Why do we multiply the magnetic flux by the winding number? A current I flows through a solenoid (with n turns). Self inductance is $L$. If the magnetic flux is $\phi$, I am taught that $-n\phi=LI$
I dont understand the part with $n$, why do we have multiply by $n$? $\phi$ itself represents the whole magnetic flux there, if $\phi$ implied the flux only occurred by one coil, then it would be correct. 
Moreover, flux change by a coil ($n$ turns) in $dt$ time is $d\phi$, so the induced electromagnetic force is 
$$dE=n \cdot \frac{d\phi}{dt}$$ 
Why do we multiply by $n$ here? If $\phi$ means the whole flux, then why multiply?  
 A: I agree that this seems mysterious on first meeting.
Multiplication by $n$ is shown to be correct by something called Stokes's Theorem, which you won't have met yet, and which lets us translate the basic equations of electromagnetism (called their Maxwell Equations) between their local and "spread out" forms. 
But at an easier level, think of a very long, time varying flux tube, like the one I've drawn below.

Now imagine separate loops around the flux tube. Each of these loops has flux $\phi$ through them, and each separately will have an EMF $-\frac{{\rm d}}{{\rm d}\,t}\phi$ between their ends.
You link them together in series, and it is just like linking voltage sources, or batteries together in series. The total EMF across the series cells is $-n\,\frac{{\rm d}}{{\rm d}\,t}\phi$. Now imagine bringing the loops together in a neatly wound solenoid. Electrically nothing changes: it doesn't matter if you put long, zero resistance wires between batteries in series, their EMFs still add in exactly the same way.
Since the whole purpose of an inductor is react to a change in current with a back EMF, the physics doesn't change if you state the equation $V = -n\,\frac{{\rm d}}{{\rm d}\,t}\phi$ as $L I = -n\,\phi$; you simply differentiate this one to get the form I just derived.
A: $\Phi$ is the flux of $\vec B$ through one surface
$$\Phi = BA$$
where A is the area bounded by a loop (turn).
But there are $n$ turns and thus $n$ surfaces that are pierced by $\vec B$ so the the flux linkage $\lambda$ is 
$$\lambda = n \Phi$$
A: The basic Faraday's law cares only about the total flux linked, in the form of a line integral. It does not care about details like the number of turns.
So then how do you account for turns?
It has to do with the topological equivalence of n turns linking some amount of flux and a single turn linking n times this flux.
This can be easily verified by having two threads, making one loop of the first thread passing through n loops of the second thread. Without breaking the threads, we can rearrange the loops so that n loops of the first thread pass through a single loop of the second thread.
So n turns linking some amount of flux can be considered as a single turn through which the same flux flows n times instead. Hence the total flux linked by this equivalent single turn is n times the flux, hence the induced EMF is now n times as much as a single turn.
