A small square loop of side $a$ and one turn is placed inside a larger square loop of side $b$ and one turn $(b\gg a).$ The two loops are coplanar with their centres coinciding. If a current $I$ is passed in the square loop of side  $b,$ then the coefficient of mutual inductance between the two loops is :

• Correct answer
  A

  $\frac{{\mu }_0}{4\pi }8\sqrt{2}\frac{{\mathrm{a}}^2}{ \mathrm{b}}$
• B

  $\frac{{\mu }_0}{4\pi }8\sqrt{2}\frac{{\mathrm{b}}^2}{\mathrm{a}}$
• C

  $\frac{{\mu }_0}{4\pi }\frac{8\sqrt{2}}{ \mathrm{b}}$
• D

  $\frac{{\mu }_0}{4\pi }\frac{8\sqrt{2}}{a}$

Solution

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$B=4\times \frac{{\mu }_{0}i}{4\pi \left(\frac{b}{2}\right)}\sqrt{2}$ $\Rightarrow B=\frac{2\sqrt{2}{\mu }_{0}i}{\pi b}$
$\Rightarrow \varphi =MI$ $\Rightarrow BA=MI$
$\Rightarrow \frac{2\sqrt{2}{\mu }_{0}i}{\pi b}·a^2=Mi$ $\Rightarrow M=\frac{2\sqrt{2}{\mu }_0{a}^2}{\pi b}=\frac{{\mu }_0}{4\pi }\frac{8\sqrt{2}{a}^2}{b}$
 

Hence, the correct answer is option: 'A'.