Past exam of the condensed matter physics major option course of the University of Oxford / 2016 / 8 / d / Solution

For a laterally infinite slab in a parallel field there is no demagnetizing correction. The field has only its applied direction, varies with $x$, and satisfies $B^{\prime \prime }=B/{\lambda }^2$. Reflection symmetry gives $B^{\prime }(0)=0$, and continuity gives $B(\pm a)=B_a$. ThusUniformly for $|x|\le a\ll \lambda$, expanding numerator and denominator to second order yieldsOnly a small fraction of the applied field is screened in this limit; the almost uniform internal field is the origin of the enhanced parallel critical field.