Question

A circular wire loop of radius $R$ is placed in
the x -y plane centered at the origin $O$ . A square loop of side $a(a<R)$ having two turns is placed with its center at $z=\sqrt{3}R$ along the axis of the circular wire loop, as shown in figure. The plane of the square loop makes an angle of ${45}^{\circ }$ with the positive $z$ -axis. If the mutual inductance between the loops is given by $\frac{{\mu }_0{a}^2}{2^{p/2}R},$ then value of p is (Answer upto two digit after decimal places)

Answer 1

At the position of square loop
$B=\frac{{\mu }_{0}i{R}^2}{2{(R^2+(\sqrt{3}R{)}^2)}^{3/2}}=\frac{{\mu }_{0}i}{16R}$

Magnetic flux is given by
$\varphi =BA\cos \theta$
$=\frac{{\mu }_{0}i}{16R}\times a^2\times \frac{1}{\sqrt{2}}$
$=\frac{{\mu }_{0}i{a}^2}{16\sqrt{2}R}$

Magnetic inductane is given by
$M=N\frac{\varphi }{i}=\frac{{\mu }_0{a}^2}{8\sqrt{2R}}=\frac{{\mu }_0{a}^2}{2^{7/2}R}$
(Where N is no of turns which is 2 for square loop)
$\Rightarrow \text{p}=7$