Magnetic-slab transmission threshold ($mv/(|q|B)>b$)

A nonrelativistic charged particle entering a uniform magnetic region normally at one plane follows a circle of radius $r=mv/(|q|B)$. If the field extends through a normal thickness $b$, the particle enters the strict region beyond the far plane only for $r>b$. Equality gives tangency, not crossing. For transmission, its deflection angle obeys $\sin \theta =b/r$. The result assumes a sharp field boundary, fixed speed in the magnetic region and no electric field there.