# 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 ?

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 $$L=\frac{n\Phi}{i}=\frac{n\frac{\mu_0 \pi nir}{2}}{i}=\frac{\mu_0 \pi n^2r}{2}$$ It is obvious that the self inductance $\color{red}{L}$ of a coil is directly proportional to $\color{red}{n^2}$