the Biot-Savart law
/ BEE-oh sah-VAR /
The Biot-Savart law tells you the magnetic field that a small piece of current-carrying wire creates at any point nearby. If Ampere's law answers 'what field does a whole symmetric arrangement make?', the Biot-Savart law works from the ground up, letting you add up the contributions of every little segment of wire, no matter how the wire twists. It answers, given a current, exactly what magnetic field does it produce, point by point?
Each short segment of wire, of length dL carrying current I, contributes a tiny magnetic field at a point a distance r away. That contribution has magnitude dB = (mu_0 / (4 pi)) times I dL sin(theta) / r^2, where theta is the angle between the segment and the line to the point, and mu_0 is the permeability of free space, a constant equal to 4 pi times 10^-7 in SI units. The direction of each contribution is perpendicular to both the segment and the line to the point, set by the right-hand rule. Adding (integrating) over the whole wire gives the total field. Notice the inverse-square 1/r^2 falloff, echoing Coulomb's law for electric charge.
From this one law you can derive the standard results: the field a distance r from a long straight wire is B = mu_0 I / (2 pi r), circling the wire; the field at the center of a circular loop of radius R is B = mu_0 I / (2 R). The Biot-Savart law is the magnetic counterpart of Coulomb's law, and it is the fundamental recipe from which all steady-current magnetic fields can, in principle, be computed.
For a long straight wire, summing the Biot-Savart contributions of all its segments gives B = mu_0 I / (2 pi r). A 10 A current produces a field of about 0.00002 T (0.2 gauss) at 10 cm, comparable to the Earth's field.
Integrating the tiny dB from every segment yields the field of any current shape.
The Biot-Savart law applies to steady (unchanging) currents. When currents change with time, a full treatment needs Maxwell's equations, which add the effects of changing fields.