A steady electric current produces a magnetic field. The Biot–Savart law builds that field by adding the contributions from small current elements.

For a current element $I,d\boldsymbol\ell$ and a field point separated by vector $\mathbf r$,

$$d\mathbf B=\frac{\mu_0}{4\pi} \frac{I,d\boldsymbol\ell\times\hat{\mathbf r}}{r^2}.$$

Direction

The cross product makes each contribution perpendicular to both the current element and the direction toward the field point. The orientation follows the right-hand rule.

Integrating along a wire

For a continuous current path, the total field is obtained by summing the contributions along the conductor:

$$\mathbf B=\frac{\mu_0 I}{4\pi} \int\frac{d\boldsymbol\ell\times\hat{\mathbf r}}{r^2}.$$

The geometry of the path determines the integral.

Example: long straight wire

For an ideal infinitely long straight wire, symmetry and the Biot–Savart law give

$$B=\frac{\mu_0 I}{2\pi r}.$$

The field circles the wire, with direction given by the right-hand rule.

Biot–Savart is a direct source law for steady currents. Ampère's law can provide a shorter route when the current geometry has strong symmetry.