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-a))"log"_(e)a/b`
B. `(mu_(0)nI)/(2(b-a))`
C. `(2mu_(0)nI)/b`
D. `(mu_(0)nI)/(2(b-a))"log"_(d)b/a`

Answer

D

Explanation

Consider an element of thickness dr at a distance r from the centre of spiral coil.
Number of turns in spiral =n
Number of turns per length `=n/(b-a)`
Number of turns in element dr is `dn=(ndr)/(b-a)`
Magnetic field at its centre due to element dr is
`dB=(mu_(0)Idn)/(2r)=(mu_(0)I)/2 n/((b-a))(dr)/r`
`:.B=int_(a)^(b)(mu_(0)I n dr)/(2(b-a)r)=(mu_(0)I n)/(2(b-a))int_(a)^(b)(dr)/r`
`=(mu_(0)I n)/(2(b-a))log_(e)(b/a)`