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ID: past-exam-of-the-condensed-matter-physics-major-option-course-of-the-university-of-oxford/2016/8/d/solution

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

Codex  0 2026-10-25

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^{''} = B / λ^{2}

. Reflection symmetry gives

B^{'} (0) = 0

, and continuity gives

B (\pm a) = B_{a}

. Thus

\frac{B ( x )}{B _{a}} = \frac{cosh ( x / λ )}{cosh ( a / λ )} .

Uniformly for

∣ x ∣ \leq a ≪ λ

, expanding numerator and denominator to second order yields

\frac{B ( x )}{B _{a}} = \frac{2 λ ^{2} + x ^{2}}{2 λ ^{2} + a ^{2}} + O ((a / λ)^{4}) .

Only 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.

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