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Two coplanar circular coils of equal radius carrying currents $i_{1}, i_{2}$ in opposite directions are at a large distance ‘d’. The distance from the first coil where the resultant magnetic induction is zero is

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$\frac{d}{1 + {(\frac{i_{2}}{i_{1}})}^{2}}$

$\frac{d}{1 + \sqrt{\frac{i_{2}}{i_{1}}}}$

$\frac{d}{1 + \frac{i_{2}}{i_{1}}}$

$\frac{d}{1 + {(\frac{i_{2}}{i_{1}})}^{\frac{1}{3}}}$

answer is C.

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Detailed Solution

Considering two loops

$Magnetic field on axis of coil = \frac{μ_{0} i a^{2}}{2 {(a^{2} + x^{2})}^{3 / 2}}$

here $d > > x and a > > x$

$B = \frac{μ_{0} {ia}^{2}}{2 x^{3}}$

At point P

$B_{1} = \frac{μ_{0} i_{1} a^{2}}{2 x^{3}}, B_{2} = \frac{μ_{0} i_{2} a^{2}}{2 (d - x)^{3}}$

$B_{1} = B_{2}$

$\Rightarrow \frac{μ_{0} i_{1} a^{2}}{2 x^{3}} = \frac{μ_{0} i_{2} a^{2}}{2 (d - x)^{3}} \Rightarrow \frac{i_{2}}{i_{1}} = {(\frac{d - x}{x})}^{3}$

$\Rightarrow \frac{d - x}{x} = {(\frac{i_{2}}{i_{1}})}^{1 / 3}$

$x = \frac{d}{1 + {(\frac{i_{2}}{i_{1}})}^{1 / 3}}$

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