c
\(B_{C}=\frac{\mu_{0} I}{2 r}, B_{a}=\frac{\mu_{0} I^{2}}{2\left(x^{2}+r^{2}\right)^{3 / 2}}\)

\(At \,\,x =\frac{r}{2}\)

\(B _{ a }=\frac{\mu_{0} Ir ^{2}}{2\left(\frac{ r ^{2}}{4}+ r ^{2}\right)^{3 / 2}}\)

\(=\frac{\mu_{0} Ir ^{2}}{2\left(\frac{5}{4} r ^{2}\right)^{3 / 2}}=\frac{\mu_{0} I }{2 r }\left(\frac{4}{5}\right)^{3 / 2}\)

\(=\frac{\mu_{0} I }{2 r }\left(\frac{2}{\sqrt{5}}\right)^{3}\)