A long insulated copper wire is closely wound as a spiral of `N` turns. The spiral has inner radius a and outer radius `b`. The spiral lies in the `xy`-plane and a steady current I flows through the wire. The`z`-component of the magetic field at the centre of the spiral is

Explanation

(a) Number of turns in separation `(b - a)` is `N`.
Number of turns per unit length `((N)/(b - a))`
Number of turns in separation `dx, N' = ((N)/(b - a))dx`
Consider a ring of radius `x` anf thickness `dx`

Small magnetic field at centre due to this ring
`dB = (mu_(0)N'i)/(2x) = (mu_(0) N i d x)/(2(b - a)x)`
`B_(0) = (mu_(0)Ni)/(2(b - a)) int_(a)^(b)(dx)/(x)`
`= (mu_(0)Ni)/(2(b - a))|log_(e) x|_(a)^(b)`
`= (mu_(0)Ni)/(2(b - a))log_(e)(b//a), ox`
(b) Small magnetic moment of ring
`dM = N'i pi x^(2) = (Ni pi x^(2) dx)/((b - a))`
`M = (Ni pi)/((b - a))int_(a)^(b)x^(2) dx = (Ni pi)/((b - a))|(x^(3))/(3)|_(a)^(b)`
`= (Ni pi)/(3(b - a))(b^(3) - a^(3)) = (Ni pi)/(3(b - a))(b^(2) + ba+a^(2))(b-a)`
`= (Ni pi(a^(2) + ab + b^(2)))/(3), ox`