An otherwise infinite, straight wire has two concentric loops of radii $a$ and $b$ carrying equal currents in opposite directions as shown in figure. The magnetic field at the common center is zero for:

\frac{a}{b} = \frac{π - 1}{π}

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\frac{a}{b} = \frac{π}{π + 1}

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\frac{a}{b} = \frac{π - 1}{π + 1}

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\frac{a}{b} = \frac{π + 1}{π - 1}

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Solution

The correct option is B $\frac{a}{b} = \frac{π}{π + 1}$
$B_{c e n t r e} = 0$
$\frac{μ_{0} I}{4 π b} ⨀ + \frac{μ_{0} I}{4 π b} ⨀ + \frac{μ_{0} I}{2 b} ⨀ + \frac{μ_{0} I}{2 a} ⨂ = 0$
$\Rightarrow \frac{μ_{0} I}{2 π b} + \frac{μ_{0} I}{2 b} - \frac{μ_{0} I}{2 a} \Rightarrow \frac{1}{2 π b} + \frac{1}{2 b} = \frac{1}{2 a}$
$\Rightarrow \frac{a}{b} = \frac{π}{π + 1}$

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