Confusion about flux linkage

Question

When looking for a proper definition of flux linkage, I come across situations involving coils. with N turns, where it is $$N * flux per coil(p)$$ I understand that in this situation, the induced emf across the coil equals -d(N*p)/dt , hence it makes sense to discuss this 'flux linkage',which seems to capture the net effect of the field on induced EMF.However, I don't think this definition is rigorous.Also, I wasn't able to explain the calculations used in other instances using similar logic. I want to know how it is exactly defined.

Answer 1

If the same flux, $\Phi$, passes through each turn of a coil, the Faraday's-Lenz law can be written $$\mathscr E=-N\frac{d\Phi}{dt}.$$ This can correctly be regarded as the sum of $N$ single turn emfs, $\mathscr E_1=-\frac{d\Phi}{dt}$, since these are in series. Since $N$ is a constant we can clearly re-arrange our first equation as $$\mathscr E=-\frac{d}{dt}\left(N\Phi \right).$$ and we can give the name 'flux linkage' to $N\Phi$. Looked at this way, the concept of flux linkage seems to be more mathematically than physically inspired.

But there is a way of viewing the emf induced in a coil that doesn't use the idea of flux linkage, or at least doesn't distinguish it from flux. The idea is that the area, $A$, over which the normal flux density is of magnitude $B$ is not $A_1$, the area of a turn, but $NA_1$. This is because the coil is in fact not a collection of closed loops, but a helix, so there is really only one surface! From this point of view, even for a coil of $N$ turns, we can write simply $$\mathscr E=-\frac{d\Phi}{dt}.$$ We must, though, remember that in this case $\Phi=BNA_1$.