The Biot–Savart law

Coulomb's law gives the electric field from a point charge; the Biot–Savart law is its magnetic counterpart, giving the magnetic field produced by a small element of current-carrying wire. It shares the inverse-square falloff with distance — both describe an influence spreading through three-dimensional space from a source — but the direction rule is completely different, and that difference is the entire content worth understanding.

A point charge's field points radially, straight out along the line from charge to observation point. A current element's field is perpendicular to two directions at once: the direction of the current itself, and the direction from the element to the observation point. The field magnitude involves sin θ between those two directions, exactly as it did for the Lorentz force, and for the same underlying reason — cross products govern magnetic effects throughout electromagnetism. Point the current element straight at the observation point (θ = 0) and the field there is zero, however large the current; the field is largest directly out to the side of the wire, where θ = 90°.

Building the total field of a real wire means summing the Biot–Savart contribution from every infinitesimal segment along its length — an integral, not a single evaluation, since each segment sits a different distance and angle from the observation point. This is more laborious than Ampère's law, but it works for any current geometry, symmetric or not, where Ampère's law only simplifies calculations when enough symmetry exists to choose a convenient loop. The two are not competing laws; Biot–Savart is the general tool and Ampère's law is the shortcut available in the symmetric cases — current in an infinite straight wire, a long solenoid, a toroid — where it applies.