= Solution
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 $\mu_0K$ and symmetry gives equal opposite fields on the two sides. Hence \b[$B\simeq\mu_0I/(2w)$] 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.
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