Law

Biot-Savart law

d B = \frac{μ _{0}}{4 π} \frac{I d ℓ \times r}{r ^{2}}

Statement

The field contribution from a small current element Idℓ, falling as the inverse square of distance, perpendicular to both the current direction and the line to the field point.

Symbols and units

$d B$	field contribution from element d\ell	T
$I$	current	A
$r$	unit vector to the field point	—
$μ_{0}$	permeability of free space	4π×10⁻⁷ T·m/A
$d ℓ$	current-carrying length element, along the current	m
$r$	distance from the element to the field point	m

When it applies

Steady (or quasi-steady) currents; the general building block for any current distribution's field.

Don't use when

Requires steady or quasi-steady currents — a rapidly time-varying current needs the full retarded-time electromagnetic treatment instead.

Derivation

An empirical law, confirmed by comparison with the observed fields of currents in various shapes — the magnetic analogue of Coulomb's law for a source element rather than a point charge.

d B = \frac{μ _{0}}{4 π} \frac{I d ℓ \times r}{r ^{2}}

Watch out for

Unlike an electric field from a point charge, a magnetic field element requires a CURRENT element (Idℓ) — an isolated magnetic charge with its own radial field has never been observed.

Connects to

Coulomb's law