Inductance of a coil when the core has a gap 
A coil of $N$ turns is wound on an iron core occupying a length $l$ of the core that has an air gap as shown in the figure.
Mean length of iron in the core is $l_i$ and the length of the air gap is $l_g$ ($l_g << l_i$
). Denoting relative permeability of the
iron by $\mu_r$ and that of the free space by $\mu_0$, deduce the suitable expression for the self-inductance of the coil


I am not able to understand why the inductance of the coil would depend on the gap or the rest of the core at all, since there is a gap and hence no loop will be completed, leading to any significant eddy currents.
TL;DR I just want to know how the self-inductance $L$ of the coil would depend on the rest of the core and especially the length of the gap
 A: Simplified, and wrong but very intuitive way to look at it, is to imagine that magnetic field is a current, that passes through resistors.
This current passes through steel core easily, resistance is about 10 000 times less than that of air.
In this model because air resistance is so much higher, even small air gap can easily make total circuit resistance much bigger. So we can say that air resistance dominates. And thats why we can take just it into account.
This resistance in our thought experiment has a proper name
https://en.m.wikipedia.org/wiki/Permeability_(electromagnetism)
Through it you can calculate how much effect on a circuit will be from the air gap and from the magnetic core, for other materials and shapes too.
A: The problem has nothing to do with eddy currents. It is simply that the magnetic field inside the coil is considerably reduced by the presence of the air gap. More precisely, the reluctance $R$ of the magnetic circuit is greater and the magnetic flux $Φ=Ni/R$ is lower.
You just have to express the reluctance of the magnetic circuit and write that $L=NΦ/i=N^2/R$.
