Inductance and coils

There is always something confusing about "the flux through a coil".
We know that a closed loop spans a surface, and this is not different for a coil. However, this is a "multi-layered" surface. The flux through this "multi-layered" surface will be approximately N times the flux through a simple surface spanning the coil "only once", when N is the number of windings.
The simple surface would be a disk, and the "multi-layered" surface would be some kind of helicoidal surface with a very small step (the step of the windings).

If we call [tex] \phi_{multi} [/tex] the correct, multi-layered surface attached to the coil, then we have that the EMF is the time variation of this flux. However, we usually call the flux through a coil, the flux through the simple surface [tex] \phi_{simple} [/tex]. So we have:

[tex] EMF = \frac{d \phi_{multi}}{dt} = N \frac{d \phi_{simple}}{dt} [/tex]

We also know that [tex]EMF_1 = L \frac{d i_1}{dt} + M \frac{d i_2}{dt}[/tex]

From this, it follows that
[tex] \phi_{simple} = \frac{L i_1 + M i_2}{N} [/tex]

 

[tex] L i_1[/tex] is the flux seen by the (multilayered) surface on coil 1 by the current in coil 1. [tex] M i_2 [/tex] is the flux seen by the multilayered surface on coil 1 by the current in coil 2.
Now, we usually call the flux, not the flux through the multilayered surface, but by the simple surface, which is then N times less, with N the number of windings of coil 1.

EDIT: you can put in minus signs. That depends on some conventions.

EDIT 2: I was taking coil 1 here as an abstract example. Of course something similar can be written about coil 2.

 

No. The flux (on the helicoidal surface) induced in coil 1 by the current in coil 2 is M I2.
The flux induced in coil 1 by the current in coil 1, is L1 I1.
The total flux is the sum of both contributions.

And if you want the flux in a single "disk" surface, you have to divide by the number of windings of coil 1.

In the same way:

The flux (on the helicoidal surface) induced in coil 2 by the current in coil 1 is M I1.
The flux induced in coil 2 by the current in coil 2, is L2 I2.
The total flux is the sum of both contributions.

And if you want the flux in a single "disk" surface, you have to divide by the number of windings of coil 2.

No: obviously, what you need are the currents (you have them), the coefficients of induction (L1, L2 and M) and the number of windings. All the geometry is already included in the coefficients of induction.