Question

Figure shows a spiral of inner radius a and outer radius b with total N turns wound in it. If a current I flows in it, find magnetic induction at the centre of spiral.

No of turns per unit radial length = $\frac{N}{b-a}$

Answer 1

The magnetic induction at the centre of the spiral is $\frac{\mu_0 I N}{2(b-a)} \ln\left(\frac{b}{a}\right)$.

Answer 2

Consider an annular ring of radius $r$ and thickness $dr$. The number of turns in this ring is $dN = \frac{N}{b-a} dr$. The magnetic field at the center due to this ring is $dB = \frac{\mu_0 I}{2r} dN$. Integrating $dB$ from $r=a$ to $r=b$ yields the total magnetic induction.