= Solution
For a long cylindrical wire carrying axial current $I$, <Ampère's law> gives $B_s=\mu_0I/(2\pi a)$ 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 $B_s=B_{c1}$:
$$
\boxed{I_{c,\rm entry}=\frac{2\pi aB_{c1}}{\mu_0},\qquad
\overline J_{c,\rm entry}=\frac{I_c}{\pi a^2}=\frac{2B_{c1}}{\mu_0a}
\simeq\frac{9.5\times10^4}{a[\mathrm m]}\,\mathrm{A\,m^{-2}}.}
$$
The dependence on $a$ is necessary because no numerical radius is supplied. The \i[local] surface <current density> at this entry scale is different:
$$
\boxed{j_{s,\rm entry}\simeq\frac{B_{c1}}{\mu_0\lambda}\sim6.8\times10^{11}\,\mathrm{A\,m^{-2}}.}
$$
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 $a/(2\lambda)$.
\b[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 <superconducting depairing current density>[depairing] scale follows from $j(q)=\Phi_0q(1-\xi^2q^2)/(2\pi\mu_0\lambda^2)$. Maximizing at $q=1/(\sqrt3\xi)$ gives
$$
\boxed{j_d=\frac{\Phi_0}{3\sqrt3\pi\mu_0\lambda^2\xi}\sim1.8\times10^{12}\,\mathrm{A\,m^{-2}}.}
$$
This 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 <superconducting depairing current density>[depairing] limit answer distinct physical criteria; the source's word “maximum” does not supply the missing pinning information.
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