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

Options

A. `(mu_(0)NI)/(2(b - a)) " In " (b/a)`
B. `(mu_(0)NI)/(2(b - a)) " In " ((b + a)/(b - a))`
C. `(mu_(0)NI)/(2b) " In " (b/a)`
D. `(mu_(0)NI)/(2b) " In" ((b+a)/(b -a))`

Answer

a

Explanation

Number of turns per unit radial thickness is `N//(b -a)`. Let us select one ring segment of radius x and thickness dx then number of turns in this segment will be `N/((b - a)) dx`.
Magnetic field at the centre due to this segment can be written as follows:
`dB = [N/((b - a))dx] xx (mu_(0)I)/(2x) = (mu_(0)NI)/(2(b - a)) (dx)/x`
Net magnetic field can be calculated by integrating the above result:
`B = (mu_(0)NI)/(2(b - a)) underset(a) overset(b) int (dx)/x = (mu_(0)NI)/(2(b - a)) " In" b/a`
Hence , option (a) is correct.