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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 ?

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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 $\color{red}{n^2}$

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  • $\begingroup$ Oh, forgot to see the n hiding inside flux... thanks $\endgroup$ – Shubham Nov 26 '15 at 7:55

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