Consider an elemental ring of radius \(x\) and with thickness \(dx\)

Number of turns in the ring \( = dN = \frac{{Ndx}}{{b - a}}\)

Magnetic field at the centre due to the ring element \(dB = \frac{{{\mu _0}(dN)i}}{{2x}} = \frac{{{\mu _0}i}}{2}.\frac{{Ndx}}{{(b - a)}}.\frac{1}{x}\)

\(\therefore \) Field at the centre \( = \int_{}^{} {dB = \frac{{{\mu _0}Ni}}{{2(b - a)}}} \int_a^b {\frac{{dx}}{x}} \) \( = \frac{{{\mu _0}Ni}}{{2(b - a)}}\ln \frac{b}{a}.\)