Past exam of the condensed matter physics major option course of the University of Oxford / 2018 / 5 / d / Solution

For a long cylindrical wire carrying axial current $I$, Ampère's law gives $B_s={\mu }_{0}I/(2\pi a)$ at its surface. With $a\gg \lambda$, the current resides mainly in a shell of thickness $\lambda$. If “critical” denotes the first equilibrium entry of vortices in an otherwise Meissner wire, set $B_s=B_{c1}$:The dependence on $a$ is necessary because no numerical radius is supplied. The local surface current density at this entry scale is different:Its conversion to the cross-section average is $\overline{J}\simeq (2\lambda /a)j_s$. Treating the shell current density as uniform over the entire thick wire would overestimate its total current by $a/(2\lambda )$.

Meaning of a maximum: in a type-II material, first vortex entry need not be the measured dissipative transport limit. Pinned vortices can sustain greater current, whereas freely moving vortices generate voltage. The hysteresis loop indicates pinning, but no geometry-dependent pinning force or transport criterion is given, so a unique practical maximum cannot be deduced solely from $\lambda ,\xi$ and $a$.

For comparison, the intrinsic homogeneous Ginzburg–Landau depairing scale follows from $j(q)={\Phi }_{0}q(1-{\xi }^2{q}^2)/(2\pi {\mu }_0{\lambda }^2)$. Maximizing at $q=1/(\sqrt{3}\xi )$ givesThis is a local, approximately uniform-flow theoretical scale, not the average current density throughout a thick self-field wire. Surface barriers, nonlinear screening, vortex motion and temperature determine how that scale relates to its actual critical current. Thus the entry estimate, local shell density and intrinsic depairing limit answer distinct physical criteria; the source's word “maximum” does not supply the missing pinning information.