Question 131 of 153

A small square loop of side $a$ and one turn is placed inside a larger square loop of side $b$ and one turn $(b≫a).$ 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 :

A$\dfrac{{\mu }_{0}}{4\pi }8\sqrt{2}\dfrac{{a}^{2}}{b}$B$\dfrac{{\mu }_{0}}{4\pi }8\sqrt{2}\dfrac{{b}^{2}}{a}$C$\dfrac{{\mu }_{0}}{4\pi }\dfrac{8\sqrt{2}}{b}$D$\dfrac{{\mu }_{0}}{4\pi }\dfrac{8\sqrt{2}}{a}$

Show answer and solution▾

Answer: (A) $\dfrac{{\mu }_{0}}{4\pi }8\sqrt{2}\dfrac{{a}^{2}}{b}$

|
$B=4\times \dfrac{{\mu }_{0}i}{4\pi \left(\dfrac{b}{2}\right)}\sqrt{2}$ $\Rightarrow B=\dfrac{2\sqrt{2}{\mu }_{0}i}{\pi b}$
$\Rightarrow \phi =MI$ $\Rightarrow BA=MI$
$\Rightarrow \dfrac{2\sqrt{2}{\mu }_{0}i}{\pi b}\cdot {a}^{2}=Mi$ $\Rightarrow M=\dfrac{2\sqrt{2}{\mu }_{0}{a}^{2}}{\pi b}=\dfrac{{\mu }_{0}}{4\pi }\dfrac{8\sqrt{2}{a}^{2}}{b}$