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)IN)/((b-a))"ln"(b)/(a)`.

Answer

C

Explanation

Let us consider a thickness `dx` of wire. Let it be at a distance `x` from the centre `O`.
Number of turns unit length `= (N)/(b-a)`
`:.` number of turns in thickness `dx = (N)/(b-a)d x`
Small amount of magnetic field produced at `O` due to thickness `d x` of the wire
`:. dB=mu_(0)/2 (NI)/(b-a) (dx)/x=mu_(0)/2 (NI)/((b-a))`
On integrating we get
`B = underset(a)overset(b)int (mu_(0))/(2)(NI)/(b-a) (dx)/(x)= (mu_(0))/(2)(NI)/(2(b-a))`
`underset(a)overset(b)int(dx)/(x)(mu_(0))/(2)(NI)/((b-a))[log_(e) x]_(a)^(b), B = (mu_(0))/(2)(NI)/((b-a))"log"_(e)(b)/(a)`
Hence correct option is (c )