Considering magnetic field along the axis of a circular loop of radius R, at what distance from the centre of the loop is the field one eighth of its value at the centre?

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$\sqrt{3} R$

answer is B.

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Detailed Solution

$\begin{array}{l} B_{axs} = \frac{μ_{0} {IR}^{2}}{2 {(R^{2} + x^{2})}^{3 / 2}} = \frac{μ_{0} I}{2 R} \frac{1}{{(1 + \frac{x^{2}}{R^{2}})}^{3 / 2}} \\ B_{centre} = \frac{μ_{0} I}{2 R} \\ \Rightarrow B_{axis} = \frac{B_{centre}}{{(1 + \frac{x^{2}}{R^{2}})}^{3 / 2}} = \frac{B_{centre}}{8} \\ {(1 + \frac{x^{2}}{R^{2}})}^{3 / 2} = 8 \Rightarrow x = \sqrt{3} R \end{array}$

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