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
[tex]i_c = H 2\pi r[/tex]
[tex]j_c = \frac{H 2\pi r}{\pi r^2} = \frac{2H}{r} [/tex]
[tex]\therefore B = \mu_0 \frac{r j_c}{2}[/tex]
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?