The fractional change in the magnetic field intensity at a distance

$\mathrm{B}_{\text {axis }}=\frac{\mu_{0} 1 R^{2}}{2\left(R^{2}+x^{2}\right)^{3 / 2}}$

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

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

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

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

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

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

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

[True only if $\mathrm{r} \ll<\mathrm{a}$ ]

Hence option (4) is the most suitable option