An infinite straight wire carrying a current I runs through the centers of two opposite faces of a regular octahedron

of an edge length $a($Fig$.2).$

Compute the line integral ${\displaystyle \underset{P}{\overset{\phantom{}}{}}}B*dl,$ where $B(r)$ is the magnetic field produced by the current, and $P$ is a non$-$closed

path along the as the thick red line.

Hint: There exist two possible solutions for this problem: one is elegant and short, another is boring and long. Choose

either.

FIG. $2$: A current and an octahedron.