ₀ / 4π) (I dℓ × r̂) / r², where r̂ is the unit vector from the source element to the field point and μ₀ is the permeability of free space. To get the total field, integrate this contribution over the entire current-carrying region.

Three things to notice about the formula. First, the inverse-square structure is the same as Coulomb's law for electric fields — the magnetic interaction falls off with distance the same way. Second, the cross-product I dℓ × r̂ encodes the rotational geometry that Ørsted observed in 1820: the magnetic field circulates around the current rather than pointing toward or away from it. Third, the factor μ₀/4π = 10⁻⁷ T·m/A in SI units (exactly, before the 2019 redefinition) is built into the unit of the ampere itself — Biot–Savart was the law used to define the SI ampere from 1948 to 2019.

Biot–Savart is to Ampère's law what Coulomb's law is to Gauss's law: an integral expression that always works, but is rarely the most efficient way to compute. For symmetric current geometries (long straight wire, infinite sheet, solenoid), Ampère's law collapses the integration to arithmetic. For everything else — finite wires, current loops, irregular distributions — Biot–Savart is the workhorse, and its line integral is the calculation that physics students perform a hundred times in their undergraduate years.