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Im having some trouble really understanding this concept. I'll summarize my difficulty in 2 situations that I struggle with:

1)

__: The inductance of a coil with N turns is known to be proportional to N__

**Coil**^{2}. 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)

__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?__

**Infinitely long wire:**That's all for now :) Thanks!