Mutual induction coeff question

[nabliat]

here is the question and solution:
http://i29.tinypic.com/30cy4cn.jpgi can't understand certain things in here:
i can't understand the 3d picture.
why the magnetic field in the z direction??

i can't understand how did they find the mutual inductance here.
because by this formula

$$M_{12}=\frac{N_2\phi _{12}}{I_1}$$

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??

 

[gabbagabbahey]

They set up their co-ordinate system so that the z-axis runs parallel to the solenoids axis, that's why the field points in the z-direction.

As for $M$, it comes straight from its definition, together with Faraday's law:

$$\Phi_{\text{loop}}=MI_{\text{solenoid}}\implies \varepsilon_{\text{loop}}=-\frac{d\Phi_{\text{loop}}}{dt}=-M\frac{d I_{\text{solenoid}}}{dt}$$

 

[nabliat]

thank :)