A coll having N turns is wound tightly in the form of a spiral with inner and outer radii a and b respectively when a current I passes through the coll , 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))"in" ((b)/(a))`
D. `(mu_(0)NI)/(2(b-a))"in"((b)/(a))`

Answer

C

Explanation

Given : No of turns = N . Inner radius of the coil = a and outer radius of the coil = b .
We know that number of turns in an element of thickness dr at a distance from the centre of spiral
loop (dN) ` = ((N)/(b - a))` dr. And mag-netic field due to this element at the centre of the coil
`(dB) = (mu_(0)(dN)I)/(2r) = (mu_(0)NI)/(2(b-a))(dr)/(r)`
`:.` Magnetic field at the centre of
coil `(B) = underset(r = a)overset(r = b)intdB`
`= underset (r = a) overset(r = b) int (mu_(0)NI)/(2(b-a))(dr)/(r)=(mu_(0)NI)/(2(b-a)) "In"((b)/(a))`