b
(b) \(\frac{{{B_{centre}}}}{{{B_{axis}}}} = {\left( {1 + \frac{{{x^2}}}{{{R^2}}}} \right)^{3/2}},\)

also \({B_{axis}} = \frac{1}{8}{B_{centre}}\)
\( \Rightarrow \frac{8}{1} = {\left( {1 + \frac{{{x^2}}}{{{R^2}}}} \right)^{3/2}} \Rightarrow 2 = {\left( {1 + \frac{{{x^2}}}{{{R^2}}}} \right)^{1/2}}\)
\( \Rightarrow 4 = 1 + \frac{{{x^2}}}{{{R^2}}} \Rightarrow 3 = \frac{{{x^2}}}{{{R^2}}} \Rightarrow {x^2} = 3{R^2}\)\( \Rightarrow x = \sqrt 3 R\)