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 $z=\surd 3R$ along the axis of the circular wire loop, as shown in figure. The plane of the square loop makes an angle of ${45}^0$ with respect to the z -axis. If the mutual inductance between the loops is given by $\frac{{\mu }_0{a}^2}{2^{\frac{P}{2}}R}$, then the value of p is

Answer 1

At the position of square loop
$B=\frac{{\mu }_{0}i{R}^2}{2{(R^2+3R^2)}^{3/2}}=\frac{{\mu }_{0}i}{16R}$
Flux through the loop,
$\varphi =BA\cos \theta$

$=\frac{{\mu }_{0}i}{16R}\times 2a^2\times \frac{1}{\sqrt{2}}$

$=\frac{{\mu }_{0}i{a}^2}{8\sqrt{2}R}$
$M=\frac{\varphi }{i}=\frac{{\mu }_0{a}^2}{8\sqrt{2R}}=\frac{{\mu }_0{a}^2}{2^{7/2}R}$
$P=7$