$B _{\text {centre }}=\frac{\mu_{0} i }{2 R }$

$\therefore B _{\text {centre }}=\frac{\mu_{0} i }{2 a }$

$\therefore B _{ axis }=\frac{\mu_{0} ia ^{2}}{2\left( a ^{2}+ r ^{2}\right)^{3 / 2}}$

$\therefore$ fractional change in magnetic field $=$

$\frac{\frac{\mu_{0} i }{2 a }-\frac{\mu_{0} ia ^{2}}{2\left( a ^{2}+ r ^{2}\right)^{3 / 2}}}{{\frac{\mu_{0} i }{2 a }}}=1-\frac{1}{\left[1+\left(\frac{ r ^{2}}{ a ^{2}}\right)\right]^{3 / 2}}$

$\approx 1-\left[1-\frac{3}{2} \frac{ r ^{2}}{ a ^{2}}\right]=\frac{3}{2} \frac{ r ^{2}}{ a ^{2}}$

Note : $\left(1+\frac{ r ^{2}}{ a ^{2}}\right)^{-3 / 2} \approx\left(1-\frac{3}{2} \frac{ r ^{2}}{ a ^{2}}\right)$

[True only if $r << a ]$