The magnetic field intensity at a distance r from centre on the axis of a coil of radius a carrying current i is given by
Baxis= ...(i)
The magnetic field intensity at the centre of the coil is given by
Bcentre= ...(ii)
According to question, the fractional change in the magnetic fieldintensity is given by
=1− ...(iii)
Substituting the values of
Baxis and
Bcentre from Eqs. (i) and (ii) respectively in Eq. (iii), we get
=1− =1− =1− =1−(1+)− Given that, r < a, applying Binomial expansion,
(1−x)−n=1−x+−...(−1)nxn, =(1+nx) we get
=1−(1−) =1−1+ =, which is the required expression.