Magnetic Fields of Current-Carrying Wires and the Biot-Savart Law

A current's magnetic field wraps around it. For a long straight wire $B = \mu_0 I/(2\pi r)$ on circles centered on the wire, with the direction given by gripping the wire with the thumb along the current. The general rule is Biot-Savart, $d\vec{B} = (\mu_0/4\pi)\,I\,d\vec{\ell}\times\hat{r}/r^2$, an element integral in which each piece has its own $r$ and its own direction, and in which an element pointing straight at the field point contributes nothing. At the center of a loop every element contributes the same axial direction, so they add to $\mu_0 I/(2R)$, and far along the axis the field falls as $1/z^3$ like a magnetic dipole.

Five errors dominate. Aiming a wire's field radially outward instead of tangentially around, which inherits the wrong geometry into every force and flux that follows. Importing like-repels-like from charges, when parallel currents in the same direction attract. Treating Biot-Savart as Coulomb by putting the whole length at the closest distance, or by giving $d\vec{B}$ a radial direction. Adding two wires' field magnitudes as plain numbers, when at the midpoint of equal same-direction currents they cancel. And giving a loop zero field at its center by imagining opposite sides cancel, when the loop is the shape whose elements reinforce.