Clarifying the notion of Flux Linkage

[SKo]

Hello!
Im having some trouble really understanding this concept. I'll summarize my difficulty in 2 situations that I struggle with:
1) Coil: The inductance of a coil with N turns is known to be proportional to N². My question is why is that true? I know that the H-field (and therefore the B-field) is proportional to N because the current of the coil is crossing the closed contour N times (from Ampere's Law). Then we calculate the magnetic flux and then multiply that flux by N to get the flux linkage. But why do we multiply the flux by N? Doesnt this flux that we calculated already taking into acount the contribution of all the N turns?

2) Infinitely long wire: Assume an infinitely long wire with radius r. The flux linkage inside the wire (i.e. x<r) is proprtional to the ratio of the areas $\frac{\pi x^{2}}{\pi r^{2}}$. In my book it is explained that this can be thought of as a fractional turn and I do understand the logic behind it but I cannot find any rigorous mathematical proof the proves that the flux linkage is $\lambda =\frac{\pi x^{2}}{\pi r^{2}}\varphi$. Can someone prove it or direct me to a source that proves it?

That's all for now :) Thanks!

 

[Hesch]

1) The inductance is by definition:

L = ψ / I
or to be more exact:
L = ψw / I , where ψw is the flux-winding-number (Ψ*N).

So as both ψ and ψw are proportional to N, the unductance L is propotional to N². (That was a load of nonsense, but I hope you will understand it anyway).

 

[SKo]

Well I understand the math but failing to see why this is true physically. Flux linkage physically means the total amount of flux linking the N turns of the coil and is generally calculated as the sum of flux contribution of each turn. But when we calculate the flux we already taking into account the N turns of the coil while we solve Ampere's Law. So what is the physical meaning of multiplying this flux again by N?

 

[Hesch]

Well I understand the math but failing to see why this is true physically. Flux linkage physically means the total amount of flux linking the N turns of the coil and is generally calculated as the sum of flux contribution of each turn. But when we calculate the flux we already taking into account the N turns of the coil while we solve Ampere's Law. So what is the physical meaning of multiplying this flux again by N?

Say you put an ac-current of 1A through a coil with 1 turn, and that due to frequency and dimension of the turn, an emf of 1V will be induced in the coil.

Now if you add another turn, so that you have two turns, then the flux will be doubled, and this doubled flux will pass through two turns, and an emf in the coil will be 4V, (double flux through double number of turns). The induced emf is proportional to dψ_w / dt.