A coil having `N` turns is would tightly in the form of a spiral with inner and outer radii a and `b` respectively. When a current `I` passes through the coil, the magnetic field at the centre is.

Options

A. `(mu_(0)NI)/b`
B. `(2mu_(0)NI)/(a)`
C. `(mu_(0)NI)/(2(b-a)) ln(b/a)`
D. `(mu_(0)I)/(2(b-a)) ln(b/a)`

Answer

C

Explanation

Number of turns of turns in length `(b-a)=N`
Number of turns`//` length `=N/(b-a)`
Number of turns in thickness `dx`
`N'=((N)/(b-a))dx`
Magnetic field due to ring of radius `dx`
`dB=(mu_(0))(N'i)/(2x)`
`B_(0)=(mu_(0)Ni)/(2(b-a))underset(a)overset(b)int (dx)/(x)`
`= (mu_(0)Ni)/(2 (b - a)) ln(b//a)`