A coil of n number of turns is wound tightly in the form of a...

Consider an element of thickness $d r$ at a distance $r$ from the centre of spiral coil. Number of turns in coil $=n$ Number of turns per unit length $=\frac{n}{b-a}$ Number of turns in element $d r=d n$ Number of turns per unit length in element $d r$ i.e., $d n=\frac{n d r}{b-a}$ Magnetic field at its centre due to element $d r$ is $d B=\frac{\mu_0 I d n}{2 r}=\frac{\mu_0 I}{2} \frac{n}{(b-a)} \frac{d r}{r}$ $\therefore \quad B=\int_a^b \frac{\mu_0 I n d r}{2(b-a) r}=\frac{\mu_0 I n}{2(b-a)} \int_a^b \frac{d r}{r}$ $=\frac{\mu_0 I n}{2(b-a)} \log _e\left(\frac{b}{a}\right)