cat uderstand certain parts of the solution..

here is the question and solution:
https://i29.tinypic.com/30cy4cn.jpg


i cant understand certain things in here:
i cant understand the 3d picture.
why the magnetic field in the z direction??

i cant understand how did they find the mutual inductance here.
because by this formula

[latex]M_{12}=\frac{N_2\phi _{12}}{I_1}[/latex]

in our situation we need to find the current in the celonoid
but they for some weird reason find the emf
and right after without any calculation
they show the mutual induction coeff

why??