Derive an expression for the applied field of a superconducting wire

[TheBigDig]

Homework Statement

A piece of superconducting wire of length l and radius r is made from a material of critical current density $j_c$. It is placed in a magnetic field $B$ which is applied parallel to the axis of the wire. The magnetic moment m is measured as B increases from zero up to a maximum $B_{max}$ and back again to zero. If a supercurrent flows in a thin layer of thickness $\lambda$ at the surface of the wire, derive an expression for the applied field at which the moment m changes discontinuously. You may assume that l >> r.

Relevant Equations

$\vec{B} = \mu_0 \vec{H}$
$\oint \vec{H}\cdot d\vec{l} = i_c$

So far the best I've been able to come up with is to use $\vec{B} = \mu_0 \vec{H}$ which gives me
i_c = H 2\pi r
j_c = \frac{H 2\pi r}{\pi r^2} = \frac{2H}{r}
\therefore B = \mu_0 \frac{r j_c}{2}
I'm fairly confident this is just terrible math and physics on my behalf but I'm struggling to see how to relate all the given variables into one unifying equation/

 

[mfb]

Why do you divide by the whole cross section? Current will only flow close to the boundary.

Did you draw a sketch? Where is current is flowing in which direction to make the internal field zero?