b
We know, the magnetic field on the axis of a current carrying circular ring is given by

\(B =\frac{\mu_{0}}{4 \pi} \frac{2 NIA }{\left( R ^{2}+ x ^{2}\right)^{3 / 2}}\)

\(\therefore \frac{ B _{1}}{ B _{2}}=\frac{8}{1}=\left[\frac{ R ^{2}+(0.2)^{2}}{ R ^{2}+(0.05)^{2}}\right]^{3 / 2}\)

\(4\left[ R ^{2}+(0.05)^{2}\right]=\left[ R ^{2}+(0.2)^{2}\right]\)

\(4 R ^{2}- R ^{2}=(0.2)^{2}-4 \times(0.05)^{2}\)

\(4 R ^{2}- R ^{2}=(0.2)^{2}-(0.1)^{2}\)

\(3 R ^{2}=0.3 \times 0.1\)

\(R ^{2}=(0.1)^{2} \Rightarrow R =0.1\)