The Field of a Long Straight Wire

Point your thumb along the current and your fingers curl the way B points: in circles around the wire, never toward or away from it. That shape comes straight out of Biot-Savart: each tiny piece of current contributes a field element dB = (mu0/4pi) I dl x r-hat / r², and the cross product dl x r-hat always points tangent to a circle around the wire, never radially. Add up every piece along an infinite wire and the radial components would have to come from somewhere that breaks the symmetry, but nothing does, so the net field only ever circles.

Doing the full Biot-Savart integral for an infinite wire takes some trig substitution, but it collapses to B = mu0 I/(2 π r), the same answer Ampere's law gives in one line using a circular loop of radius r centered on the wire. That agreement is the point of this topic: it previews Ampere's law the way a single point charge's field previewed Gauss's law. The 1/r falloff, not 1/r², is the signature of a line source, just like the 1/r potential pattern you will see again with current sheets and solenoids.

Two wires carrying opposite currents push their fields in the same direction at the midpoint between them, because each wire's circular field curls oppositely there; parallel currents instead cancel at the midpoint. A finite wire segment gives a weaker field than the infinite formula, since you are only picking up the tangent contributions from a limited arc of angles instead of integrating over the full range from -90 to 90 degrees relative to the wire's ends.