Q:

Self-induction in a circular coil

There's a formula for self inductance: $$ L=\frac{n\Phi}{i}$$ where n is the number of loops.
But the book also says self inductance is directly proportional to $ n^2 $
I totally agree with the second sentence. But the formula seems to show $L$ is directly proprtional to $n $ and not $ n^2 $.
I am confused. Can you help?

A:

Notice, the magnetic magnetic field $B$ at the center of  a coil carrying current $i$, with radius $r$ & having $n$ no. of turns  $$B=\frac{\mu_0}{2}\frac{ni}{r}$$
hence, magnetic flux $\phi$ linked to the coil is given as $$\Phi=BA=\frac{\mu_0}{2}\frac{ni}{r}\pi r^2=\frac{\mu_0 \pi nir}{2}$$ 
Now, setting the value of $\phi$, we get 
$$L=\frac{n\Phi}{i}=\frac{n\frac{\mu_0 \pi nir}{2}}{i}=\frac{\mu_0 \pi  n^2r}{2}$$
$$L\propto n^2$$
It is obvious that keeping other parameters constant, the self inductance $\color{red}{L}$ of a coil is directly proportional to $\color{red}{n^2}$

