The Biot-Savart Law

A single moving charge makes a magnetic field, so a wire full of moving charges is really a chain of tiny sources lined up end to end. Chop the wire into infinitesimal pieces of length ds, each carrying current I, and each piece contributes its own scrap of field dB at any point in space. Add up all those scraps with an integral and you get the total field from the whole wire. This is the magnetic cousin of chopping a charged rod into dq pieces and integrating Coulomb's law to get E.

The formula is dB = (mu0/4pi) I (ds x r-hat)/r², where ds is a vector along the current's direction, r-hat points from the segment to the field point, and r is the distance between them. The cross product is the whole story: it forces dB to be perpendicular to both the wire segment and the line to the point, so the field wraps around the wire in circles instead of shooting straight outward like an electric field does. The sine of the angle between ds and r-hat controls the strength, maxed out at 90 degrees and zero when the field point sits right on the line of the wire.

This sits right next to Ampere's law the way Coulomb's law sits next to Gauss's law for electricity. Biot-Savart is the brute-force, point-by-point tool that works for any current shape, while Ampere's law exploits symmetry to find B quickly for simple geometries like straight wires, solenoids, and toroids. Knowing both lets you pick the fast method when symmetry is kind and fall back on direct integration when it isn't.