Question

A coil having N turns is wound tightly in the form of a spiral with inner and outer radii a and b respectively. When a current I passes through the coil, find the magnetic field at the centre

Answer 1

No. of turns in dx, $n=\frac{N\times dx}{(b-a)}$
The magnetic field induction at the centre O due to current I through the entire spiral
$B={\int }_a^b\frac{{\mu }_0}{4\pi }\frac{2\pi nI}{x}dx=\frac{{\mu }_0}{4\pi }2\pi {\int }_a^b\left(\frac{N}{b-a}\right)\frac{dx}{x}I$
$=\frac{{\mu }_0}{4\pi }\frac{NI}{(b-a)}(lo{g}_{e}x{)}_a^b=\frac{{\mu }_0}{2}\frac{NI}{(b-a)}lo{g}_e\frac{b}{a}$