We know that currents create magnetic fields. The Biot-Savart law tells you exactly how much field, and in which direction, for any configuration of current.

The law

A small piece of current-carrying wire $I\, d\mathbf{l}$ creates a magnetic field:

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

where: - $\mu_0 \approx 4\pi \times 10^{-7}$ T·m/A is the permeability of free space - $d\mathbf{l}$ is a tiny vector along the wire in the direction of current - $\hat{\mathbf{r}}$ is the unit vector from the wire piece to the point where you’re measuring - $r$ is the distance

The cross product ($\times$) is the key — it’s what makes the field wrap around the wire instead of pointing away from it.

A long straight wire

For an infinitely long straight wire carrying current $I$, the Biot-Savart integral gives:

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

The field: - Circles the wire (right-hand rule) - Falls off as $1/r$ — not $1/r^2$ like the electric field from a point charge - Is proportional to the current

This $1/r$ vs $1/r^2$ difference makes sense: a point charge is zero-dimensional, but a wire extends in one dimension. The extra dimension changes the geometry.

Other classic results

The Biot-Savart law can be integrated for any current distribution:

• Circular loop on axis: $B = \mu_0 I R^2 / (2(R^2 + z^2)^{3/2})$ — strongest at the centre, falls off along the axis
• Solenoid (many loops): $B = \mu_0 n I$ inside (uniform!), nearly zero outside
• Helmholtz coils (two loops separated by their radius): nearly uniform field in the gap

Relation to Ampère’s law

The Biot-Savart law and Ampère’s law ($\nabla \times \mathbf{B} = \mu_0 \mathbf{J}$) contain the same physics. Ampère’s law is the differential (local) form; Biot-Savart is the integral (global) form — it tells you the total field at a point by summing contributions from all currents.

For problems with high symmetry (infinite wire, solenoid), Ampère’s law is simpler. For arbitrary current geometries, Biot-Savart is the tool.