Past exam of the physics B course of the University of Cambridge / 2016 / Paper 1 / Section A / A1

Take a long strip along $y$, occupying $-w/2<x<w/2$ in $z=0$, with uniform surface current $K=I/w$. Above the strip's centre, the contributions odd in $x$ cancel, while the Biot-Savart law gives the magnetic field of a uniformly conducting stripThe field reverses below the strip. Its field lines wrap around the longitudinal current: nearly parallel to the faces near the centre, then curving round the edges and approaching circles at large distances. The sketch is an original field integration; the conductor thickness is idealized as negligible.

At distance $r\gg w$, the strip is indistinguishable to leading order from a straight line current. Ampère's law on a large circular contour gives $B\simeq {\mu }_{0}I/(2\pi r)$, directed tangentially around the strip's longitudinal axis by the right-hand rule. On the perpendicular centreline this also follows by expanding ${\tan }^{-1}(w/(2z))\simeq w/(2z)$.

At perpendicular distance $z\ll w$ from the middle of a face, but outside the conductor, the strip approximates an infinite current sheet. The tangential field jump is ${\mu }_{0}K$ and symmetry gives equal opposite fields on the two sides. Hence $B\simeq {\mu }_{0}I/(2w)$ on either side. The centreline integral gives the same limit as ${\tan }^{-1}(w/(2z))\to \pi /2$. Uniform current across the width and distance from the strip edges are implicit in this sheet approximation; a nonuniform surface current would require its local value instead.