A circular current carrying coil has a radius R. The distance from the centre of the coil, on the axis, where B will be $\frac{1}{8}$ of its value at the centre of the coil is

$\frac{R}{\sqrt{3}}$

$\sqrt{3} R$

2 $\sqrt{3}$ R

$\frac{2 R}{\sqrt{3}}$

Solution

$\sqrt{3} R$

B = $\frac{μ_{0} Ni R^{2}}{2 {(R^{2} + x^{2})}^{\frac{3}{2}}}$

= $\frac{μ_{0} N i}{2 R} \frac{1}{{[1 + \frac{x^{2}}{R^{2}}]}^{\frac{3}{2}}}$

Let B_C = $\frac{μ_{0} Ni}{2 R}$

B = B_C $\frac{1}{{[1 + \frac{x^{2}}{R^{2}}]}^{\frac{3}{2}}}$

⇒ $\frac{B_{C}}{8} = \frac{B_{C}}{{(1 + \frac{x^{2}}{R^{2}})}^{\frac{3}{2}}}$ $[∵ B = \frac{B_{C}}{8}]$

⇒ ${(1 + \frac{x^{2}}{R^{2}})}^{\frac{3}{2}}$ = 8

⇒ 1 + $\frac{x^{2}}{R^{2}}$ = 4

⇒ $\frac{x^{2}}{R^{2}}$ = 3

⇒ x = R $\sqrt{3}$

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