A coil having N turns is wound tightly in the form of a spiral with inner and outer radii ${}^{\text{'}}a^{\text{'}} \mathrm{and} {}^{\text{'}}b^{\text{'}}$ respectively. Find the magnetic field at centre, when a current I passes through coil

Given data: 

inner and outer $radii \text{'}a\text{'}\text{ and }\mathrm{b}\text{'}respectively,current i.$

Concept used: moving charges and magnetism

Detailed solution:

Assume about an element with thickness dr that is located r away from the center. The number of turns in the element will be.  $dN=\left(\frac{N}{b-a}\right)dr$

$\mathrm{dB}=\frac{{\mu }_{0}I}{2}\left(\frac{N}{b-a}\right)\left(\frac{dr}{r}\right)$

$\mathrm{B}={\int }_{r=a}^{r=b}dB$

$B=\frac{{\mu }_{0}NI}{2(b-a)}\ln \frac{b}{a}$